Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.3% → 81.7%

Time: 10.6s

Alternatives: 23

Speedup: 0.6×

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: 66.3% 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: 81.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ t_1 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_2 := \frac{t\_1}{d + d}\\ t_3 := {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\\ t_4 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_5 := t\_4 \cdot t\_1\\ t_6 := t\_3 \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_5}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_7 := t\_2 \cdot t\_4\\ \mathbf{if}\;t\_6 \leq -5 \cdot 10^{+42}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(t\_7 \cdot \left(t\_7 \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;t\_6 \leq 0:\\ \;\;\;\;t\_0 \cdot \left(1 - \left(0.5 \cdot {\left(t\_1 \cdot \frac{1}{\frac{d}{t\_4 \cdot 0.5}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;t\_6 \leq 5 \cdot 10^{+107}:\\ \;\;\;\;t\_3 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(1 - \frac{\left(\left(\left(0.25 \cdot \frac{t\_5}{d}\right) \cdot t\_2\right) \cdot t\_4\right) \cdot h}{\ell}\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ (fabs d) (sqrt (* l h))))
        (t_1 (fmax (fabs M) (fabs D)))
        (t_2 (/ t_1 (+ d d)))
        (t_3 (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))))
        (t_4 (fmin (fabs M) (fabs D)))
        (t_5 (* t_4 t_1))
        (t_6
         (*
          t_3
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ t_5 (* 2.0 d)) 2.0)) (/ h l)))))
        (t_7 (* t_2 t_4)))
   (if (<= t_6 -5e+42)
     (*
      (* (sqrt (/ d l)) (sqrt (/ d h)))
      (- 1.0 (* (* t_7 (* t_7 0.5)) (/ h l))))
     (if (<= t_6 0.0)
       (*
        t_0
        (-
         1.0
         (* (* 0.5 (pow (* t_1 (/ 1.0 (/ d (* t_4 0.5)))) 2.0)) (/ h l))))
       (if (<= t_6 5e+107)
         (* t_3 1.0)
         (* t_0 (- 1.0 (/ (* (* (* (* 0.25 (/ t_5 d)) t_2) t_4) h) l))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fabs(d) / sqrt((l * h));
	double t_1 = fmax(fabs(M), fabs(D));
	double t_2 = t_1 / (d + d);
	double t_3 = pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0));
	double t_4 = fmin(fabs(M), fabs(D));
	double t_5 = t_4 * t_1;
	double t_6 = t_3 * (1.0 - (((1.0 / 2.0) * pow((t_5 / (2.0 * d)), 2.0)) * (h / l)));
	double t_7 = t_2 * t_4;
	double tmp;
	if (t_6 <= -5e+42) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((t_7 * (t_7 * 0.5)) * (h / l)));
	} else if (t_6 <= 0.0) {
		tmp = t_0 * (1.0 - ((0.5 * pow((t_1 * (1.0 / (d / (t_4 * 0.5)))), 2.0)) * (h / l)));
	} else if (t_6 <= 5e+107) {
		tmp = t_3 * 1.0;
	} else {
		tmp = t_0 * (1.0 - (((((0.25 * (t_5 / d)) * t_2) * t_4) * 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) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: tmp
    t_0 = abs(d) / sqrt((l * h))
    t_1 = fmax(abs(m), abs(d_1))
    t_2 = t_1 / (d + d)
    t_3 = ((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))
    t_4 = fmin(abs(m), abs(d_1))
    t_5 = t_4 * t_1
    t_6 = t_3 * (1.0d0 - (((1.0d0 / 2.0d0) * ((t_5 / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    t_7 = t_2 * t_4
    if (t_6 <= (-5d+42)) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - ((t_7 * (t_7 * 0.5d0)) * (h / l)))
    else if (t_6 <= 0.0d0) then
        tmp = t_0 * (1.0d0 - ((0.5d0 * ((t_1 * (1.0d0 / (d / (t_4 * 0.5d0)))) ** 2.0d0)) * (h / l)))
    else if (t_6 <= 5d+107) then
        tmp = t_3 * 1.0d0
    else
        tmp = t_0 * (1.0d0 - (((((0.25d0 * (t_5 / d)) * t_2) * t_4) * 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 = Math.abs(d) / Math.sqrt((l * h));
	double t_1 = fmax(Math.abs(M), Math.abs(D));
	double t_2 = t_1 / (d + d);
	double t_3 = Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0));
