Henrywood and Agarwal, Equation (12)

Percentage Accurate: 65.7% → 83.4%

Time: 10.4s

Alternatives: 16

Speedup: 2.1×

Specification

Math

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

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: 65.7% accurate, 1.0× speedup

Math

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

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: 83.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_1 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_2 := t\_0 \cdot t\_1\\ t_3 := \frac{t\_0}{d + d} \cdot t\_1\\ t_4 := \frac{-0.5}{d} \cdot t\_2\\ \mathbf{if}\;\ell \leq -1.3 \cdot 10^{-307}:\\ \;\;\;\;\left(\left(-d\right) \cdot \frac{\sqrt{-\frac{1}{\ell}}}{\sqrt{-h}}\right) \cdot \left(1 - \frac{\left(\left(\frac{t\_1 \cdot t\_0}{d} \cdot 0.25\right) \cdot t\_3\right) \cdot h}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 3 \cdot 10^{-96}:\\ \;\;\;\;\left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(0.25 \cdot \frac{t\_2}{d}\right) \cdot \frac{t\_3 \cdot h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\frac{-d}{\sqrt{\ell}}\right|}{\sqrt{h}} \cdot \left(1 - t\_4 \cdot \left(t\_4 \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0 (fmax (fabs M) (fabs D)))
       (t_1 (fmin (fabs M) (fabs D)))
       (t_2 (* t_0 t_1))
       (t_3 (* (/ t_0 (+ d d)) t_1))
       (t_4 (* (/ -0.5 d) t_2)))
  (if (<= l -1.3e-307)
    (*
     (* (- d) (/ (sqrt (- (/ 1.0 l))) (sqrt (- h))))
     (- 1.0 (/ (* (* (* (/ (* t_1 t_0) d) 0.25) t_3) h) l)))
    (if (<= l 3e-96)
      (*
       (fabs (/ (- d) (sqrt (* l h))))
       (- 1.0 (* (* 0.25 (/ t_2 d)) (/ (* t_3 h) l))))
      (*
       (/ (fabs (/ (- d) (sqrt l))) (sqrt h))
       (- 1.0 (* t_4 (* t_4 (* 0.5 (/ h l))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fmax(fabs(M), fabs(D));
	double t_1 = fmin(fabs(M), fabs(D));
	double t_2 = t_0 * t_1;
	double t_3 = (t_0 / (d + d)) * t_1;
	double t_4 = (-0.5 / d) * t_2;
	double tmp;
	if (l <= -1.3e-307) {
		tmp = (-d * (sqrt(-(1.0 / l)) / sqrt(-h))) * (1.0 - ((((((t_1 * t_0) / d) * 0.25) * t_3) * h) / l));
	} else if (l <= 3e-96) {
		tmp = fabs((-d / sqrt((l * h)))) * (1.0 - ((0.25 * (t_2 / d)) * ((t_3 * h) / l)));
	} else {
		tmp = (fabs((-d / sqrt(l))) / sqrt(h)) * (1.0 - (t_4 * (t_4 * (0.5 * (h / l)))));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = fmax(abs(m), abs(d_1))
    t_1 = fmin(abs(m), abs(d_1))
    t_2 = t_0 * t_1
    t_3 = (t_0 / (d + d)) * t_1
    t_4 = ((-0.5d0) / d) * t_2
    if (l <= (-1.3d-307)) then
        tmp = (-d * (sqrt(-(1.0d0 / l)) / sqrt(-h))) * (1.0d0 - ((((((t_1 * t_0) / d) * 0.25d0) * t_3) * h) / l))
    else if (l <= 3d-96) then
        tmp = abs((-d / sqrt((l * h)))) * (1.0d0 - ((0.25d0 * (t_2 / d)) * ((t_3 * h) / l)))
    else
        tmp = (abs((-d / sqrt(l))) / sqrt(h)) * (1.0d0 - (t_4 * (t_4 * (0.5d0 * (h / l)))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = fmax(Math.abs(M), Math.abs(D));
	double t_1 = fmin(Math.abs(M), Math.abs(D));
	double t_2 = t_0 * t_1;
	double t_3 = (t_0 / (d + d)) * t_1;
	double t_4 = (-0.5 / d) * t_2;
	double tmp;
	if (l <= -1.3e-307) {
		tmp = (-d * (Math.sqrt(-(1.0 / l)) / Math.sqrt(-h))) * (1.0 - ((((((t_1 * t_0) / d) * 0.25) * t_3) * h) / l));
	} else if (l <= 3e-96) {
		tmp = Math.abs((-d / Math.sqrt((l * h)))) * (1.0 - ((0.25 * (t_2 / d)) * ((t_3 * h) / l)));
	} else {
		tmp = (Math.abs((-d / Math.sqrt(l))) / Math.sqrt(h)) * (1.0 - (t_4 * (t_4 * (0.5 * (h / l)))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = fmax(math.fabs(M), math.fabs(D))
	t_1 = fmin(math.fabs(M), math.fabs(D))
	t_2 = t_0 * t_1
	t_3 = (t_0 / (d + d)) * t_1
	t_4 = (-0.5 / d) * t_2
	tmp = 0
	if l <= -1.3e-307:
		tmp = (-d * (math.sqrt(-(1.0 / l)) / math.sqrt(-h))) * (1.0 - ((((((t_1 * t_0) / d) * 0.25) * t_3) * h) / l))
	elif l <= 3e-96:
		tmp = math.fabs((-d / math.sqrt((l * h)))) * (1.0 - ((0.25 * (t_2 / d)) * ((t_3 * h) / l)))
	else:
		tmp = (math.fabs((-d / math.sqrt(l))) / math.sqrt(h)) * (1.0 - (t_4 * (t_4 * (0.5 * (h / l)))))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = fmax(abs(M), abs(D))
	t_1 = fmin(abs(M), abs(D))
	t_2 = Float64(t_0 * t_1)
	t_3 = Float64(Float64(t_0 / Float64(d + d)) * t_1)
	t_4 = Float64(Float64(-0.5 / d) * t_2)
	tmp = 0.0
	if (l <= -1.3e-307)
		tmp = Float64(Float64(Float64(-d) * Float64(sqrt(Float64(-Float64(1.0 / l))) / sqrt(Float64(-h)))) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(t_1 * t_0) / d) * 0.25) * t_3) * h) / l)));
	elseif (l <= 3e-96)
		tmp = Float64(abs(Float64(Float64(-d) / sqrt(Float64(l * h)))) * Float64(1.0 - Float64(Float64(0.25 * Float64(t_2 / d)) * Float64(Float64(t_3 * h) / l))));
	else
		tmp = Float64(Float64(abs(Float64(Float64(-d) / sqrt(l))) / sqrt(h)) * Float64(1.0 - Float64(t_4 * Float64(t_4 * Float64(0.5 * Float64(h / l))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = max(abs(M), abs(D));
	t_1 = min(abs(M), abs(D));
	t_2 = t_0 * t_1;
	t_3 = (t_0 / (d + d)) * t_1;
	t_4 = (-0.5 / d) * t_2;
	tmp = 0.0;
	if (l <= -1.3e-307)
		tmp = (-d * (sqrt(-(1.0 / l)) / sqrt(-h))) * (1.0 - ((((((t_1 * t_0) / d) * 0.25) * t_3) * h) / l));
	elseif (l <= 3e-96)
		tmp = abs((-d / sqrt((l * h)))) * (1.0 - ((0.25 * (t_2 / d)) * ((t_3 * h) / l)));
	else
		tmp = (abs((-d / sqrt(l))) / sqrt(h)) * (1.0 - (t_4 * (t_4 * (0.5 * (h / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Max[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$0 / N[(d + d), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(N[(-0.5 / d), $MachinePrecision] * t$95$2), $MachinePrecision]}, If[LessEqual[l, -1.3e-307], N[(N[((-d) * N[(N[Sqrt[(-N[(1.0 / l), $MachinePrecision])], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(N[(t$95$1 * t$95$0), $MachinePrecision] / d), $MachinePrecision] * 0.25), $MachinePrecision] * t$95$3), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3e-96], N[(N[Abs[N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(0.25 * N[(t$95$2 / d), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$3 * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[N[((-d) / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(t$95$4 * N[(t$95$4 * N[(0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.3e-307

   1. Initial program 65.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-neg-rev N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   3. Applied rewrites 67.4%

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

   5. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. mult-flip N/A

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

      14. sqrt-prod N/A

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

      15. lower-unsound-sqrt.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. sqr-neg-rev N/A

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

      19. lift-neg.f64 N/A

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

      20. lift-neg.f64 N/A

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

      21. sqrt-unprod N/A

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

      22. rem-square-sqrt N/A

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

      23. lower-unsound-*.f64 N/A

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

      24. lower-unsound-sqrt.f64 N/A

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

   7. Applied rewrites 37.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. frac-2neg N/A

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

      6. lift-neg.f64 N/A

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

      7. sqrt-div N/A

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

      8. lower-unsound-sqrt.f64 N/A

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

      9. lower-unsound-/.f64 N/A

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

      10. lower-unsound-sqrt.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 41.6%

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

   9. Applied rewrites 41.6%

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

   if -1.3e-307 < l < 3.0000000000000001e-96

   1. Initial program 65.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-neg-rev N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   3. Applied rewrites 67.4%

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

   5. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. sqrt-fabs-rev N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lower-fabs.f64 53.0%

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

      18. lift-sqrt.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-/.f64 N/A

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

      21. associate-*r/ N/A

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

      22. lift-*.f64 N/A

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

      23. sqrt-div N/A

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

   7. Applied rewrites 73.3%

      \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

   9. Applied rewrites 76.1%

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

   if 3.0000000000000001e-96 < l

   1. Initial program 65.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-neg-rev N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   3. Applied rewrites 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow1/2 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-undiv N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. pow1/2 N/A

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

      17. lift-sqrt.f64 N/A

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

      18. sqrt-unprod N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lower-*.f64 30.9%

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

   5. Applied rewrites 30.9%

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

      1. lift-sqrt.f64 N/A

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

      2. sqrt-fabs-rev N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lower-fabs.f64 30.9%

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lift-*.f64 N/A

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

      10. sqrt-div N/A

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

      11. lower-unsound-sqrt.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. sqr-neg-rev N/A

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

      15. lift-neg.f64 N/A

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

      16. lift-neg.f64 N/A

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

      17. sqrt-unprod N/A

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

      18. rem-square-sqrt N/A

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

      19. lower-unsound-/.f64 N/A

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

      20. lower-unsound-sqrt.f64 40.9%

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

   7. Applied rewrites 40.9%

      \[\leadsto \frac{\color{blue}{\left|\frac{-d}{\sqrt{\ell}}\right|}}{\sqrt{h}} \cdot \left(1 - \left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 83.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{D \cdot M}{d}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \frac{D}{d + d} \cdot M\\ t_3 := \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot t\_2\right) \cdot \frac{h}{\ell}\right)\\ t_4 := \left|\frac{-d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{if}\;t\_1 \leq -200000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 10^{-307}:\\ \;\;\;\;t\_4 \cdot \left(1 - t\_0 \cdot \frac{\left(\left(0.25 \cdot M\right) \cdot D\right) \cdot h}{\left(d + d\right) \cdot \ell}\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+219}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_4 \cdot \left(1 - \left(0.25 \cdot t\_0\right) \cdot \frac{t\_2 \cdot h}{\ell}\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0 (/ (* D M) d))
       (t_1
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (-
          1.0
          (*
           (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0))
           (/ h l)))))
       (t_2 (* (/ D (+ d d)) M))
       (t_3
        (*
         (* (sqrt (/ d l)) (sqrt (/ d h)))
         (- 1.0 (* (* (* (/ (* M D) d) 0.25) t_2) (/ h l)))))
       (t_4 (fabs (/ (- d) (sqrt (* l h))))))
  (if (<= t_1 -200000.0)
    t_3
    (if (<= t_1 1e-307)
      (* t_4 (- 1.0 (* t_0 (/ (* (* (* 0.25 M) D) h) (* (+ d d) l)))))
      (if (<= t_1 2e+219)
        t_3
        (* t_4 (- 1.0 (* (* 0.25 t_0) (/ (* t_2 h) l)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (D * M) / d;
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = (D / (d + d)) * M;
	double t_3 = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (((((M * D) / d) * 0.25) * t_2) * (h / l)));
	double t_4 = fabs((-d / sqrt((l * h))));
	double tmp;
	if (t_1 <= -200000.0) {
		tmp = t_3;
	} else if (t_1 <= 1e-307) {
		tmp = t_4 * (1.0 - (t_0 * ((((0.25 * M) * D) * h) / ((d + d) * l))));
	} else if (t_1 <= 2e+219) {
		tmp = t_3;
	} else {
		tmp = t_4 * (1.0 - ((0.25 * t_0) * ((t_2 * h) / l)));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = (d_1 * m) / d
    t_1 = (((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)))
    t_2 = (d_1 / (d + d)) * m
    t_3 = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - (((((m * d_1) / d) * 0.25d0) * t_2) * (h / l)))
    t_4 = abs((-d / sqrt((l * h))))
    if (t_1 <= (-200000.0d0)) then
        tmp = t_3
    else if (t_1 <= 1d-307) then
        tmp = t_4 * (1.0d0 - (t_0 * ((((0.25d0 * m) * d_1) * h) / ((d + d) * l))))
    else if (t_1 <= 2d+219) then
        tmp = t_3
    else
        tmp = t_4 * (1.0d0 - ((0.25d0 * t_0) * ((t_2 * h) / l)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (D * M) / d;
	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 * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = (D / (d + d)) * M;
	double t_3 = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - (((((M * D) / d) * 0.25) * t_2) * (h / l)));
	double t_4 = Math.abs((-d / Math.sqrt((l * h))));
	double tmp;
	if (t_1 <= -200000.0) {
		tmp = t_3;
	} else if (t_1 <= 1e-307) {
		tmp = t_4 * (1.0 - (t_0 * ((((0.25 * M) * D) * h) / ((d + d) * l))));
	} else if (t_1 <= 2e+219) {
		tmp = t_3;
	} else {
		tmp = t_4 * (1.0 - ((0.25 * t_0) * ((t_2 * h) / l)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (D * M) / d
	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 * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_2 = (D / (d + d)) * M
	t_3 = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - (((((M * D) / d) * 0.25) * t_2) * (h / l)))
	t_4 = math.fabs((-d / math.sqrt((l * h))))
	tmp = 0
	if t_1 <= -200000.0:
		tmp = t_3
	elif t_1 <= 1e-307:
		tmp = t_4 * (1.0 - (t_0 * ((((0.25 * M) * D) * h) / ((d + d) * l))))
	elif t_1 <= 2e+219:
		tmp = t_3
	else:
		tmp = t_4 * (1.0 - ((0.25 * t_0) * ((t_2 * h) / l)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(D * M) / d)
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_2 = Float64(Float64(D / Float64(d + d)) * M)
	t_3 = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(M * D) / d) * 0.25) * t_2) * Float64(h / l))))
	t_4 = abs(Float64(Float64(-d) / sqrt(Float64(l * h))))
	tmp = 0.0
	if (t_1 <= -200000.0)
		tmp = t_3;
	elseif (t_1 <= 1e-307)
		tmp = Float64(t_4 * Float64(1.0 - Float64(t_0 * Float64(Float64(Float64(Float64(0.25 * M) * D) * h) / Float64(Float64(d + d) * l)))));
	elseif (t_1 <= 2e+219)
		tmp = t_3;
	else
		tmp = Float64(t_4 * Float64(1.0 - Float64(Float64(0.25 * t_0) * Float64(Float64(t_2 * h) / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (D * M) / d;
	t_1 = (((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)));
	t_2 = (D / (d + d)) * M;
	t_3 = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (((((M * D) / d) * 0.25) * t_2) * (h / l)));
	t_4 = abs((-d / sqrt((l * h))));
	tmp = 0.0;
	if (t_1 <= -200000.0)
		tmp = t_3;
	elseif (t_1 <= 1e-307)
		tmp = t_4 * (1.0 - (t_0 * ((((0.25 * M) * D) * h) / ((d + d) * l))));
	elseif (t_1 <= 2e+219)
		tmp = t_3;
	else
		tmp = t_4 * (1.0 - ((0.25 * t_0) * ((t_2 * h) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(D * M), $MachinePrecision] / d), $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 * D), $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 / N[(d + d), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision] * 0.25), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Abs[N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -200000.0], t$95$3, If[LessEqual[t$95$1, 1e-307], N[(t$95$4 * N[(1.0 - N[(t$95$0 * N[(N[(N[(N[(0.25 * M), $MachinePrecision] * D), $MachinePrecision] * h), $MachinePrecision] / N[(N[(d + d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+219], t$95$3, N[(t$95$4 * N[(1.0 - N[(N[(0.25 * t$95$0), $MachinePrecision] * N[(N[(t$95$2 * h), $MachinePrecision] / l), $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)))) < -2e5 or 9.9999999999999991e-308 < (*.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)))) < 1.9999999999999999e219