	double t_4 = fmin(Math.abs(M), Math.abs(D));
	double t_5 = t_4 * t_1;
	double t_6 = t_3 * (1.0 - (((1.0 / 2.0) * Math.pow((t_5 / (2.0 * d)), 2.0)) * (h / l)));
	double t_7 = t_2 * t_4;
	double tmp;
	if (t_6 <= -5e+42) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - ((t_7 * (t_7 * 0.5)) * (h / l)));
	} else if (t_6 <= 0.0) {
		tmp = t_0 * (1.0 - ((0.5 * Math.pow((t_1 * (1.0 / (d / (t_4 * 0.5)))), 2.0)) * (h / l)));
	} else if (t_6 <= 5e+107) {
		tmp = t_3 * 1.0;
	} else {
		tmp = t_0 * (1.0 - (((((0.25 * (t_5 / d)) * t_2) * t_4) * h) / l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.fabs(d) / math.sqrt((l * h))
	t_1 = fmax(math.fabs(M), math.fabs(D))
	t_2 = t_1 / (d + d)
	t_3 = math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))
	t_4 = fmin(math.fabs(M), math.fabs(D))
	t_5 = t_4 * t_1
	t_6 = t_3 * (1.0 - (((1.0 / 2.0) * math.pow((t_5 / (2.0 * d)), 2.0)) * (h / l)))
	t_7 = t_2 * t_4
	tmp = 0
	if t_6 <= -5e+42:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - ((t_7 * (t_7 * 0.5)) * (h / l)))
	elif t_6 <= 0.0:
		tmp = t_0 * (1.0 - ((0.5 * math.pow((t_1 * (1.0 / (d / (t_4 * 0.5)))), 2.0)) * (h / l)))
	elif t_6 <= 5e+107:
		tmp = t_3 * 1.0
	else:
		tmp = t_0 * (1.0 - (((((0.25 * (t_5 / d)) * t_2) * t_4) * h) / l))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(abs(d) / sqrt(Float64(l * h)))
	t_1 = fmax(abs(M), abs(D))
	t_2 = Float64(t_1 / Float64(d + d))
	t_3 = Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0)))
	t_4 = fmin(abs(M), abs(D))
	t_5 = Float64(t_4 * t_1)
	t_6 = Float64(t_3 * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(t_5 / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_7 = Float64(t_2 * t_4)
	tmp = 0.0
	if (t_6 <= -5e+42)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(t_7 * Float64(t_7 * 0.5)) * Float64(h / l))));
	elseif (t_6 <= 0.0)
		tmp = Float64(t_0 * Float64(1.0 - Float64(Float64(0.5 * (Float64(t_1 * Float64(1.0 / Float64(d / Float64(t_4 * 0.5)))) ^ 2.0)) * Float64(h / l))));
	elseif (t_6 <= 5e+107)
		tmp = Float64(t_3 * 1.0);
	else
		tmp = Float64(t_0 * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(0.25 * Float64(t_5 / d)) * t_2) * t_4) * h) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = abs(d) / sqrt((l * h));
	t_1 = max(abs(M), abs(D));
	t_2 = t_1 / (d + d);
	t_3 = ((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0));
	t_4 = min(abs(M), abs(D));
	t_5 = t_4 * t_1;
	t_6 = t_3 * (1.0 - (((1.0 / 2.0) * ((t_5 / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_7 = t_2 * t_4;
	tmp = 0.0;
	if (t_6 <= -5e+42)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((t_7 * (t_7 * 0.5)) * (h / l)));
	elseif (t_6 <= 0.0)
		tmp = t_0 * (1.0 - ((0.5 * ((t_1 * (1.0 / (d / (t_4 * 0.5)))) ^ 2.0)) * (h / l)));
	elseif (t_6 <= 5e+107)
		tmp = t_3 * 1.0;
	else
		tmp = t_0 * (1.0 - (((((0.25 * (t_5 / d)) * t_2) * t_4) * h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(d + d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = 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]}, Block[{t$95$4 = N[Min[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 * t$95$1), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$3 * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(t$95$5 / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(t$95$2 * t$95$4), $MachinePrecision]}, If[LessEqual[t$95$6, -5e+42], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(t$95$7 * N[(t$95$7 * 0.5), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 0.0], N[(t$95$0 * N[(1.0 - N[(N[(0.5 * N[Power[N[(t$95$1 * N[(1.0 / N[(d / N[(t$95$4 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 5e+107], N[(t$95$3 * 1.0), $MachinePrecision], N[(t$95$0 * N[(1.0 - N[(N[(N[(N[(N[(0.25 * N[(t$95$5 / d), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$4), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5.00000000000000007e42

   1. Initial program 66.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lift-/.f64 N/A

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

      12. lower-/.f64 65.8

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

      13. lift-/.f64 N/A

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

      14. metadata-eval 65.8

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

   3. Applied rewrites 65.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-prod-down N/A

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

      9. unpow1/2 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. frac-times N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. sqrt-div N/A

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

      16. lower-unsound-sqrt.f32 N/A

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

      17. lower-sqrt.f32 N/A

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

      18. lift-*.f64 N/A

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

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

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

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

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

      21. lower-fabs.f64 N/A

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

      22. lower-unsound-sqrt.f64 70.2

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-*.f64 70.2

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

   5. Applied rewrites 70.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. associate-/r* N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. mult-flip N/A

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

      19. lift-/.f64 N/A

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

   7. Applied rewrites 70.3%

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

      1. lift-/.f64 N/A

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

      2. lift-fabs.f64 N/A

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

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

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. sqrt-undiv N/A

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

      9. lift-*.f64 N/A

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

      10. frac-times N/A

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

      11. sqrt-unprod N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-/.f64 66.1

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

   9. Applied rewrites 66.1%

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

   if -5.00000000000000007e42 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0

   1. Initial program 66.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lift-/.f64 N/A

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

      12. lower-/.f64 65.8

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

      13. lift-/.f64 N/A

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

      14. metadata-eval 65.8

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

   3. Applied rewrites 65.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-prod-down N/A

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

      9. unpow1/2 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. frac-times N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. sqrt-div N/A

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

      16. lower-unsound-sqrt.f32 N/A

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

      17. lower-sqrt.f32 N/A

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

      18. lift-*.f64 N/A

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

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

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

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

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

      21. lower-fabs.f64 N/A

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

      22. lower-unsound-sqrt.f64 70.2

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-*.f64 70.2

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

   5. Applied rewrites 70.2%

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

   6. Evaluated real constant 70.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. div-flip N/A

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

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

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. mult-flip N/A

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

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

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

      10. mult-flip N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 70.1

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

   8. Applied rewrites 70.1%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000002e107

   1. Initial program 66.3%

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

   2. Taylor expanded in d around inf

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

   4. Applied rewrites 39.0%

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

   if 5.0000000000000002e107 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lift-/.f64 N/A

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

      12. lower-/.f64 65.8

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

      13. lift-/.f64 N/A

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

      14. metadata-eval 65.8

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

   3. Applied rewrites 65.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-prod-down N/A

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

      9. unpow1/2 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. frac-times N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. sqrt-div N/A