   1. Initial program 65.7%

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

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

      3. lower-*.f64 65.7%

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 65.7%

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 65.7%

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

   3. Applied rewrites 65.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lift-+.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. lift-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. lower-*.f64 65.7%

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

   5. Applied rewrites 65.1%

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

   if -2e5 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.9999999999999991e-308

   1. Initial program 65.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-neg-rev N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   3. Applied rewrites 67.4%

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

   5. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. sqrt-fabs-rev N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lower-fabs.f64 53.0%

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

      18. lift-sqrt.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-/.f64 N/A

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

      21. associate-*r/ N/A

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

      22. lift-*.f64 N/A

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

      23. sqrt-div N/A

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

   7. Applied rewrites 73.3%

      \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

   9. Applied rewrites 75.1%

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

   if 1.9999999999999999e219 < (*.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 65.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-neg-rev N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   3. Applied rewrites 67.4%

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

   5. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. sqrt-fabs-rev N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lower-fabs.f64 53.0%

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

      18. lift-sqrt.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-/.f64 N/A

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

      21. associate-*r/ N/A

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

      22. lift-*.f64 N/A

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

      23. sqrt-div N/A

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

   7. Applied rewrites 73.3%

      \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

   9. Applied rewrites 76.1%

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

Alternative 3: 82.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{-0.5}{d} \cdot \left(D \cdot M\right)\\ t_1 := {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\\ \mathbf{if}\;t\_1 \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2 \cdot 10^{+219}:\\ \;\;\;\;t\_1 \cdot \left(1 - t\_0 \cdot \left(t\_0 \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(0.25 \cdot \frac{D \cdot M}{d}\right) \cdot \frac{\left(\frac{D}{d + d} \cdot M\right) \cdot h}{\ell}\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0 (* (/ -0.5 d) (* D M)))
       (t_1 (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))))
  (if (<=
       (*
        t_1
        (-
         1.0
         (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))
       2e+219)
    (* t_1 (- 1.0 (* t_0 (* t_0 (* 0.5 (/ h l))))))
    (*
     (fabs (/ (- d) (sqrt (* l h))))
     (-
      1.0
      (* (* 0.25 (/ (* D M) d)) (/ (* (* (/ D (+ d d)) M) h) l)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (-0.5 / d) * (D * M);
	double t_1 = pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0));
	double tmp;
	if ((t_1 * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= 2e+219) {
		tmp = t_1 * (1.0 - (t_0 * (t_0 * (0.5 * (h / l)))));
	} else {
		tmp = fabs((-d / sqrt((l * h)))) * (1.0 - ((0.25 * ((D * M) / d)) * ((((D / (d + d)) * M) * h) / l)));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((-0.5d0) / d) * (d_1 * m)
    t_1 = ((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))
    if ((t_1 * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))) <= 2d+219) then
        tmp = t_1 * (1.0d0 - (t_0 * (t_0 * (0.5d0 * (h / l)))))
    else
        tmp = abs((-d / sqrt((l * h)))) * (1.0d0 - ((0.25d0 * ((d_1 * m) / d)) * ((((d_1 / (d + d)) * m) * h) / l)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (-0.5 / d) * (D * M);
	double t_1 = Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0));
	double tmp;
	if ((t_1 * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= 2e+219) {
		tmp = t_1 * (1.0 - (t_0 * (t_0 * (0.5 * (h / l)))));
	} else {
		tmp = Math.abs((-d / Math.sqrt((l * h)))) * (1.0 - ((0.25 * ((D * M) / d)) * ((((D / (d + d)) * M) * h) / l)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (-0.5 / d) * (D * M)
	t_1 = math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))
	tmp = 0
	if (t_1 * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= 2e+219:
		tmp = t_1 * (1.0 - (t_0 * (t_0 * (0.5 * (h / l)))))
	else:
		tmp = math.fabs((-d / math.sqrt((l * h)))) * (1.0 - ((0.25 * ((D * M) / d)) * ((((D / (d + d)) * M) * h) / l)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(-0.5 / d) * Float64(D * M))
	t_1 = Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0)))
	tmp = 0.0
	if (Float64(t_1 * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= 2e+219)
		tmp = Float64(t_1 * Float64(1.0 - Float64(t_0 * Float64(t_0 * Float64(0.5 * Float64(h / l))))));
	else
		tmp = Float64(abs(Float64(Float64(-d) / sqrt(Float64(l * h)))) * Float64(1.0 - Float64(Float64(0.25 * Float64(Float64(D * M) / d)) * Float64(Float64(Float64(Float64(D / Float64(d + d)) * M) * h) / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (-0.5 / d) * (D * M);
	t_1 = ((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0));
	tmp = 0.0;
	if ((t_1 * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)))) <= 2e+219)
		tmp = t_1 * (1.0 - (t_0 * (t_0 * (0.5 * (h / l)))));
	else
		tmp = abs((-d / sqrt((l * h)))) * (1.0 - ((0.25 * ((D * M) / d)) * ((((D / (d + d)) * M) * h) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(-0.5 / d), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[N[(t$95$1 * 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], 2e+219], N[(t$95$1 * N[(1.0 - N[(t$95$0 * N[(t$95$0 * N[(0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(0.25 * N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(D / N[(d + d), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision] * h), $MachinePrecision] / 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)))) < 1.9999999999999999e219

   1. Initial program 65.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-neg-rev N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   3. Applied rewrites 67.4%

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

   if 1.9999999999999999e219 < (*.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 65.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-neg-rev N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   3. Applied rewrites 67.4%

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

   5. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. sqrt-fabs-rev N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lower-fabs.f64 53.0%

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

      18. lift-sqrt.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-/.f64 N/A

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

      21. associate-*r/ N/A

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

      22. lift-*.f64 N/A

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

      23. sqrt-div N/A

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

   7. Applied rewrites 73.3%

      \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

   9. Applied rewrites 76.1%

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

Alternative 4: 78.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|M\right|, D\right)\\ t_1 := \frac{-d}{\sqrt{\ell \cdot h}}\\ t_2 := \mathsf{max}\left(\left|M\right|, D\right)\\ t_3 := 1 - \frac{\left(\left(\frac{t\_0 \cdot t\_2}{d} \cdot 0.25\right) \cdot \left(\frac{t\_2}{d + d} \cdot t\_0\right)\right) \cdot h}{\ell}\\ \mathbf{if}\;h \leq -1 \cdot 10^{+70}:\\ \;\;\;\;t\_1 \cdot t\_3\\ \mathbf{elif}\;h \leq 4.6 \cdot 10^{-242}:\\ \;\;\;\;\left|t\_1\right| \cdot \left(1 - \frac{t\_2 \cdot t\_0}{d} \cdot \frac{\left(\left(0.25 \cdot t\_0\right) \cdot t\_2\right) \cdot h}{\left(d + d\right) \cdot \ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\frac{-d}{\sqrt{\ell}}}{-\sqrt{h}}\right| \cdot t\_3\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0 (fmin (fabs M) D))
       (t_1 (/ (- d) (sqrt (* l h))))
       (t_2 (fmax (fabs M) D))
       (t_3
        (-
         1.0
         (/
          (* (* (* (/ (* t_0 t_2) d) 0.25) (* (/ t_2 (+ d d)) t_0)) h)
          l))))
  (if (<= h -1e+70)
    (* t_1 t_3)
    (if (<= h 4.6e-242)
      (*
       (fabs t_1)
       (-
        1.0
        (*
         (/ (* t_2 t_0) d)
         (/ (* (* (* 0.25 t_0) t_2) h) (* (+ d d) l)))))
      (* (fabs (/ (/ (- d) (sqrt l)) (- (sqrt h)))) t_3)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fmin(fabs(M), D);
	double t_1 = -d / sqrt((l * h));
	double t_2 = fmax(fabs(M), D);
	double t_3 = 1.0 - ((((((t_0 * t_2) / d) * 0.25) * ((t_2 / (d + d)) * t_0)) * h) / l);
	double tmp;
	if (h <= -1e+70) {
		tmp = t_1 * t_3;
	} else if (h <= 4.6e-242) {
		tmp = fabs(t_1) * (1.0 - (((t_2 * t_0) / d) * ((((0.25 * t_0) * t_2) * h) / ((d + d) * l))));
	} else {
		tmp = fabs(((-d / sqrt(l)) / -sqrt(h))) * t_3;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = fmin(abs(m), d_1)
    t_1 = -d / sqrt((l * h))
    t_2 = fmax(abs(m), d_1)
    t_3 = 1.0d0 - ((((((t_0 * t_2) / d) * 0.25d0) * ((t_2 / (d + d)) * t_0)) * h) / l)
    if (h <= (-1d+70)) then
        tmp = t_1 * t_3
    else if (h <= 4.6d-242) then
        tmp = abs(t_1) * (1.0d0 - (((t_2 * t_0) / d) * ((((0.25d0 * t_0) * t_2) * h) / ((d + d) * l))))
    else
        tmp = abs(((-d / sqrt(l)) / -sqrt(h))) * t_3
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = fmin(Math.abs(M), D);
	double t_1 = -d / Math.sqrt((l * h));
	double t_2 = fmax(Math.abs(M), D);
	double t_3 = 1.0 - ((((((t_0 * t_2) / d) * 0.25) * ((t_2 / (d + d)) * t_0)) * h) / l);
	double tmp;
	if (h <= -1e+70) {
		tmp = t_1 * t_3;
	} else if (h <= 4.6e-242) {
		tmp = Math.abs(t_1) * (1.0 - (((t_2 * t_0) / d) * ((((0.25 * t_0) * t_2) * h) / ((d + d) * l))));
	} else {
		tmp = Math.abs(((-d / Math.sqrt(l)) / -Math.sqrt(h))) * t_3;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = fmin(math.fabs(M), D)
	t_1 = -d / math.sqrt((l * h))
	t_2 = fmax(math.fabs(M), D)
	t_3 = 1.0 - ((((((t_0 * t_2) / d) * 0.25) * ((t_2 / (d + d)) * t_0)) * h) / l)
	tmp = 0
	if h <= -1e+70:
		tmp = t_1 * t_3
	elif h <= 4.6e-242:
		tmp = math.fabs(t_1) * (1.0 - (((t_2 * t_0) / d) * ((((0.25 * t_0) * t_2) * h) / ((d + d) * l))))
	else:
		tmp = math.fabs(((-d / math.sqrt(l)) / -math.sqrt(h))) * t_3
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = fmin(abs(M), D)
	t_1 = Float64(Float64(-d) / sqrt(Float64(l * h)))
	t_2 = fmax(abs(M), D)
	t_3 = Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(t_0 * t_2) / d) * 0.25) * Float64(Float64(t_2 / Float64(d + d)) * t_0)) * h) / l))
	tmp = 0.0
	if (h <= -1e+70)
		tmp = Float64(t_1 * t_3);
	elseif (h <= 4.6e-242)
		tmp = Float64(abs(t_1) * Float64(1.0 - Float64(Float64(Float64(t_2 * t_0) / d) * Float64(Float64(Float64(Float64(0.25 * t_0) * t_2) * h) / Float64(Float64(d + d) * l)))));
	else
		tmp = Float64(abs(Float64(Float64(Float64(-d) / sqrt(l)) / Float64(-sqrt(h)))) * t_3);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = min(abs(M), D);
	t_1 = -d / sqrt((l * h));
	t_2 = max(abs(M), D);
	t_3 = 1.0 - ((((((t_0 * t_2) / d) * 0.25) * ((t_2 / (d + d)) * t_0)) * h) / l);
	tmp = 0.0;
	if (h <= -1e+70)
		tmp = t_1 * t_3;
	elseif (h <= 4.6e-242)
		tmp = abs(t_1) * (1.0 - (((t_2 * t_0) / d) * ((((0.25 * t_0) * t_2) * h) / ((d + d) * l))));
	else
		tmp = abs(((-d / sqrt(l)) / -sqrt(h))) * t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Min[N[Abs[M], $MachinePrecision], D], $MachinePrecision]}, Block[{t$95$1 = N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Abs[M], $MachinePrecision], D], $MachinePrecision]}, Block[{t$95$3 = N[(1.0 - N[(N[(N[(N[(N[(N[(t$95$0 * t$95$2), $MachinePrecision] / d), $MachinePrecision] * 0.25), $MachinePrecision] * N[(N[(t$95$2 / N[(d + d), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -1e+70], N[(t$95$1 * t$95$3), $MachinePrecision], If[LessEqual[h, 4.6e-242], N[(N[Abs[t$95$1], $MachinePrecision] * N[(1.0 - N[(N[(N[(t$95$2 * t$95$0), $MachinePrecision] / d), $MachinePrecision] * N[(N[(N[(N[(0.25 * t$95$0), $MachinePrecision] * t$95$2), $MachinePrecision] * h), $MachinePrecision] / N[(N[(d + d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[N[(N[((-d) / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / (-N[Sqrt[h], $MachinePrecision])), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.0000000000000001e70