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

      16. lower-unsound-sqrt.f32 N/A

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

      17. lower-sqrt.f32 N/A

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

      18. lift-*.f64 N/A

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

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

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

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

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

      21. lower-fabs.f64 N/A

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

      22. lower-unsound-sqrt.f64 70.2

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-*.f64 70.2

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

   5. Applied rewrites 70.2%

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

   6. Evaluated real constant 70.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

   8. Applied rewrites 73.3%

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

Alternative 2: 81.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\\ t_1 := t\_0 \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ t_3 := \frac{D}{d + d}\\ t_4 := t\_3 \cdot M\\ t_5 := 0.25 \cdot \frac{M \cdot D}{d}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+42}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(t\_4 \cdot \left(t\_4 \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;t\_2 \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot t\_5}{d + d} \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+107}:\\ \;\;\;\;t\_0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left(1 - \frac{\left(\left(t\_5 \cdot t\_3\right) \cdot M\right) \cdot h}{\ell}\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))))
        (t_1
         (*
          t_0
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_2 (/ (fabs d) (sqrt (* l h))))
        (t_3 (/ D (+ d d)))
        (t_4 (* t_3 M))
        (t_5 (* 0.25 (/ (* M D) d))))
   (if (<= t_1 -5e+42)
     (*
      (* (sqrt (/ d l)) (sqrt (/ d h)))
      (- 1.0 (* (* t_4 (* t_4 0.5)) (/ h l))))
     (if (<= t_1 0.0)
       (* t_2 (- 1.0 (* (/ (* (* M D) t_5) (+ d d)) (/ h l))))
       (if (<= t_1 5e+107)
         (* t_0 1.0)
         (* t_2 (- 1.0 (/ (* (* (* t_5 t_3) M) h) l))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0));
	double t_1 = t_0 * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = fabs(d) / sqrt((l * h));
	double t_3 = D / (d + d);
	double t_4 = t_3 * M;
	double t_5 = 0.25 * ((M * D) / d);
	double tmp;
	if (t_1 <= -5e+42) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((t_4 * (t_4 * 0.5)) * (h / l)));
	} else if (t_1 <= 0.0) {
		tmp = t_2 * (1.0 - ((((M * D) * t_5) / (d + d)) * (h / l)));
	} else if (t_1 <= 5e+107) {
		tmp = t_0 * 1.0;
	} else {
		tmp = t_2 * (1.0 - ((((t_5 * t_3) * 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) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_0 = ((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))
    t_1 = t_0 * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    t_2 = abs(d) / sqrt((l * h))
    t_3 = d_1 / (d + d)
    t_4 = t_3 * m
    t_5 = 0.25d0 * ((m * d_1) / d)
    if (t_1 <= (-5d+42)) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - ((t_4 * (t_4 * 0.5d0)) * (h / l)))
    else if (t_1 <= 0.0d0) then
        tmp = t_2 * (1.0d0 - ((((m * d_1) * t_5) / (d + d)) * (h / l)))
    else if (t_1 <= 5d+107) then
        tmp = t_0 * 1.0d0
    else
        tmp = t_2 * (1.0d0 - ((((t_5 * t_3) * 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 = Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0));
	double t_1 = t_0 * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = Math.abs(d) / Math.sqrt((l * h));
	double t_3 = D / (d + d);
	double t_4 = t_3 * M;
	double t_5 = 0.25 * ((M * D) / d);
	double tmp;
	if (t_1 <= -5e+42) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - ((t_4 * (t_4 * 0.5)) * (h / l)));
	} else if (t_1 <= 0.0) {
		tmp = t_2 * (1.0 - ((((M * D) * t_5) / (d + d)) * (h / l)));
	} else if (t_1 <= 5e+107) {
		tmp = t_0 * 1.0;
	} else {
		tmp = t_2 * (1.0 - ((((t_5 * t_3) * M) * h) / l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))
	t_1 = t_0 * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_2 = math.fabs(d) / math.sqrt((l * h))
	t_3 = D / (d + d)
	t_4 = t_3 * M
	t_5 = 0.25 * ((M * D) / d)
	tmp = 0
	if t_1 <= -5e+42:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - ((t_4 * (t_4 * 0.5)) * (h / l)))
	elif t_1 <= 0.0:
		tmp = t_2 * (1.0 - ((((M * D) * t_5) / (d + d)) * (h / l)))
	elif t_1 <= 5e+107:
		tmp = t_0 * 1.0
	else:
		tmp = t_2 * (1.0 - ((((t_5 * t_3) * M) * h) / l))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0)))
	t_1 = Float64(t_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(abs(d) / sqrt(Float64(l * h)))
	t_3 = Float64(D / Float64(d + d))
	t_4 = Float64(t_3 * M)
	t_5 = Float64(0.25 * Float64(Float64(M * D) / d))
	tmp = 0.0
	if (t_1 <= -5e+42)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(t_4 * Float64(t_4 * 0.5)) * Float64(h / l))));
	elseif (t_1 <= 0.0)
		tmp = Float64(t_2 * Float64(1.0 - Float64(Float64(Float64(Float64(M * D) * t_5) / Float64(d + d)) * Float64(h / l))));
	elseif (t_1 <= 5e+107)
		tmp = Float64(t_0 * 1.0);
	else
		tmp = Float64(t_2 * Float64(1.0 - Float64(Float64(Float64(Float64(t_5 * t_3) * M) * h) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = ((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0));
	t_1 = t_0 * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_2 = abs(d) / sqrt((l * h));
	t_3 = D / (d + d);
	t_4 = t_3 * M;
	t_5 = 0.25 * ((M * D) / d);
	tmp = 0.0;
	if (t_1 <= -5e+42)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((t_4 * (t_4 * 0.5)) * (h / l)));
	elseif (t_1 <= 0.0)
		tmp = t_2 * (1.0 - ((((M * D) * t_5) / (d + d)) * (h / l)));
	elseif (t_1 <= 5e+107)
		tmp = t_0 * 1.0;
	else
		tmp = t_2 * (1.0 - ((((t_5 * t_3) * M) * h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = 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]}, Block[{t$95$1 = N[(t$95$0 * 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[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(D / N[(d + d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * M), $MachinePrecision]}, Block[{t$95$5 = N[(0.25 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+42], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(t$95$4 * N[(t$95$4 * 0.5), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(t$95$2 * N[(1.0 - N[(N[(N[(N[(M * D), $MachinePrecision] * t$95$5), $MachinePrecision] / N[(d + d), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+107], N[(t$95$0 * 1.0), $MachinePrecision], N[(t$95$2 * N[(1.0 - N[(N[(N[(N[(t$95$5 * t$95$3), $MachinePrecision] * M), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5.00000000000000007e42