   1. Initial program 65.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-neg-rev N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   3. Applied rewrites 67.4%

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

   5. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. sqrt-div N/A

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

      14. lower-unsound-sqrt.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. sqr-neg-rev N/A

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

      18. lift-neg.f64 N/A

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

      19. lift-neg.f64 N/A

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

      20. sqrt-unprod N/A

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

      21. rem-square-sqrt N/A

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

      22. lower-unsound-/.f64 N/A

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

      23. lower-unsound-sqrt.f64 37.7%

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

      24. lift-*.f64 N/A

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

      25. *-commutative N/A

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

      26. lower-*.f64 37.7%

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

   7. Applied rewrites 37.7%

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

   if -1.0000000000000001e70 < h < 4.5999999999999997e-242

   1. Initial program 65.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-neg-rev N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   3. Applied rewrites 67.4%

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

   5. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. sqrt-fabs-rev N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lower-fabs.f64 53.0%

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

      18. lift-sqrt.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-/.f64 N/A

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

      21. associate-*r/ N/A

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

      22. lift-*.f64 N/A

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

      23. sqrt-div N/A

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

   7. Applied rewrites 73.3%

      \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

   9. Applied rewrites 75.1%

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

   if 4.5999999999999997e-242 < h

   1. Initial program 65.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-neg-rev N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   3. Applied rewrites 67.4%

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

   5. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow-prod-down N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow1/2 N/A

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

      12. sqrt-undiv N/A

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

      13. sqrt-fabs-rev N/A

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

      14. lift-sqrt.f64 N/A

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

      15. sqrt-fabs-rev N/A

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

      16. lift-sqrt.f64 N/A

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

      17. neg-fabs N/A

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

      18. div-fabs N/A

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

      19. lower-fabs.f64 N/A

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

      20. lower-/.f64 N/A

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

   7. Applied rewrites 41.1%

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

Alternative 5: 76.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|M\right|, D\right)\\ t_1 := \frac{-d}{\sqrt{\ell \cdot h}}\\ t_2 := \mathsf{max}\left(\left|M\right|, D\right)\\ \mathbf{if}\;h \leq -1 \cdot 10^{+70}:\\ \;\;\;\;t\_1 \cdot \left(1 - \frac{\left(\left(\frac{t\_0 \cdot t\_2}{d} \cdot 0.25\right) \cdot \left(\frac{t\_2}{d + d} \cdot t\_0\right)\right) \cdot h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|t\_1\right| \cdot \left(1 - \frac{t\_2 \cdot t\_0}{d} \cdot \frac{\left(\left(0.25 \cdot t\_0\right) \cdot t\_2\right) \cdot h}{\left(d + d\right) \cdot \ell}\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0 (fmin (fabs M) D))
       (t_1 (/ (- d) (sqrt (* l h))))
       (t_2 (fmax (fabs M) D)))
  (if (<= h -1e+70)
    (*
     t_1
     (-
      1.0
      (/
       (* (* (* (/ (* t_0 t_2) d) 0.25) (* (/ t_2 (+ d d)) t_0)) h)
       l)))
    (*
     (fabs t_1)
     (-
      1.0
      (*
       (/ (* t_2 t_0) d)
       (/ (* (* (* 0.25 t_0) t_2) h) (* (+ d d) l))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fmin(fabs(M), D);
	double t_1 = -d / sqrt((l * h));
	double t_2 = fmax(fabs(M), D);
	double tmp;
	if (h <= -1e+70) {
		tmp = t_1 * (1.0 - ((((((t_0 * t_2) / d) * 0.25) * ((t_2 / (d + d)) * t_0)) * h) / l));
	} else {
		tmp = fabs(t_1) * (1.0 - (((t_2 * t_0) / d) * ((((0.25 * t_0) * t_2) * h) / ((d + d) * l))));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = fmin(abs(m), d_1)
    t_1 = -d / sqrt((l * h))
    t_2 = fmax(abs(m), d_1)
    if (h <= (-1d+70)) then
        tmp = t_1 * (1.0d0 - ((((((t_0 * t_2) / d) * 0.25d0) * ((t_2 / (d + d)) * t_0)) * h) / l))
    else
        tmp = abs(t_1) * (1.0d0 - (((t_2 * t_0) / d) * ((((0.25d0 * t_0) * t_2) * h) / ((d + d) * l))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = fmin(Math.abs(M), D);
	double t_1 = -d / Math.sqrt((l * h));
	double t_2 = fmax(Math.abs(M), D);
	double tmp;
	if (h <= -1e+70) {
		tmp = t_1 * (1.0 - ((((((t_0 * t_2) / d) * 0.25) * ((t_2 / (d + d)) * t_0)) * h) / l));
	} else {
		tmp = Math.abs(t_1) * (1.0 - (((t_2 * t_0) / d) * ((((0.25 * t_0) * t_2) * h) / ((d + d) * l))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = fmin(math.fabs(M), D)
	t_1 = -d / math.sqrt((l * h))
	t_2 = fmax(math.fabs(M), D)
	tmp = 0
	if h <= -1e+70:
		tmp = t_1 * (1.0 - ((((((t_0 * t_2) / d) * 0.25) * ((t_2 / (d + d)) * t_0)) * h) / l))
	else:
		tmp = math.fabs(t_1) * (1.0 - (((t_2 * t_0) / d) * ((((0.25 * t_0) * t_2) * h) / ((d + d) * l))))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = fmin(abs(M), D)
	t_1 = Float64(Float64(-d) / sqrt(Float64(l * h)))
	t_2 = fmax(abs(M), D)
	tmp = 0.0
	if (h <= -1e+70)
		tmp = Float64(t_1 * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(t_0 * t_2) / d) * 0.25) * Float64(Float64(t_2 / Float64(d + d)) * t_0)) * h) / l)));
	else
		tmp = Float64(abs(t_1) * Float64(1.0 - Float64(Float64(Float64(t_2 * t_0) / d) * Float64(Float64(Float64(Float64(0.25 * t_0) * t_2) * h) / Float64(Float64(d + d) * l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = min(abs(M), D);
	t_1 = -d / sqrt((l * h));
	t_2 = max(abs(M), D);
	tmp = 0.0;
	if (h <= -1e+70)
		tmp = t_1 * (1.0 - ((((((t_0 * t_2) / d) * 0.25) * ((t_2 / (d + d)) * t_0)) * h) / l));
	else
		tmp = abs(t_1) * (1.0 - (((t_2 * t_0) / d) * ((((0.25 * t_0) * t_2) * h) / ((d + d) * l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Min[N[Abs[M], $MachinePrecision], D], $MachinePrecision]}, Block[{t$95$1 = N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Abs[M], $MachinePrecision], D], $MachinePrecision]}, If[LessEqual[h, -1e+70], N[(t$95$1 * N[(1.0 - N[(N[(N[(N[(N[(N[(t$95$0 * t$95$2), $MachinePrecision] / d), $MachinePrecision] * 0.25), $MachinePrecision] * N[(N[(t$95$2 / N[(d + d), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[t$95$1], $MachinePrecision] * N[(1.0 - N[(N[(N[(t$95$2 * t$95$0), $MachinePrecision] / d), $MachinePrecision] * N[(N[(N[(N[(0.25 * t$95$0), $MachinePrecision] * t$95$2), $MachinePrecision] * h), $MachinePrecision] / N[(N[(d + d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if h < -1.0000000000000001e70

   1. Initial program 65.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-neg-rev N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   3. Applied rewrites 67.4%

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

   5. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. sqrt-div N/A

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

      14. lower-unsound-sqrt.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. sqr-neg-rev N/A

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

      18. lift-neg.f64 N/A

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

      19. lift-neg.f64 N/A

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

      20. sqrt-unprod N/A

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

      21. rem-square-sqrt N/A

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

      22. lower-unsound-/.f64 N/A

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

      23. lower-unsound-sqrt.f64 37.7%

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

      24. lift-*.f64 N/A

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

      25. *-commutative N/A

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

      26. lower-*.f64 37.7%

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

   7. Applied rewrites 37.7%

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

   if -1.0000000000000001e70 < h

   1. Initial program 65.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-neg-rev N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   3. Applied rewrites 67.4%

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

   5. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. sqrt-fabs-rev N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lower-fabs.f64 53.0%

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

      18. lift-sqrt.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-/.f64 N/A

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

      21. associate-*r/ N/A

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

      22. lift-*.f64 N/A

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

      23. sqrt-div N/A

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

   7. Applied rewrites 73.3%

      \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

   9. Applied rewrites 75.1%

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

Alternative 6: 75.2% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (d h l M D)
  :precision binary64
  (*
 (fabs (/ (- d) (sqrt (* l h))))
 (- 1.0 (* (* 0.25 (/ (* D M) d)) (/ (* (* (/ D (+ d d)) M) h) l)))))

C

double code(double d, double h, double l, double M, double D) {
	return fabs((-d / sqrt((l * h)))) * (1.0 - ((0.25 * ((D * M) / d)) * ((((D / (d + d)) * M) * 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 = abs((-d / sqrt((l * h)))) * (1.0d0 - ((0.25d0 * ((d_1 * m) / d)) * ((((d_1 / (d + d)) * m) * h) / l)))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return Math.abs((-d / Math.sqrt((l * h)))) * (1.0 - ((0.25 * ((D * M) / d)) * ((((D / (d + d)) * M) * h) / l)));
}

Python

def code(d, h, l, M, D):
	return math.fabs((-d / math.sqrt((l * h)))) * (1.0 - ((0.25 * ((D * M) / d)) * ((((D / (d + d)) * M) * h) / l)))

Julia

function code(d, h, l, M, D)
	return Float64(abs(Float64(Float64(-d) / sqrt(Float64(l * h)))) * Float64(1.0 - Float64(Float64(0.25 * Float64(Float64(D * M) / d)) * Float64(Float64(Float64(Float64(D / Float64(d + d)) * M) * h) / l))))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = abs((-d / sqrt((l * h)))) * (1.0 - ((0.25 * ((D * M) / d)) * ((((D / (d + d)) * M) * h) / l)));
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(N[Abs[N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(0.25 * N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(D / N[(d + d), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lift-pow.f64 N/A

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

   6. unpow2 N/A

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

   7. sqr-neg-rev N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f64 N/A

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

3. Applied rewrites 67.4%

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

5. Applied rewrites 66.9%

   \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}}\right) \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. pow-prod-down N/A

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

   5. lift-/.f64 N/A

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

   6. metadata-eval N/A

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

   7. unpow1/2 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. frac-times N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*r/ N/A

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

   13. lift-/.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. sqrt-fabs-rev N/A

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

   16. lift-sqrt.f64 N/A

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

   17. lower-fabs.f64 53.0%

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

   18. lift-sqrt.f64 N/A

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

   19. lift-*.f64 N/A

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

   20. lift-/.f64 N/A

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

   21. associate-*r/ N/A

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

   22. lift-*.f64 N/A

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

   23. sqrt-div N/A

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

7. Applied rewrites 73.3%

   \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/l* N/A

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

   10. lift-/.f64 N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. associate-/l* N/A

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

   15. lower-*.f64 N/A

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

9. Applied rewrites 76.1%

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

Alternative 7: 74.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{-d}{\sqrt{\ell \cdot h}}\\ t_1 := \frac{D}{d + d} \cdot M\\ \mathbf{if}\;h \leq -1 \cdot 10^{-90}:\\ \;\;\;\;t\_0 \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot t\_1\right) \cdot h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|t\_0\right| \cdot \left(1 - \frac{\left(\left(0.25 \cdot M\right) \cdot D\right) \cdot \left(t\_1 \cdot h\right)}{\ell \cdot d}\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0 (/ (- d) (sqrt (* l h)))) (t_1 (* (/ D (+ d d)) M)))
  (if (<= h -1e-90)
    (* t_0 (- 1.0 (/ (* (* (* (/ (* M D) d) 0.25) t_1) h) l)))
    (*
     (fabs t_0)
     (- 1.0 (/ (* (* (* 0.25 M) D) (* t_1 h)) (* l d)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = -d / sqrt((l * h));
	double t_1 = (D / (d + d)) * M;
	double tmp;
	if (h <= -1e-90) {
		tmp = t_0 * (1.0 - ((((((M * D) / d) * 0.25) * t_1) * h) / l));
	} else {
		tmp = fabs(t_0) * (1.0 - ((((0.25 * M) * D) * (t_1 * h)) / (l * d)));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = -d / sqrt((l * h))
    t_1 = (d_1 / (d + d)) * m
    if (h <= (-1d-90)) then
        tmp = t_0 * (1.0d0 - ((((((m * d_1) / d) * 0.25d0) * t_1) * h) / l))
    else
        tmp = abs(t_0) * (1.0d0 - ((((0.25d0 * m) * d_1) * (t_1 * h)) / (l * d)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = -d / Math.sqrt((l * h));
	double t_1 = (D / (d + d)) * M;
	double tmp;
	if (h <= -1e-90) {
		tmp = t_0 * (1.0 - ((((((M * D) / d) * 0.25) * t_1) * h) / l));
	} else {
		tmp = Math.abs(t_0) * (1.0 - ((((0.25 * M) * D) * (t_1 * h)) / (l * d)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = -d / math.sqrt((l * h))
	t_1 = (D / (d + d)) * M
	tmp = 0
	if h <= -1e-90:
		tmp = t_0 * (1.0 - ((((((M * D) / d) * 0.25) * t_1) * h) / l))
	else:
		tmp = math.fabs(t_0) * (1.0 - ((((0.25 * M) * D) * (t_1 * h)) / (l * d)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(-d) / sqrt(Float64(l * h)))
	t_1 = Float64(Float64(D / Float64(d + d)) * M)
	tmp = 0.0
	if (h <= -1e-90)
		tmp = Float64(t_0 * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(M * D) / d) * 0.25) * t_1) * h) / l)));
	else
		tmp = Float64(abs(t_0) * Float64(1.0 - Float64(Float64(Float64(Float64(0.25 * M) * D) * Float64(t_1 * h)) / Float64(l * d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = -d / sqrt((l * h));
	t_1 = (D / (d + d)) * M;
	tmp = 0.0;
	if (h <= -1e-90)
		tmp = t_0 * (1.0 - ((((((M * D) / d) * 0.25) * t_1) * h) / l));
	else
		tmp = abs(t_0) * (1.0 - ((((0.25 * M) * D) * (t_1 * h)) / (l * d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(D / N[(d + d), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision]}, If[LessEqual[h, -1e-90], N[(t$95$0 * N[(1.0 - N[(N[(N[(N[(N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision] * 0.25), $MachinePrecision] * t$95$1), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[t$95$0], $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(0.25 * M), $MachinePrecision] * D), $MachinePrecision] * N[(t$95$1 * h), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if h < -9.9999999999999999e-91

   1. Initial program 65.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-neg-rev N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   3. Applied rewrites 67.4%

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

   5. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. sqrt-div N/A

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

      14. lower-unsound-sqrt.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. sqr-neg-rev N/A

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

      18. lift-neg.f64 N/A

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

      19. lift-neg.f64 N/A

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

      20. sqrt-unprod N/A

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

      21. rem-square-sqrt N/A

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

      22. lower-unsound-/.f64 N/A

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

      23. lower-unsound-sqrt.f64 37.7%

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

      24. lift-*.f64 N/A

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

      25. *-commutative N/A

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

      26. lower-*.f64 37.7%

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

   7. Applied rewrites 37.7%

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

   if -9.9999999999999999e-91 < h

   1. Initial program 65.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-neg-rev N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   3. Applied rewrites 67.4%

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

   5. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. sqrt-fabs-rev N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lower-fabs.f64 53.0%

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

      18. lift-sqrt.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-/.f64 N/A

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

      21. associate-*r/ N/A

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

      22. lift-*.f64 N/A

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

      23. sqrt-div N/A

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

   7. Applied rewrites 73.3%

      \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lift-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*r/ N/A

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

   9. Applied rewrites 73.7%

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

Alternative 8: 71.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|M\right|, D\right)\\ t_1 := \frac{-d}{\sqrt{\ell \cdot h}}\\ t_2 := \mathsf{max}\left(\left|M\right|, D\right)\\ \mathbf{if}\;h \leq -1 \cdot 10^{-90}:\\ \;\;\;\;t\_1 \cdot \left(1 - \frac{\left(\left(\frac{t\_0 \cdot t\_2}{d} \cdot 0.25\right) \cdot \left(\frac{t\_2}{d + d} \cdot t\_0\right)\right) \cdot h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|t\_1\right| \cdot \left(1 - \frac{h \cdot \left(\left(0.25 \cdot t\_0\right) \cdot t\_2\right)}{\left(\left(d + d\right) \cdot \ell\right) \cdot d} \cdot \left(t\_2 \cdot t\_0\right)\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0 (fmin (fabs M) D))
       (t_1 (/ (- d) (sqrt (* l h))))
       (t_2 (fmax (fabs M) D)))
  (if (<= h -1e-90)
    (*
     t_1
     (-
      1.0
      (/
       (* (* (* (/ (* t_0 t_2) d) 0.25) (* (/ t_2 (+ d d)) t_0)) h)
       l)))
    (*
     (fabs t_1)
     (-
      1.0
      (*
       (/ (* h (* (* 0.25 t_0) t_2)) (* (* (+ d d) l) d))
       (* t_2 t_0)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fmin(fabs(M), D);
	double t_1 = -d / sqrt((l * h));
	double t_2 = fmax(fabs(M), D);
	double tmp;
	if (h <= -1e-90) {
		tmp = t_1 * (1.0 - ((((((t_0 * t_2) / d) * 0.25) * ((t_2 / (d + d)) * t_0)) * h) / l));
	} else {
		tmp = fabs(t_1) * (1.0 - (((h * ((0.25 * t_0) * t_2)) / (((d + d) * l) * d)) * (t_2 * t_0)));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = fmin(abs(m), d_1)
    t_1 = -d / sqrt((l * h))
    t_2 = fmax(abs(m), d_1)
    if (h <= (-1d-90)) then
        tmp = t_1 * (1.0d0 - ((((((t_0 * t_2) / d) * 0.25d0) * ((t_2 / (d + d)) * t_0)) * h) / l))
    else
        tmp = abs(t_1) * (1.0d0 - (((h * ((0.25d0 * t_0) * t_2)) / (((d + d) * l) * d)) * (t_2 * t_0)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = fmin(Math.abs(M), D);
	double t_1 = -d / Math.sqrt((l * h));
	double t_2 = fmax(Math.abs(M), D);
	double tmp;
	if (h <= -1e-90) {
		tmp = t_1 * (1.0 - ((((((t_0 * t_2) / d) * 0.25) * ((t_2 / (d + d)) * t_0)) * h) / l));
	} else {
		tmp = Math.abs(t_1) * (1.0 - (((h * ((0.25 * t_0) * t_2)) / (((d + d) * l) * d)) * (t_2 * t_0)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = fmin(math.fabs(M), D)
	t_1 = -d / math.sqrt((l * h))
	t_2 = fmax(math.fabs(M), D)
	tmp = 0
	if h <= -1e-90:
		tmp = t_1 * (1.0 - ((((((t_0 * t_2) / d) * 0.25) * ((t_2 / (d + d)) * t_0)) * h) / l))
	else:
		tmp = math.fabs(t_1) * (1.0 - (((h * ((0.25 * t_0) * t_2)) / (((d + d) * l) * d)) * (t_2 * t_0)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = fmin(abs(M), D)
	t_1 = Float64(Float64(-d) / sqrt(Float64(l * h)))
	t_2 = fmax(abs(M), D)
	tmp = 0.0
	if (h <= -1e-90)
		tmp = Float64(t_1 * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(t_0 * t_2) / d) * 0.25) * Float64(Float64(t_2 / Float64(d + d)) * t_0)) * h) / l)));
	else
		tmp = Float64(abs(t_1) * Float64(1.0 - Float64(Float64(Float64(h * Float64(Float64(0.25 * t_0) * t_2)) / Float64(Float64(Float64(d + d) * l) * d)) * Float64(t_2 * t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = min(abs(M), D);
	t_1 = -d / sqrt((l * h));
	t_2 = max(abs(M), D);
	tmp = 0.0;
	if (h <= -1e-90)
		tmp = t_1 * (1.0 - ((((((t_0 * t_2) / d) * 0.25) * ((t_2 / (d + d)) * t_0)) * h) / l));
	else
		tmp = abs(t_1) * (1.0 - (((h * ((0.25 * t_0) * t_2)) / (((d + d) * l) * d)) * (t_2 * t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Min[N[Abs[M], $MachinePrecision], D], $MachinePrecision]}, Block[{t$95$1 = N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Abs[M], $MachinePrecision], D], $MachinePrecision]}, If[LessEqual[h, -1e-90], N[(t$95$1 * N[(1.0 - N[(N[(N[(N[(N[(N[(t$95$0 * t$95$2), $MachinePrecision] / d), $MachinePrecision] * 0.25), $MachinePrecision] * N[(N[(t$95$2 / N[(d + d), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[t$95$1], $MachinePrecision] * N[(1.0 - N[(N[(N[(h * N[(N[(0.25 * t$95$0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(d + d), $MachinePrecision] * l), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if h < -9.9999999999999999e-91

   1. Initial program 65.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-neg-rev N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   3. Applied rewrites 67.4%

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

   5. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. sqrt-div N/A

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

      14. lower-unsound-sqrt.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. sqr-neg-rev N/A

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

      18. lift-neg.f64 N/A

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

      19. lift-neg.f64 N/A

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

      20. sqrt-unprod N/A

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

      21. rem-square-sqrt N/A

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

      22. lower-unsound-/.f64 N/A

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

      23. lower-unsound-sqrt.f64 37.7%

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

      24. lift-*.f64 N/A

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

      25. *-commutative N/A

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

      26. lower-*.f64 37.7%

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

   7. Applied rewrites 37.7%

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

   if -9.9999999999999999e-91 < h

   1. Initial program 65.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-neg-rev N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   3. Applied rewrites 67.4%

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

   5. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. sqrt-fabs-rev N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lower-fabs.f64 53.0%

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

      18. lift-sqrt.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-/.f64 N/A

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

      21. associate-*r/ N/A

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

      22. lift-*.f64 N/A

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

      23. sqrt-div N/A

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

   7. Applied rewrites 73.3%

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

   9. Applied rewrites 69.4%

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

Alternative 9: 71.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_0 := \left|\frac{-d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{if}\;\left|M\right| \cdot \left|D\right| \leq 10^{-129}:\\ \;\;\;\;t\_0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(1 - \frac{h \cdot \left(\left(0.25 \cdot \left|M\right|\right) \cdot \left|D\right|\right)}{\left(\left(d + d\right) \cdot \ell\right) \cdot d} \cdot \left(\left|D\right| \cdot \left|M\right|\right)\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0 (fabs (/ (- d) (sqrt (* l h))))))
  (if (<= (* (fabs M) (fabs D)) 1e-129)
    (* t_0 1.0)
    (*
     t_0
     (-
      1.0
      (*
       (/ (* h (* (* 0.25 (fabs M)) (fabs D))) (* (* (+ d d) l) d))
       (* (fabs D) (fabs M))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fabs((-d / sqrt((l * h))));
	double tmp;
	if ((fabs(M) * fabs(D)) <= 1e-129) {
		tmp = t_0 * 1.0;
	} else {
		tmp = t_0 * (1.0 - (((h * ((0.25 * fabs(M)) * fabs(D))) / (((d + d) * l) * d)) * (fabs(D) * fabs(M))));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((-d / sqrt((l * h))))
    if ((abs(m) * abs(d_1)) <= 1d-129) then
        tmp = t_0 * 1.0d0
    else
        tmp = t_0 * (1.0d0 - (((h * ((0.25d0 * abs(m)) * abs(d_1))) / (((d + d) * l) * d)) * (abs(d_1) * abs(m))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.abs((-d / Math.sqrt((l * h))));
	double tmp;
	if ((Math.abs(M) * Math.abs(D)) <= 1e-129) {
		tmp = t_0 * 1.0;
	} else {
		tmp = t_0 * (1.0 - (((h * ((0.25 * Math.abs(M)) * Math.abs(D))) / (((d + d) * l) * d)) * (Math.abs(D) * Math.abs(M))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.fabs((-d / math.sqrt((l * h))))
	tmp = 0
	if (math.fabs(M) * math.fabs(D)) <= 1e-129:
		tmp = t_0 * 1.0
	else:
		tmp = t_0 * (1.0 - (((h * ((0.25 * math.fabs(M)) * math.fabs(D))) / (((d + d) * l) * d)) * (math.fabs(D) * math.fabs(M))))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = abs(Float64(Float64(-d) / sqrt(Float64(l * h))))
	tmp = 0.0
	if (Float64(abs(M) * abs(D)) <= 1e-129)
		tmp = Float64(t_0 * 1.0);
	else
		tmp = Float64(t_0 * Float64(1.0 - Float64(Float64(Float64(h * Float64(Float64(0.25 * abs(M)) * abs(D))) / Float64(Float64(Float64(d + d) * l) * d)) * Float64(abs(D) * abs(M)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = abs((-d / sqrt((l * h))));
	tmp = 0.0;
	if ((abs(M) * abs(D)) <= 1e-129)
		tmp = t_0 * 1.0;
	else
		tmp = t_0 * (1.0 - (((h * ((0.25 * abs(M)) * abs(D))) / (((d + d) * l) * d)) * (abs(D) * abs(M))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Abs[N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Abs[M], $MachinePrecision] * N[Abs[D], $MachinePrecision]), $MachinePrecision], 1e-129], N[(t$95$0 * 1.0), $MachinePrecision], N[(t$95$0 * N[(1.0 - N[(N[(N[(h * N[(N[(0.25 * N[Abs[M], $MachinePrecision]), $MachinePrecision] * N[Abs[D], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(d + d), $MachinePrecision] * l), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[D], $MachinePrecision] * N[Abs[M], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 9.9999999999999993e-130

   1. Initial program 65.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-neg-rev N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   3. Applied rewrites 67.4%

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

   5. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. sqrt-fabs-rev N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lower-fabs.f64 53.0%

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

      18. lift-sqrt.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-/.f64 N/A

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

      21. associate-*r/ N/A

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

      22. lift-*.f64 N/A

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

      23. sqrt-div N/A

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

   7. Applied rewrites 73.3%

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

   8. Taylor expanded in d around inf

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

   10. Applied rewrites 42.5%

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

   if 9.9999999999999993e-130 < (*.f64 M D)

   1. Initial program 65.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      7. sqr-neg-rev N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right)\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