   1. Initial program 66.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lift-/.f64 N/A

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

      12. lower-/.f64 65.8

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

      13. lift-/.f64 N/A

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

      14. metadata-eval 65.8

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

   3. Applied rewrites 65.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-prod-down N/A

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

      9. unpow1/2 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. frac-times N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. sqrt-div N/A

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

      16. lower-unsound-sqrt.f32 N/A

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

      17. lower-sqrt.f32 N/A

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

      18. lift-*.f64 N/A

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

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

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

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

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

      21. lower-fabs.f64 N/A

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

      22. lower-unsound-sqrt.f64 70.2

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-*.f64 70.2

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

   5. Applied rewrites 70.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. associate-/r* N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. mult-flip N/A

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

      19. lift-/.f64 N/A

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

   7. Applied rewrites 70.3%

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

      1. lift-/.f64 N/A

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

      2. lift-fabs.f64 N/A

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

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

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. sqrt-undiv N/A

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

      9. lift-*.f64 N/A

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

      10. frac-times N/A

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

      11. sqrt-unprod N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-/.f64 66.1

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

   9. Applied rewrites 66.1%

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

   if -5.00000000000000007e42 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0

   1. Initial program 66.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lift-/.f64 N/A

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

      12. lower-/.f64 65.8

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

      13. lift-/.f64 N/A

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

      14. metadata-eval 65.8

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

   3. Applied rewrites 65.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-prod-down N/A

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

      9. unpow1/2 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. frac-times N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. sqrt-div N/A

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

      16. lower-unsound-sqrt.f32 N/A

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

      17. lower-sqrt.f32 N/A

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

      18. lift-*.f64 N/A

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

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

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

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

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

      21. lower-fabs.f64 N/A

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

      22. lower-unsound-sqrt.f64 70.2

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-*.f64 70.2

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

   5. Applied rewrites 70.2%

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

   6. Evaluated real constant 70.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. associate-/r* N/A

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

      10. count-2-rev N/A

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

      11. lift-+.f64 N/A

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

      12. mult-flip N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. pow2 N/A

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

      16. associate-*r* N/A

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

      17. lift-*.f64 N/A

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

   8. Applied rewrites 69.9%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000002e107

   1. Initial program 66.3%

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

   2. Taylor expanded in d around inf

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

   4. Applied rewrites 39.0%

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

   if 5.0000000000000002e107 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lift-/.f64 N/A

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

      12. lower-/.f64 65.8

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

      13. lift-/.f64 N/A

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

      14. metadata-eval 65.8

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

   3. Applied rewrites 65.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-prod-down N/A

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

      9. unpow1/2 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. frac-times N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. sqrt-div N/A