   3. Applied rewrites 67.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)}\right) \]

   5. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      4. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      10. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      12. associate-*r/ N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{d \cdot \color{blue}{\frac{d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      15. sqrt-fabs-rev N/A

          \[\leadsto \color{blue}{\left|\sqrt{d \cdot \frac{d}{h \cdot \ell}}\right|} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \left|\color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      17. lower-fabs.f64 53.0%

          \[\leadsto \color{blue}{\left|\sqrt{d \cdot \frac{d}{h \cdot \ell}}\right|} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      18. lift-sqrt.f64 N/A

          \[\leadsto \left|\color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \left|\sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      20. lift-/.f64 N/A

          \[\leadsto \left|\sqrt{d \cdot \color{blue}{\frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      21. associate-*r/ N/A

          \[\leadsto \left|\sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      22. lift-*.f64 N/A

          \[\leadsto \left|\sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      23. sqrt-div N/A

          \[\leadsto \left|\color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

   7. Applied rewrites 73.3%

      \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

   9. Applied rewrites 69.4%

      \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\frac{h \cdot \left(\left(0.25 \cdot M\right) \cdot D\right)}{\left(\left(d + d\right) \cdot \ell\right) \cdot d} \cdot \left(D \cdot M\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 67.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \left(d + d\right) \cdot \ell\\ t_1 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_2 := 0.25 \cdot t\_1\\ t_3 := \frac{-d}{\sqrt{\ell \cdot h}}\\ t_4 := \left|t\_3\right|\\ t_5 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_6 := t\_5 \cdot t\_1\\ \mathbf{if}\;d \leq -1.1 \cdot 10^{+71}:\\ \;\;\;\;t\_3 \cdot \left(1 - \left(\left(t\_2 \cdot \frac{t\_5}{d}\right) \cdot t\_6\right) \cdot \frac{h}{t\_0}\right)\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{+197}:\\ \;\;\;\;t\_4 \cdot \left(1 - t\_2 \cdot \frac{t\_5 \cdot \left(t\_6 \cdot h\right)}{d \cdot t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_4 \cdot 1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0 (* (+ d d) l))
       (t_1 (fmin (fabs M) (fabs D)))
       (t_2 (* 0.25 t_1))
       (t_3 (/ (- d) (sqrt (* l h))))
       (t_4 (fabs t_3))
       (t_5 (fmax (fabs M) (fabs D)))
       (t_6 (* t_5 t_1)))
  (if (<= d -1.1e+71)
    (* t_3 (- 1.0 (* (* (* t_2 (/ t_5 d)) t_6) (/ h t_0))))
    (if (<= d 3.2e+197)
      (* t_4 (- 1.0 (* t_2 (/ (* t_5 (* t_6 h)) (* d t_0)))))
      (* t_4 1.0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (d + d) * l;
	double t_1 = fmin(fabs(M), fabs(D));
	double t_2 = 0.25 * t_1;
	double t_3 = -d / sqrt((l * h));
	double t_4 = fabs(t_3);
	double t_5 = fmax(fabs(M), fabs(D));
	double t_6 = t_5 * t_1;
	double tmp;
	if (d <= -1.1e+71) {
		tmp = t_3 * (1.0 - (((t_2 * (t_5 / d)) * t_6) * (h / t_0)));
	} else if (d <= 3.2e+197) {
		tmp = t_4 * (1.0 - (t_2 * ((t_5 * (t_6 * h)) / (d * t_0))));
	} else {
		tmp = t_4 * 1.0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: tmp
    t_0 = (d + d) * l
    t_1 = fmin(abs(m), abs(d_1))
    t_2 = 0.25d0 * t_1
    t_3 = -d / sqrt((l * h))
    t_4 = abs(t_3)
    t_5 = fmax(abs(m), abs(d_1))
    t_6 = t_5 * t_1
    if (d <= (-1.1d+71)) then
        tmp = t_3 * (1.0d0 - (((t_2 * (t_5 / d)) * t_6) * (h / t_0)))
    else if (d <= 3.2d+197) then
        tmp = t_4 * (1.0d0 - (t_2 * ((t_5 * (t_6 * h)) / (d * t_0))))
    else
        tmp = t_4 * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (d + d) * l;
	double t_1 = fmin(Math.abs(M), Math.abs(D));
	double t_2 = 0.25 * t_1;
	double t_3 = -d / Math.sqrt((l * h));
	double t_4 = Math.abs(t_3);
	double t_5 = fmax(Math.abs(M), Math.abs(D));
	double t_6 = t_5 * t_1;
	double tmp;
	if (d <= -1.1e+71) {
		tmp = t_3 * (1.0 - (((t_2 * (t_5 / d)) * t_6) * (h / t_0)));
	} else if (d <= 3.2e+197) {
		tmp = t_4 * (1.0 - (t_2 * ((t_5 * (t_6 * h)) / (d * t_0))));
	} else {
		tmp = t_4 * 1.0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (d + d) * l
	t_1 = fmin(math.fabs(M), math.fabs(D))
	t_2 = 0.25 * t_1
	t_3 = -d / math.sqrt((l * h))
	t_4 = math.fabs(t_3)
	t_5 = fmax(math.fabs(M), math.fabs(D))
	t_6 = t_5 * t_1
	tmp = 0
	if d <= -1.1e+71:
		tmp = t_3 * (1.0 - (((t_2 * (t_5 / d)) * t_6) * (h / t_0)))
	elif d <= 3.2e+197:
		tmp = t_4 * (1.0 - (t_2 * ((t_5 * (t_6 * h)) / (d * t_0))))
	else:
		tmp = t_4 * 1.0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(d + d) * l)
	t_1 = fmin(abs(M), abs(D))
	t_2 = Float64(0.25 * t_1)
	t_3 = Float64(Float64(-d) / sqrt(Float64(l * h)))
	t_4 = abs(t_3)
	t_5 = fmax(abs(M), abs(D))
	t_6 = Float64(t_5 * t_1)
	tmp = 0.0
	if (d <= -1.1e+71)
		tmp = Float64(t_3 * Float64(1.0 - Float64(Float64(Float64(t_2 * Float64(t_5 / d)) * t_6) * Float64(h / t_0))));
	elseif (d <= 3.2e+197)
		tmp = Float64(t_4 * Float64(1.0 - Float64(t_2 * Float64(Float64(t_5 * Float64(t_6 * h)) / Float64(d * t_0)))));
	else
		tmp = Float64(t_4 * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (d + d) * l;
	t_1 = min(abs(M), abs(D));
	t_2 = 0.25 * t_1;
	t_3 = -d / sqrt((l * h));
	t_4 = abs(t_3);
	t_5 = max(abs(M), abs(D));
	t_6 = t_5 * t_1;
	tmp = 0.0;
	if (d <= -1.1e+71)
		tmp = t_3 * (1.0 - (((t_2 * (t_5 / d)) * t_6) * (h / t_0)));
	elseif (d <= 3.2e+197)
		tmp = t_4 * (1.0 - (t_2 * ((t_5 * (t_6 * h)) / (d * t_0))));
	else
		tmp = t_4 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(d + d), $MachinePrecision] * l), $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(0.25 * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Abs[t$95$3], $MachinePrecision]}, Block[{t$95$5 = N[Max[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(t$95$5 * t$95$1), $MachinePrecision]}, If[LessEqual[d, -1.1e+71], N[(t$95$3 * N[(1.0 - N[(N[(N[(t$95$2 * N[(t$95$5 / d), $MachinePrecision]), $MachinePrecision] * t$95$6), $MachinePrecision] * N[(h / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.2e+197], N[(t$95$4 * N[(1.0 - N[(t$95$2 * N[(N[(t$95$5 * N[(t$95$6 * h), $MachinePrecision]), $MachinePrecision] / N[(d * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$4 * 1.0), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.1e71

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      7. sqr-neg-rev N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right)\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

   3. Applied rewrites 67.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)}\right) \]

   5. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}}\right) \]

   7. Applied rewrites 35.5%

      \[\leadsto \color{blue}{\frac{-d}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left(\left(\left(0.25 \cdot M\right) \cdot \frac{D}{d}\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right)} \]

   if -1.1e71 < d < 3.1999999999999998e197

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      7. sqr-neg-rev N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right)\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

   3. Applied rewrites 67.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)}\right) \]

   5. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      4. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      10. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      12. associate-*r/ N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{d \cdot \color{blue}{\frac{d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      15. sqrt-fabs-rev N/A

          \[\leadsto \color{blue}{\left|\sqrt{d \cdot \frac{d}{h \cdot \ell}}\right|} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \left|\color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      17. lower-fabs.f64 53.0%

          \[\leadsto \color{blue}{\left|\sqrt{d \cdot \frac{d}{h \cdot \ell}}\right|} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      18. lift-sqrt.f64 N/A

          \[\leadsto \left|\color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \left|\sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      20. lift-/.f64 N/A

          \[\leadsto \left|\sqrt{d \cdot \color{blue}{\frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      21. associate-*r/ N/A

          \[\leadsto \left|\sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      22. lift-*.f64 N/A

          \[\leadsto \left|\sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      23. sqrt-div N/A

          \[\leadsto \left|\color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

   7. Applied rewrites 73.3%

      \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\color{blue}{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}}{\ell}\right) \]

      3. associate-/l* N/A

         \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot \frac{h}{\ell}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right)} \cdot \frac{h}{\ell}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot \frac{h}{\ell}\right) \]

      6. *-commutative N/A

         \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\color{blue}{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right)} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\left(\frac{1}{4} \cdot \color{blue}{\frac{M \cdot D}{d}}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\left(\frac{1}{4} \cdot \frac{\color{blue}{M \cdot D}}{d}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot \frac{h}{\ell}\right) \]

      9. associate-/l* N/A

         \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\left(\frac{1}{4} \cdot \color{blue}{\left(M \cdot \frac{D}{d}\right)}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\left(\frac{1}{4} \cdot \left(M \cdot \color{blue}{\frac{D}{d}}\right)\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot \frac{h}{\ell}\right) \]

      11. associate-*l* N/A

          \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\color{blue}{\left(\left(\frac{1}{4} \cdot M\right) \cdot \frac{D}{d}\right)} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot \frac{h}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\left(\color{blue}{\left(\frac{1}{4} \cdot M\right)} \cdot \frac{D}{d}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\color{blue}{\left(\left(\frac{1}{4} \cdot M\right) \cdot \frac{D}{d}\right)} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot \frac{h}{\ell}\right) \]

      14. associate-*l* N/A

          \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{4} \cdot M\right) \cdot \frac{D}{d}\right) \cdot \left(\left(\frac{D}{d + d} \cdot M\right) \cdot \frac{h}{\ell}\right)}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{4} \cdot M\right) \cdot \frac{D}{d}\right)} \cdot \left(\left(\frac{D}{d + d} \cdot M\right) \cdot \frac{h}{\ell}\right)\right) \]

   9. Applied rewrites 66.6%

      \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(0.25 \cdot M\right) \cdot \frac{D \cdot \left(\left(D \cdot M\right) \cdot h\right)}{d \cdot \left(\left(d + d\right) \cdot \ell\right)}}\right) \]

   if 3.1999999999999998e197 < d

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      7. sqr-neg-rev N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right)\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

   3. Applied rewrites 67.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)}\right) \]

   5. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      4. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      10. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      12. associate-*r/ N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{d \cdot \color{blue}{\frac{d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      15. sqrt-fabs-rev N/A

          \[\leadsto \color{blue}{\left|\sqrt{d \cdot \frac{d}{h \cdot \ell}}\right|} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \left|\color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      17. lower-fabs.f64 53.0%

          \[\leadsto \color{blue}{\left|\sqrt{d \cdot \frac{d}{h \cdot \ell}}\right|} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      18. lift-sqrt.f64 N/A

          \[\leadsto \left|\color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \left|\sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      20. lift-/.f64 N/A

          \[\leadsto \left|\sqrt{d \cdot \color{blue}{\frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      21. associate-*r/ N/A

          \[\leadsto \left|\sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      22. lift-*.f64 N/A

          \[\leadsto \left|\sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      23. sqrt-div N/A

          \[\leadsto \left|\color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

   7. Applied rewrites 73.3%

      \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

   8. Taylor expanded in d around inf

      \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{1} \]
   9. Step-by-step derivation

   10. Applied rewrites 42.5%

       \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 64.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \left(d + d\right) \cdot \ell\\ t_1 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_2 := 0.25 \cdot t\_1\\ t_3 := \sqrt{\ell \cdot h}\\ t_4 := \frac{-d}{t\_3}\\ t_5 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ \mathbf{if}\;d \leq -1.95 \cdot 10^{+71}:\\ \;\;\;\;t\_4 \cdot \left(1 - \left(\left(t\_2 \cdot \frac{t\_5}{d}\right) \cdot \left(t\_5 \cdot t\_1\right)\right) \cdot \frac{h}{t\_0}\right)\\ \mathbf{elif}\;d \leq -8.8 \cdot 10^{-302}:\\ \;\;\;\;\frac{\left(1 - \frac{h \cdot \left(\left(t\_2 \cdot t\_5\right) \cdot t\_5\right)}{t\_0 \cdot d} \cdot t\_1\right) \cdot \left(-d\right)}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;d \leq 2.15 \cdot 10^{+120}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(t\_5 \cdot t\_5\right) \cdot h\right) \cdot -0.5, t\_2 \cdot \frac{t\_1}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{\left|d\right|}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\left|t\_4\right| \cdot 1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0 (* (+ d d) l))
       (t_1 (fmin (fabs M) (fabs D)))
       (t_2 (* 0.25 t_1))
       (t_3 (sqrt (* l h)))
       (t_4 (/ (- d) t_3))
       (t_5 (fmax (fabs M) (fabs D))))
  (if (<= d -1.95e+71)
    (* t_4 (- 1.0 (* (* (* t_2 (/ t_5 d)) (* t_5 t_1)) (/ h t_0))))
    (if (<= d -8.8e-302)
      (/
       (*
        (- 1.0 (* (/ (* h (* (* t_2 t_5) t_5)) (* t_0 d)) t_1))
        (- d))
       (sqrt (* h l)))
      (if (<= d 2.15e+120)
        (*
         (fma
          (* (* (* t_5 t_5) h) -0.5)
          (* t_2 (/ t_1 (* (* d d) l)))
          1.0)
         (/ (fabs d) t_3))
        (* (fabs t_4) 1.0))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (d + d) * l;
	double t_1 = fmin(fabs(M), fabs(D));
	double t_2 = 0.25 * t_1;
	double t_3 = sqrt((l * h));
	double t_4 = -d / t_3;
	double t_5 = fmax(fabs(M), fabs(D));
	double tmp;
	if (d <= -1.95e+71) {
		tmp = t_4 * (1.0 - (((t_2 * (t_5 / d)) * (t_5 * t_1)) * (h / t_0)));
	} else if (d <= -8.8e-302) {
		tmp = ((1.0 - (((h * ((t_2 * t_5) * t_5)) / (t_0 * d)) * t_1)) * -d) / sqrt((h * l));
	} else if (d <= 2.15e+120) {
		tmp = fma((((t_5 * t_5) * h) * -0.5), (t_2 * (t_1 / ((d * d) * l))), 1.0) * (fabs(d) / t_3);
	} else {
		tmp = fabs(t_4) * 1.0;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(d + d) * l)
	t_1 = fmin(abs(M), abs(D))
	t_2 = Float64(0.25 * t_1)
	t_3 = sqrt(Float64(l * h))
	t_4 = Float64(Float64(-d) / t_3)
	t_5 = fmax(abs(M), abs(D))
	tmp = 0.0
	if (d <= -1.95e+71)
		tmp = Float64(t_4 * Float64(1.0 - Float64(Float64(Float64(t_2 * Float64(t_5 / d)) * Float64(t_5 * t_1)) * Float64(h / t_0))));
	elseif (d <= -8.8e-302)
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(Float64(h * Float64(Float64(t_2 * t_5) * t_5)) / Float64(t_0 * d)) * t_1)) * Float64(-d)) / sqrt(Float64(h * l)));
	elseif (d <= 2.15e+120)
		tmp = Float64(fma(Float64(Float64(Float64(t_5 * t_5) * h) * -0.5), Float64(t_2 * Float64(t_1 / Float64(Float64(d * d) * l))), 1.0) * Float64(abs(d) / t_3));
	else
		tmp = Float64(abs(t_4) * 1.0);
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(d + d), $MachinePrecision] * l), $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(0.25 * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[((-d) / t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[Max[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -1.95e+71], N[(t$95$4 * N[(1.0 - N[(N[(N[(t$95$2 * N[(t$95$5 / d), $MachinePrecision]), $MachinePrecision] * N[(t$95$5 * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(h / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -8.8e-302], N[(N[(N[(1.0 - N[(N[(N[(h * N[(N[(t$95$2 * t$95$5), $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * d), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * (-d)), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.15e+120], N[(N[(N[(N[(N[(t$95$5 * t$95$5), $MachinePrecision] * h), $MachinePrecision] * -0.5), $MachinePrecision] * N[(t$95$2 * N[(t$95$1 / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision], N[(N[Abs[t$95$4], $MachinePrecision] * 1.0), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.9500000000000001e71

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      7. sqr-neg-rev N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right)\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

   3. Applied rewrites 67.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)}\right) \]

   5. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}}\right) \]

   7. Applied rewrites 35.5%

      \[\leadsto \color{blue}{\frac{-d}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left(\left(\left(0.25 \cdot M\right) \cdot \frac{D}{d}\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right)} \]

   if -1.9500000000000001e71 < d < -8.8000000000000003e-302

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      7. sqr-neg-rev N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right)\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

   3. Applied rewrites 67.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)}\right) \]

   5. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      4. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      10. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      12. associate-*r/ N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{d \cdot \color{blue}{\frac{d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      15. sqrt-fabs-rev N/A

          \[\leadsto \color{blue}{\left|\sqrt{d \cdot \frac{d}{h \cdot \ell}}\right|} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \left|\color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      17. lower-fabs.f64 53.0%

          \[\leadsto \color{blue}{\left|\sqrt{d \cdot \frac{d}{h \cdot \ell}}\right|} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      18. lift-sqrt.f64 N/A

          \[\leadsto \left|\color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \left|\sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      20. lift-/.f64 N/A

          \[\leadsto \left|\sqrt{d \cdot \color{blue}{\frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      21. associate-*r/ N/A

          \[\leadsto \left|\sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      22. lift-*.f64 N/A

          \[\leadsto \left|\sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      23. sqrt-div N/A

          \[\leadsto \left|\color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

   7. Applied rewrites 73.3%

      \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

   9. Applied rewrites 33.6%

      \[\leadsto \color{blue}{\frac{\left(1 - \frac{h \cdot \left(\left(\left(0.25 \cdot M\right) \cdot D\right) \cdot D\right)}{\left(\left(d + d\right) \cdot \ell\right) \cdot d} \cdot M\right) \cdot \left(-d\right)}{\sqrt{h \cdot \ell}}} \]

   if -8.8000000000000003e-302 < d < 2.1500000000000001e120

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 35.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right)}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}{\ell \cdot \left(d \cdot d\right)}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h \cdot \color{blue}{\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{\left(h \cdot \left(D \cdot D\right)\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)}}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      5. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\left(h \cdot \left(D \cdot D\right)\right) \cdot \frac{\left(M \cdot M\right) \cdot \frac{1}{4}}{\ell \cdot \left(d \cdot d\right)}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\left(h \cdot \left(D \cdot D\right)\right) \cdot \frac{\left(M \cdot M\right) \cdot \frac{1}{4}}{\ell \cdot \left(d \cdot d\right)}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\left(\left(D \cdot D\right) \cdot h\right)} \cdot \frac{\left(M \cdot M\right) \cdot \frac{1}{4}}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\left(\left(D \cdot D\right) \cdot h\right)} \cdot \frac{\left(M \cdot M\right) \cdot \frac{1}{4}}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      9. lower-/.f64 34.0%

         \[\leadsto \mathsf{fma}\left(-0.5, \left(\left(D \cdot D\right) \cdot h\right) \cdot \color{blue}{\frac{\left(M \cdot M\right) \cdot 0.25}{\ell \cdot \left(d \cdot d\right)}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{\color{blue}{\left(M \cdot M\right) \cdot \frac{1}{4}}}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{\color{blue}{\frac{1}{4} \cdot \left(M \cdot M\right)}}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      12. lower-*.f64 34.0%

          \[\leadsto \mathsf{fma}\left(-0.5, \left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{\color{blue}{0.25 \cdot \left(M \cdot M\right)}}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{\frac{1}{4} \cdot \left(M \cdot M\right)}{\color{blue}{\ell \cdot \left(d \cdot d\right)}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{\frac{1}{4} \cdot \left(M \cdot M\right)}{\color{blue}{\left(d \cdot d\right) \cdot \ell}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      15. lower-*.f64 34.0%

          \[\leadsto \mathsf{fma}\left(-0.5, \left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{0.25 \cdot \left(M \cdot M\right)}{\color{blue}{\left(d \cdot d\right) \cdot \ell}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

   5. Applied rewrites 34.0%

      \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{0.25 \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]
   6. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{\frac{1}{4} \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}\right) + 1\right)} \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{2} \cdot \color{blue}{\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{\frac{1}{4} \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}\right)} + 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{2} \cdot \left(\left(D \cdot D\right) \cdot h\right)\right) \cdot \frac{\frac{1}{4} \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}} + 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(D \cdot D\right) \cdot h\right), \frac{\frac{1}{4} \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right)} \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}}, \frac{\frac{1}{4} \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      6. lower-*.f64 34.0%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(D \cdot D\right) \cdot h\right) \cdot -0.5}, \frac{0.25 \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \color{blue}{\frac{\frac{1}{4} \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \frac{\color{blue}{\frac{1}{4} \cdot \left(M \cdot M\right)}}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \frac{\frac{1}{4} \cdot \color{blue}{\left(M \cdot M\right)}}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \frac{\color{blue}{\left(\frac{1}{4} \cdot M\right) \cdot M}}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \color{blue}{\left(\frac{1}{4} \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \color{blue}{\left(\frac{1}{4} \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \color{blue}{\left(\frac{1}{4} \cdot M\right)} \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      14. lower-/.f64 37.8%

          \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot -0.5, \left(0.25 \cdot M\right) \cdot \color{blue}{\frac{M}{\left(d \cdot d\right) \cdot \ell}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      15. lift-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \left(\frac{1}{4} \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}} \]

   7. Applied rewrites 27.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot -0.5, \left(0.25 \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}} \]
   8. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \left(\frac{1}{4} \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{\color{blue}{\sqrt{-d} \cdot \sqrt{-d}}}{\sqrt{\ell \cdot h}} \]

      2. sqrt-unprod N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \left(\frac{1}{4} \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{\color{blue}{\sqrt{\left(-d\right) \cdot \left(-d\right)}}}{\sqrt{\ell \cdot h}} \]

      3. rem-sqrt-square-rev N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \left(\frac{1}{4} \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{\color{blue}{\left|-d\right|}}{\sqrt{\ell \cdot h}} \]

      4. lift-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \left(\frac{1}{4} \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{\left|\color{blue}{\mathsf{neg}\left(d\right)}\right|}{\sqrt{\ell \cdot h}} \]

      5. neg-fabs N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \left(\frac{1}{4} \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{\color{blue}{\left|d\right|}}{\sqrt{\ell \cdot h}} \]

      6. lower-fabs.f64 53.7%

         \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot -0.5, \left(0.25 \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{\color{blue}{\left|d\right|}}{\sqrt{\ell \cdot h}} \]

   9. Applied rewrites 53.7%

      \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot -0.5, \left(0.25 \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{\color{blue}{\left|d\right|}}{\sqrt{\ell \cdot h}} \]

   if 2.1500000000000001e120 < d

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      7. sqr-neg-rev N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right)\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

   3. Applied rewrites 67.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)}\right) \]

   5. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      4. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      10. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      12. associate-*r/ N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{d \cdot \color{blue}{\frac{d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      15. sqrt-fabs-rev N/A

          \[\leadsto \color{blue}{\left|\sqrt{d \cdot \frac{d}{h \cdot \ell}}\right|} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \left|\color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      17. lower-fabs.f64 53.0%

          \[\leadsto \color{blue}{\left|\sqrt{d \cdot \frac{d}{h \cdot \ell}}\right|} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      18. lift-sqrt.f64 N/A

          \[\leadsto \left|\color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \left|\sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      20. lift-/.f64 N/A

          \[\leadsto \left|\sqrt{d \cdot \color{blue}{\frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      21. associate-*r/ N/A

          \[\leadsto \left|\sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      22. lift-*.f64 N/A

          \[\leadsto \left|\sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      23. sqrt-div N/A

          \[\leadsto \left|\color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

   7. Applied rewrites 73.3%

      \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

   8. Taylor expanded in d around inf

      \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{1} \]
   9. Step-by-step derivation

   10. Applied rewrites 42.5%

       \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{1} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 12: 62.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ t_1 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_2 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_3 := \mathsf{fma}\left(\left(\left(t\_2 \cdot t\_2\right) \cdot h\right) \cdot -0.5, \left(0.25 \cdot t\_1\right) \cdot \frac{t\_1}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{\left|d\right|}{t\_0}\\ t_4 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_1 \cdot t\_2}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_4 \leq -1 \cdot 10^{-27}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\left|\frac{-d}{t\_0}\right| \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0 (sqrt (* l h)))
       (t_1 (fmin (fabs M) (fabs D)))
       (t_2 (fmax (fabs M) (fabs D)))
       (t_3
        (*
         (fma
          (* (* (* t_2 t_2) h) -0.5)
          (* (* 0.25 t_1) (/ t_1 (* (* d d) l)))
          1.0)
         (/ (fabs d) t_0)))
       (t_4
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (-
          1.0
          (*
           (* (/ 1.0 2.0) (pow (/ (* t_1 t_2) (* 2.0 d)) 2.0))
           (/ h l))))))
  (if (<= t_4 -1e-27)
    t_3
    (if (<= t_4 INFINITY) (* (fabs (/ (- d) t_0)) 1.0) t_3))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((l * h));
	double t_1 = fmin(fabs(M), fabs(D));
	double t_2 = fmax(fabs(M), fabs(D));
	double t_3 = fma((((t_2 * t_2) * h) * -0.5), ((0.25 * t_1) * (t_1 / ((d * d) * l))), 1.0) * (fabs(d) / t_0);
	double t_4 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((t_1 * t_2) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_4 <= -1e-27) {
		tmp = t_3;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = fabs((-d / t_0)) * 1.0;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(l * h))
	t_1 = fmin(abs(M), abs(D))
	t_2 = fmax(abs(M), abs(D))
	t_3 = Float64(fma(Float64(Float64(Float64(t_2 * t_2) * h) * -0.5), Float64(Float64(0.25 * t_1) * Float64(t_1 / Float64(Float64(d * d) * l))), 1.0) * Float64(abs(d) / t_0))
	t_4 = 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(t_1 * t_2) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_4 <= -1e-27)
		tmp = t_3;
	elseif (t_4 <= Inf)
		tmp = Float64(abs(Float64(Float64(-d) / t_0)) * 1.0);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(t$95$2 * t$95$2), $MachinePrecision] * h), $MachinePrecision] * -0.5), $MachinePrecision] * N[(N[(0.25 * t$95$1), $MachinePrecision] * N[(t$95$1 / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = 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[(t$95$1 * t$95$2), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -1e-27], t$95$3, If[LessEqual[t$95$4, Infinity], N[(N[Abs[N[((-d) / t$95$0), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], t$95$3]]]]]]]

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)))) < -1e-27 or +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 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 35.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right)}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}{\ell \cdot \left(d \cdot d\right)}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h \cdot \color{blue}{\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{\left(h \cdot \left(D \cdot D\right)\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)}}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      5. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\left(h \cdot \left(D \cdot D\right)\right) \cdot \frac{\left(M \cdot M\right) \cdot \frac{1}{4}}{\ell \cdot \left(d \cdot d\right)}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\left(h \cdot \left(D \cdot D\right)\right) \cdot \frac{\left(M \cdot M\right) \cdot \frac{1}{4}}{\ell \cdot \left(d \cdot d\right)}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\left(\left(D \cdot D\right) \cdot h\right)} \cdot \frac{\left(M \cdot M\right) \cdot \frac{1}{4}}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\left(\left(D \cdot D\right) \cdot h\right)} \cdot \frac{\left(M \cdot M\right) \cdot \frac{1}{4}}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      9. lower-/.f64 34.0%

         \[\leadsto \mathsf{fma}\left(-0.5, \left(\left(D \cdot D\right) \cdot h\right) \cdot \color{blue}{\frac{\left(M \cdot M\right) \cdot 0.25}{\ell \cdot \left(d \cdot d\right)}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{\color{blue}{\left(M \cdot M\right) \cdot \frac{1}{4}}}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{\color{blue}{\frac{1}{4} \cdot \left(M \cdot M\right)}}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      12. lower-*.f64 34.0%

          \[\leadsto \mathsf{fma}\left(-0.5, \left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{\color{blue}{0.25 \cdot \left(M \cdot M\right)}}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{\frac{1}{4} \cdot \left(M \cdot M\right)}{\color{blue}{\ell \cdot \left(d \cdot d\right)}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{\frac{1}{4} \cdot \left(M \cdot M\right)}{\color{blue}{\left(d \cdot d\right) \cdot \ell}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      15. lower-*.f64 34.0%

          \[\leadsto \mathsf{fma}\left(-0.5, \left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{0.25 \cdot \left(M \cdot M\right)}{\color{blue}{\left(d \cdot d\right) \cdot \ell}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

   5. Applied rewrites 34.0%

      \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{0.25 \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]
   6. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{\frac{1}{4} \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}\right) + 1\right)} \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{2} \cdot \color{blue}{\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{\frac{1}{4} \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}\right)} + 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{2} \cdot \left(\left(D \cdot D\right) \cdot h\right)\right) \cdot \frac{\frac{1}{4} \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}} + 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(D \cdot D\right) \cdot h\right), \frac{\frac{1}{4} \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right)} \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}}, \frac{\frac{1}{4} \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      6. lower-*.f64 34.0%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(D \cdot D\right) \cdot h\right) \cdot -0.5}, \frac{0.25 \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \color{blue}{\frac{\frac{1}{4} \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \frac{\color{blue}{\frac{1}{4} \cdot \left(M \cdot M\right)}}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \frac{\frac{1}{4} \cdot \color{blue}{\left(M \cdot M\right)}}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \frac{\color{blue}{\left(\frac{1}{4} \cdot M\right) \cdot M}}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \color{blue}{\left(\frac{1}{4} \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \color{blue}{\left(\frac{1}{4} \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \color{blue}{\left(\frac{1}{4} \cdot M\right)} \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      14. lower-/.f64 37.8%

          \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot -0.5, \left(0.25 \cdot M\right) \cdot \color{blue}{\frac{M}{\left(d \cdot d\right) \cdot \ell}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      15. lift-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \left(\frac{1}{4} \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}} \]

   7. Applied rewrites 27.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot -0.5, \left(0.25 \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}} \]
   8. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \left(\frac{1}{4} \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{\color{blue}{\sqrt{-d} \cdot \sqrt{-d}}}{\sqrt{\ell \cdot h}} \]

      2. sqrt-unprod N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \left(\frac{1}{4} \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{\color{blue}{\sqrt{\left(-d\right) \cdot \left(-d\right)}}}{\sqrt{\ell \cdot h}} \]

      3. rem-sqrt-square-rev N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \left(\frac{1}{4} \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{\color{blue}{\left|-d\right|}}{\sqrt{\ell \cdot h}} \]

      4. lift-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \left(\frac{1}{4} \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{\left|\color{blue}{\mathsf{neg}\left(d\right)}\right|}{\sqrt{\ell \cdot h}} \]

      5. neg-fabs N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \left(\frac{1}{4} \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{\color{blue}{\left|d\right|}}{\sqrt{\ell \cdot h}} \]

      6. lower-fabs.f64 53.7%

         \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot -0.5, \left(0.25 \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{\color{blue}{\left|d\right|}}{\sqrt{\ell \cdot h}} \]

   9. Applied rewrites 53.7%

      \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot -0.5, \left(0.25 \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{\color{blue}{\left|d\right|}}{\sqrt{\ell \cdot h}} \]

   if -1e-27 < (*.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 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      7. sqr-neg-rev N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right)\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

   3. Applied rewrites 67.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)}\right) \]

   5. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      4. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      10. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      12. associate-*r/ N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{d \cdot \color{blue}{\frac{d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      15. sqrt-fabs-rev N/A

          \[\leadsto \color{blue}{\left|\sqrt{d \cdot \frac{d}{h \cdot \ell}}\right|} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \left|\color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      17. lower-fabs.f64 53.0%

          \[\leadsto \color{blue}{\left|\sqrt{d \cdot \frac{d}{h \cdot \ell}}\right|} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      18. lift-sqrt.f64 N/A

          \[\leadsto \left|\color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \left|\sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      20. lift-/.f64 N/A

          \[\leadsto \left|\sqrt{d \cdot \color{blue}{\frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      21. associate-*r/ N/A

          \[\leadsto \left|\sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      22. lift-*.f64 N/A

          \[\leadsto \left|\sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      23. sqrt-div N/A

          \[\leadsto \left|\color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

   7. Applied rewrites 73.3%

      \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

   8. Taylor expanded in d around inf

      \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{1} \]
   9. Step-by-step derivation

   10. Applied rewrites 42.5%

       \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 51.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_1 := \frac{-d}{\sqrt{\ell \cdot h}}\\ t_2 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_0 \cdot t\_2}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -1 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, \left(t\_2 \cdot h\right) \cdot \left(-0.125 \cdot \frac{t\_0 \cdot t\_0}{\left(d \cdot d\right) \cdot \ell}\right), 1\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left|t\_1\right| \cdot 1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0 (fmin (fabs M) (fabs D)))
       (t_1 (/ (- d) (sqrt (* l h))))
       (t_2 (fmax (fabs M) (fabs D))))
  (if (<=
       (*
        (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
        (-
         1.0
         (*
          (* (/ 1.0 2.0) (pow (/ (* t_0 t_2) (* 2.0 d)) 2.0))
          (/ h l))))
       -1e-27)
    (*
     (fma
      t_2
      (* (* t_2 h) (* -0.125 (/ (* t_0 t_0) (* (* d d) l))))
      1.0)
     t_1)
    (* (fabs t_1) 1.0))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fmin(fabs(M), fabs(D));
	double t_1 = -d / sqrt((l * h));
	double t_2 = fmax(fabs(M), fabs(D));
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((t_0 * t_2) / (2.0 * d)), 2.0)) * (h / l)))) <= -1e-27) {
		tmp = fma(t_2, ((t_2 * h) * (-0.125 * ((t_0 * t_0) / ((d * d) * l)))), 1.0) * t_1;
	} else {
		tmp = fabs(t_1) * 1.0;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = fmin(abs(M), abs(D))
	t_1 = Float64(Float64(-d) / sqrt(Float64(l * h)))
	t_2 = fmax(abs(M), abs(D))
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(t_0 * t_2) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= -1e-27)
		tmp = Float64(fma(t_2, Float64(Float64(t_2 * h) * Float64(-0.125 * Float64(Float64(t_0 * t_0) / Float64(Float64(d * d) * l)))), 1.0) * t_1);
	else
		tmp = Float64(abs(t_1) * 1.0);
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Min[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(t$95$0 * t$95$2), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-27], N[(N[(t$95$2 * N[(N[(t$95$2 * h), $MachinePrecision] * N[(-0.125 * N[(N[(t$95$0 * t$95$0), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[Abs[t$95$1], $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1e-27

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 35.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right)}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}{\ell \cdot \left(d \cdot d\right)}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h \cdot \color{blue}{\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{\left(h \cdot \left(D \cdot D\right)\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)}}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      5. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\left(h \cdot \left(D \cdot D\right)\right) \cdot \frac{\left(M \cdot M\right) \cdot \frac{1}{4}}{\ell \cdot \left(d \cdot d\right)}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\left(h \cdot \left(D \cdot D\right)\right) \cdot \frac{\left(M \cdot M\right) \cdot \frac{1}{4}}{\ell \cdot \left(d \cdot d\right)}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\left(\left(D \cdot D\right) \cdot h\right)} \cdot \frac{\left(M \cdot M\right) \cdot \frac{1}{4}}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\left(\left(D \cdot D\right) \cdot h\right)} \cdot \frac{\left(M \cdot M\right) \cdot \frac{1}{4}}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      9. lower-/.f64 34.0%

         \[\leadsto \mathsf{fma}\left(-0.5, \left(\left(D \cdot D\right) \cdot h\right) \cdot \color{blue}{\frac{\left(M \cdot M\right) \cdot 0.25}{\ell \cdot \left(d \cdot d\right)}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{\color{blue}{\left(M \cdot M\right) \cdot \frac{1}{4}}}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{\color{blue}{\frac{1}{4} \cdot \left(M \cdot M\right)}}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      12. lower-*.f64 34.0%

          \[\leadsto \mathsf{fma}\left(-0.5, \left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{\color{blue}{0.25 \cdot \left(M \cdot M\right)}}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{\frac{1}{4} \cdot \left(M \cdot M\right)}{\color{blue}{\ell \cdot \left(d \cdot d\right)}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{\frac{1}{4} \cdot \left(M \cdot M\right)}{\color{blue}{\left(d \cdot d\right) \cdot \ell}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      15. lower-*.f64 34.0%

          \[\leadsto \mathsf{fma}\left(-0.5, \left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{0.25 \cdot \left(M \cdot M\right)}{\color{blue}{\left(d \cdot d\right) \cdot \ell}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

   5. Applied rewrites 34.0%

      \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{0.25 \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]
   6. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{\frac{1}{4} \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}\right) + 1\right)} \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{2} \cdot \color{blue}{\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{\frac{1}{4} \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}\right)} + 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{2} \cdot \left(\left(D \cdot D\right) \cdot h\right)\right) \cdot \frac{\frac{1}{4} \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}} + 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(D \cdot D\right) \cdot h\right), \frac{\frac{1}{4} \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right)} \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}}, \frac{\frac{1}{4} \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      6. lower-*.f64 34.0%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(D \cdot D\right) \cdot h\right) \cdot -0.5}, \frac{0.25 \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \color{blue}{\frac{\frac{1}{4} \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \frac{\color{blue}{\frac{1}{4} \cdot \left(M \cdot M\right)}}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \frac{\frac{1}{4} \cdot \color{blue}{\left(M \cdot M\right)}}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \frac{\color{blue}{\left(\frac{1}{4} \cdot M\right) \cdot M}}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \color{blue}{\left(\frac{1}{4} \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \color{blue}{\left(\frac{1}{4} \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \color{blue}{\left(\frac{1}{4} \cdot M\right)} \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      14. lower-/.f64 37.8%

          \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot -0.5, \left(0.25 \cdot M\right) \cdot \color{blue}{\frac{M}{\left(d \cdot d\right) \cdot \ell}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      15. lift-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \left(\frac{1}{4} \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}} \]

   7. Applied rewrites 27.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot -0.5, \left(0.25 \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}} \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}\right) \cdot \left(\left(\frac{1}{4} \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right) + 1\right)} \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}\right)} \cdot \left(\left(\frac{1}{4} \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right) + 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      3. associate-*l* N/A

         \[\leadsto \left(\color{blue}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(\frac{-1}{2} \cdot \left(\left(\frac{1}{4} \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right)\right)} + 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(D \cdot D\right) \cdot h\right)} \cdot \left(\frac{-1}{2} \cdot \left(\left(\frac{1}{4} \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right)\right) + 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(D \cdot D\right)} \cdot h\right) \cdot \left(\frac{-1}{2} \cdot \left(\left(\frac{1}{4} \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right)\right) + 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      6. associate-*l* N/A

         \[\leadsto \left(\color{blue}{\left(D \cdot \left(D \cdot h\right)\right)} \cdot \left(\frac{-1}{2} \cdot \left(\left(\frac{1}{4} \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right)\right) + 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      7. associate-*l* N/A

         \[\leadsto \left(\color{blue}{D \cdot \left(\left(D \cdot h\right) \cdot \left(\frac{-1}{2} \cdot \left(\left(\frac{1}{4} \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right)\right)\right)} + 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(D, \left(D \cdot h\right) \cdot \left(\frac{-1}{2} \cdot \left(\left(\frac{1}{4} \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right)\right), 1\right)} \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

   9. Applied rewrites 27.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(D, \left(D \cdot h\right) \cdot \left(-0.125 \cdot \frac{M \cdot M}{\left(d \cdot d\right) \cdot \ell}\right), 1\right)} \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

   if -1e-27 < (*.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 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      7. sqr-neg-rev N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right)\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

   3. Applied rewrites 67.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)}\right) \]

   5. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      4. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      10. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      12. associate-*r/ N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{d \cdot \color{blue}{\frac{d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      15. sqrt-fabs-rev N/A

          \[\leadsto \color{blue}{\left|\sqrt{d \cdot \frac{d}{h \cdot \ell}}\right|} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \left|\color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      17. lower-fabs.f64 53.0%

          \[\leadsto \color{blue}{\left|\sqrt{d \cdot \frac{d}{h \cdot \ell}}\right|} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      18. lift-sqrt.f64 N/A

          \[\leadsto \left|\color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \left|\sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      20. lift-/.f64 N/A

          \[\leadsto \left|\sqrt{d \cdot \color{blue}{\frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      21. associate-*r/ N/A

          \[\leadsto \left|\sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      22. lift-*.f64 N/A

          \[\leadsto \left|\sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      23. sqrt-div N/A

          \[\leadsto \left|\color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

   7. Applied rewrites 73.3%

      \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

   8. Taylor expanded in d around inf

      \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{1} \]
   9. Step-by-step derivation

   10. Applied rewrites 42.5%

       \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 51.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_1 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_0 \cdot t\_1}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -1 \cdot 10^{-27}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot \frac{t\_0 \cdot t\_0}{\left(d \cdot d\right) \cdot \ell}, \left(t\_1 \cdot t\_1\right) \cdot h, 1\right) \cdot \left(-d\right)}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot 1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0 (fmin (fabs M) (fabs D))) (t_1 (fmax (fabs M) (fabs D))))
  (if (<=
       (*
        (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
        (-
         1.0
         (*
          (* (/ 1.0 2.0) (pow (/ (* t_0 t_1) (* 2.0 d)) 2.0))
          (/ h l))))
       -1e-27)
    (/
     (*
      (fma
       (* -0.125 (/ (* t_0 t_0) (* (* d d) l)))
       (* (* t_1 t_1) h)
       1.0)
      (- d))
     (sqrt (* h l)))
    (* (fabs (/ (- d) (sqrt (* l h)))) 1.0))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fmin(fabs(M), fabs(D));
	double t_1 = fmax(fabs(M), fabs(D));
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((t_0 * t_1) / (2.0 * d)), 2.0)) * (h / l)))) <= -1e-27) {
		tmp = (fma((-0.125 * ((t_0 * t_0) / ((d * d) * l))), ((t_1 * t_1) * h), 1.0) * -d) / sqrt((h * l));
	} else {
		tmp = fabs((-d / sqrt((l * h)))) * 1.0;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = fmin(abs(M), abs(D))
	t_1 = fmax(abs(M), abs(D))
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(t_0 * t_1) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= -1e-27)
		tmp = Float64(Float64(fma(Float64(-0.125 * Float64(Float64(t_0 * t_0) / Float64(Float64(d * d) * l))), Float64(Float64(t_1 * t_1) * h), 1.0) * Float64(-d)) / sqrt(Float64(h * l)));
	else
		tmp = Float64(abs(Float64(Float64(-d) / sqrt(Float64(l * h)))) * 1.0);
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Min[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(t$95$0 * t$95$1), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-27], N[(N[(N[(N[(-0.125 * N[(N[(t$95$0 * t$95$0), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 * t$95$1), $MachinePrecision] * h), $MachinePrecision] + 1.0), $MachinePrecision] * (-d)), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Abs[N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1e-27

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 35.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right)}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}{\ell \cdot \left(d \cdot d\right)}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h \cdot \color{blue}{\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{\left(h \cdot \left(D \cdot D\right)\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)}}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      5. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\left(h \cdot \left(D \cdot D\right)\right) \cdot \frac{\left(M \cdot M\right) \cdot \frac{1}{4}}{\ell \cdot \left(d \cdot d\right)}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\left(h \cdot \left(D \cdot D\right)\right) \cdot \frac{\left(M \cdot M\right) \cdot \frac{1}{4}}{\ell \cdot \left(d \cdot d\right)}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\left(\left(D \cdot D\right) \cdot h\right)} \cdot \frac{\left(M \cdot M\right) \cdot \frac{1}{4}}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\left(\left(D \cdot D\right) \cdot h\right)} \cdot \frac{\left(M \cdot M\right) \cdot \frac{1}{4}}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      9. lower-/.f64 34.0%

         \[\leadsto \mathsf{fma}\left(-0.5, \left(\left(D \cdot D\right) \cdot h\right) \cdot \color{blue}{\frac{\left(M \cdot M\right) \cdot 0.25}{\ell \cdot \left(d \cdot d\right)}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{\color{blue}{\left(M \cdot M\right) \cdot \frac{1}{4}}}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{\color{blue}{\frac{1}{4} \cdot \left(M \cdot M\right)}}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      12. lower-*.f64 34.0%

          \[\leadsto \mathsf{fma}\left(-0.5, \left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{\color{blue}{0.25 \cdot \left(M \cdot M\right)}}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{\frac{1}{4} \cdot \left(M \cdot M\right)}{\color{blue}{\ell \cdot \left(d \cdot d\right)}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{\frac{1}{4} \cdot \left(M \cdot M\right)}{\color{blue}{\left(d \cdot d\right) \cdot \ell}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      15. lower-*.f64 34.0%

          \[\leadsto \mathsf{fma}\left(-0.5, \left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{0.25 \cdot \left(M \cdot M\right)}{\color{blue}{\left(d \cdot d\right) \cdot \ell}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

   5. Applied rewrites 34.0%

      \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{0.25 \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]
   6. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{\frac{1}{4} \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}\right) + 1\right)} \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{2} \cdot \color{blue}{\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{\frac{1}{4} \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}\right)} + 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{2} \cdot \left(\left(D \cdot D\right) \cdot h\right)\right) \cdot \frac{\frac{1}{4} \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}} + 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(D \cdot D\right) \cdot h\right), \frac{\frac{1}{4} \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right)} \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}}, \frac{\frac{1}{4} \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      6. lower-*.f64 34.0%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(D \cdot D\right) \cdot h\right) \cdot -0.5}, \frac{0.25 \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \color{blue}{\frac{\frac{1}{4} \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \frac{\color{blue}{\frac{1}{4} \cdot \left(M \cdot M\right)}}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \frac{\frac{1}{4} \cdot \color{blue}{\left(M \cdot M\right)}}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \frac{\color{blue}{\left(\frac{1}{4} \cdot M\right) \cdot M}}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \color{blue}{\left(\frac{1}{4} \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \color{blue}{\left(\frac{1}{4} \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \color{blue}{\left(\frac{1}{4} \cdot M\right)} \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      14. lower-/.f64 37.8%

          \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot -0.5, \left(0.25 \cdot M\right) \cdot \color{blue}{\frac{M}{\left(d \cdot d\right) \cdot \ell}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      15. lift-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \left(\frac{1}{4} \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}} \]

   7. Applied rewrites 27.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot -0.5, \left(0.25 \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \left(\frac{1}{4} \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \left(\frac{1}{4} \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \color{blue}{\frac{-d}{\sqrt{\ell \cdot h}}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \left(\frac{1}{4} \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \left(-d\right)}{\sqrt{\ell \cdot h}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{-1}{2}, \left(\frac{1}{4} \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \left(-d\right)}{\sqrt{\ell \cdot h}}} \]

   9. Applied rewrites 24.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{M \cdot M}{\left(d \cdot d\right) \cdot \ell}, \left(D \cdot D\right) \cdot h, 1\right) \cdot \left(-d\right)}{\sqrt{h \cdot \ell}}} \]

   if -1e-27 < (*.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 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      7. sqr-neg-rev N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right)\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

   3. Applied rewrites 67.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)}\right) \]

   5. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      4. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      10. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      12. associate-*r/ N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{d \cdot \color{blue}{\frac{d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      15. sqrt-fabs-rev N/A

          \[\leadsto \color{blue}{\left|\sqrt{d \cdot \frac{d}{h \cdot \ell}}\right|} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \left|\color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      17. lower-fabs.f64 53.0%

          \[\leadsto \color{blue}{\left|\sqrt{d \cdot \frac{d}{h \cdot \ell}}\right|} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      18. lift-sqrt.f64 N/A

          \[\leadsto \left|\color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \left|\sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      20. lift-/.f64 N/A

          \[\leadsto \left|\sqrt{d \cdot \color{blue}{\frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      21. associate-*r/ N/A

          \[\leadsto \left|\sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      22. lift-*.f64 N/A

          \[\leadsto \left|\sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      23. sqrt-div N/A

          \[\leadsto \left|\color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

   7. Applied rewrites 73.3%

      \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

   8. Taylor expanded in d around inf

      \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{1} \]
   9. Step-by-step derivation

   10. Applied rewrites 42.5%

       \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 45.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -1 \cdot 10^{-47}:\\ \;\;\;\;\left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot 1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :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 D) (* 2.0 d)) 2.0)) (/ h l))))
     -1e-47)
  (* (* (- d) (sqrt (/ 1.0 (* l h)))) 1.0)
  (* (fabs (/ (- d) (sqrt (* l h)))) 1.0)))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -1e-47) {
		tmp = (-d * sqrt((1.0 / (l * h)))) * 1.0;
	} else {
		tmp = fabs((-d / sqrt((l * h)))) * 1.0;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (((((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)))) <= (-1d-47)) then
        tmp = (-d * sqrt((1.0d0 / (l * h)))) * 1.0d0
    else
        tmp = abs((-d / sqrt((l * h)))) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	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 * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -1e-47) {
		tmp = (-d * Math.sqrt((1.0 / (l * h)))) * 1.0;
	} else {
		tmp = Math.abs((-d / Math.sqrt((l * h)))) * 1.0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	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 * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -1e-47:
		tmp = (-d * math.sqrt((1.0 / (l * h)))) * 1.0
	else:
		tmp = math.fabs((-d / math.sqrt((l * h)))) * 1.0
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= -1e-47)
		tmp = Float64(Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(l * h)))) * 1.0);
	else
		tmp = Float64(abs(Float64(Float64(-d) / sqrt(Float64(l * h)))) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (((((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)))) <= -1e-47)
		tmp = (-d * sqrt((1.0 / (l * h)))) * 1.0;
	else
		tmp = abs((-d / sqrt((l * h)))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := 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 * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-47], N[(N[((-d) * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Abs[N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -9.9999999999999997e-48

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      7. sqr-neg-rev N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right)\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

   3. Applied rewrites 67.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)}\right) \]

   5. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      4. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      10. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      13. mult-flip N/A

          \[\leadsto \sqrt{\color{blue}{\left(d \cdot d\right) \cdot \frac{1}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      14. sqrt-prod N/A

          \[\leadsto \color{blue}{\left(\sqrt{d \cdot d} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      15. lower-unsound-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{d \cdot d}} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{d \cdot d}} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\color{blue}{d \cdot d}} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      18. sqr-neg-rev N/A

          \[\leadsto \left(\sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \left(\mathsf{neg}\left(d\right)\right)}} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      19. lift-neg.f64 N/A

          \[\leadsto \left(\sqrt{\color{blue}{\left(-d\right)} \cdot \left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      20. lift-neg.f64 N/A

          \[\leadsto \left(\sqrt{\left(-d\right) \cdot \color{blue}{\left(-d\right)}} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      21. sqrt-unprod N/A

          \[\leadsto \left(\color{blue}{\left(\sqrt{-d} \cdot \sqrt{-d}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      22. rem-square-sqrt N/A

          \[\leadsto \left(\color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      23. lower-unsound-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      24. lower-unsound-sqrt.f64 N/A

          \[\leadsto \left(\left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

   7. Applied rewrites 37.6%

      \[\leadsto \color{blue}{\left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

   8. Taylor expanded in d around inf

      \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \color{blue}{1} \]
   9. Step-by-step derivation

   10. Applied rewrites 26.2%

       \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \color{blue}{1} \]

   if -9.9999999999999997e-48 < (*.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 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      7. sqr-neg-rev N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right)\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

   3. Applied rewrites 67.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)}\right) \]

   5. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      4. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      10. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      12. associate-*r/ N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{d \cdot \color{blue}{\frac{d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      15. sqrt-fabs-rev N/A

          \[\leadsto \color{blue}{\left|\sqrt{d \cdot \frac{d}{h \cdot \ell}}\right|} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \left|\color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      17. lower-fabs.f64 53.0%

          \[\leadsto \color{blue}{\left|\sqrt{d \cdot \frac{d}{h \cdot \ell}}\right|} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      18. lift-sqrt.f64 N/A

          \[\leadsto \left|\color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \left|\sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      20. lift-/.f64 N/A

          \[\leadsto \left|\sqrt{d \cdot \color{blue}{\frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      21. associate-*r/ N/A

          \[\leadsto \left|\sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      22. lift-*.f64 N/A

          \[\leadsto \left|\sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

      23. sqrt-div N/A

          \[\leadsto \left|\color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

   7. Applied rewrites 73.3%

      \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

   8. Taylor expanded in d around inf

      \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{1} \]
   9. Step-by-step derivation

   10. Applied rewrites 42.5%

       \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 42.5% accurate, 6.7× speedup

Math

\[\left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot 1 \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (* (fabs (/ (- d) (sqrt (* l h)))) 1.0))

C

double code(double d, double h, double l, double M, double D) {
	return fabs((-d / sqrt((l * h)))) * 1.0;
}

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 = abs((-d / sqrt((l * h)))) * 1.0d0
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return Math.abs((-d / Math.sqrt((l * h)))) * 1.0;
}

Python

def code(d, h, l, M, D):
	return math.fabs((-d / math.sqrt((l * h)))) * 1.0

Julia

function code(d, h, l, M, D)
	return Float64(abs(Float64(Float64(-d) / sqrt(Float64(l * h)))) * 1.0)
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = abs((-d / sqrt((l * h)))) * 1.0;
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(N[Abs[N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]

Derivation

1. Initial program 65.7%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

   3. *-commutative N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

   4. associate-*l* N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

   5. lift-pow.f64 N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

   7. sqr-neg-rev N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right)\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

3. Applied rewrites 67.4%

   \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)}\right) \]

5. Applied rewrites 66.9%

   \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}}\right) \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

   2. lift-pow.f64 N/A

      \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

   3. lift-pow.f64 N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

   4. pow-prod-down N/A

      \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

   5. lift-/.f64 N/A

      \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

   6. metadata-eval N/A

      \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

   7. unpow1/2 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

   8. lift-/.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

   9. lift-/.f64 N/A

      \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

   10. frac-times N/A

       \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

   11. lift-*.f64 N/A

       \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

   12. associate-*r/ N/A

       \[\leadsto \sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

   13. lift-/.f64 N/A

       \[\leadsto \sqrt{d \cdot \color{blue}{\frac{d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

   14. lift-*.f64 N/A

       \[\leadsto \sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

   15. sqrt-fabs-rev N/A

       \[\leadsto \color{blue}{\left|\sqrt{d \cdot \frac{d}{h \cdot \ell}}\right|} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

   16. lift-sqrt.f64 N/A

       \[\leadsto \left|\color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

   17. lower-fabs.f64 53.0%

       \[\leadsto \color{blue}{\left|\sqrt{d \cdot \frac{d}{h \cdot \ell}}\right|} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

   18. lift-sqrt.f64 N/A

       \[\leadsto \left|\color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

   19. lift-*.f64 N/A

       \[\leadsto \left|\sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

   20. lift-/.f64 N/A

       \[\leadsto \left|\sqrt{d \cdot \color{blue}{\frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

   21. associate-*r/ N/A

       \[\leadsto \left|\sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

   22. lift-*.f64 N/A

       \[\leadsto \left|\sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

   23. sqrt-div N/A

       \[\leadsto \left|\color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

7. Applied rewrites 73.3%

   \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{\left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}\right) \]

8. Taylor expanded in d around inf

   \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{1} \]
9. Step-by-step derivation

10. Applied rewrites 42.5%

    \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{1} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025210 
(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)))))