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

      16. lower-unsound-sqrt.f32 N/A

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

      17. lower-sqrt.f32 N/A

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

      18. lift-*.f64 N/A

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

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

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

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

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

      21. lower-fabs.f64 N/A

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

      22. lower-unsound-sqrt.f64 70.2

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-*.f64 70.2

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

   5. Applied rewrites 70.2%

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

   6. Evaluated real constant 70.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

   8. Applied rewrites 73.3%

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

Alternative 3: 81.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_3 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ t_4 := \frac{D}{d + d}\\ t_5 := t\_4 \cdot M\\ t_6 := 0.25 \cdot \frac{M \cdot D}{d}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+42}:\\ \;\;\;\;\left(t\_0 \cdot t\_1\right) \cdot \left(1 - \left(t\_5 \cdot \left(t\_5 \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t\_3 \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot t\_6}{d + d} \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+107}:\\ \;\;\;\;t\_1 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_3 \cdot \left(1 - \frac{\left(\left(t\_6 \cdot t\_4\right) \cdot M\right) \cdot h}{\ell}\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l)))
        (t_1 (sqrt (/ d h)))
        (t_2
         (*
          (* (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_3 (/ (fabs d) (sqrt (* l h))))
        (t_4 (/ D (+ d d)))
        (t_5 (* t_4 M))
        (t_6 (* 0.25 (/ (* M D) d))))
   (if (<= t_2 -5e+42)
     (* (* t_0 t_1) (- 1.0 (* (* t_5 (* t_5 0.5)) (/ h l))))
     (if (<= t_2 0.0)
       (* t_3 (- 1.0 (* (/ (* (* M D) t_6) (+ d d)) (/ h l))))
       (if (<= t_2 5e+107)
         (* t_1 t_0)
         (* t_3 (- 1.0 (/ (* (* (* t_6 t_4) M) h) l))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / l));
	double t_1 = sqrt((d / h));
	double t_2 = (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_3 = fabs(d) / sqrt((l * h));
	double t_4 = D / (d + d);
	double t_5 = t_4 * M;
	double t_6 = 0.25 * ((M * D) / d);
	double tmp;
	if (t_2 <= -5e+42) {
		tmp = (t_0 * t_1) * (1.0 - ((t_5 * (t_5 * 0.5)) * (h / l)));
	} else if (t_2 <= 0.0) {
		tmp = t_3 * (1.0 - ((((M * D) * t_6) / (d + d)) * (h / l)));
	} else if (t_2 <= 5e+107) {
		tmp = t_1 * t_0;
	} else {
		tmp = t_3 * (1.0 - ((((t_6 * t_4) * 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) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: tmp
    t_0 = sqrt((d / l))
    t_1 = sqrt((d / h))
    t_2 = (((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_3 = abs(d) / sqrt((l * h))
    t_4 = d_1 / (d + d)
    t_5 = t_4 * m
    t_6 = 0.25d0 * ((m * d_1) / d)
    if (t_2 <= (-5d+42)) then
        tmp = (t_0 * t_1) * (1.0d0 - ((t_5 * (t_5 * 0.5d0)) * (h / l)))
    else if (t_2 <= 0.0d0) then
        tmp = t_3 * (1.0d0 - ((((m * d_1) * t_6) / (d + d)) * (h / l)))
    else if (t_2 <= 5d+107) then
        tmp = t_1 * t_0
    else
        tmp = t_3 * (1.0d0 - ((((t_6 * t_4) * 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 = Math.sqrt((d / l));
	double t_1 = Math.sqrt((d / h));
	double t_2 = (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_3 = Math.abs(d) / Math.sqrt((l * h));
	double t_4 = D / (d + d);
	double t_5 = t_4 * M;
	double t_6 = 0.25 * ((M * D) / d);
	double tmp;
	if (t_2 <= -5e+42) {
		tmp = (t_0 * t_1) * (1.0 - ((t_5 * (t_5 * 0.5)) * (h / l)));
	} else if (t_2 <= 0.0) {
		tmp = t_3 * (1.0 - ((((M * D) * t_6) / (d + d)) * (h / l)));
	} else if (t_2 <= 5e+107) {
		tmp = t_1 * t_0;
	} else {
		tmp = t_3 * (1.0 - ((((t_6 * t_4) * M) * h) / l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / l))
	t_1 = math.sqrt((d / h))
	t_2 = (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_3 = math.fabs(d) / math.sqrt((l * h))
	t_4 = D / (d + d)
	t_5 = t_4 * M
	t_6 = 0.25 * ((M * D) / d)
	tmp = 0
	if t_2 <= -5e+42:
		tmp = (t_0 * t_1) * (1.0 - ((t_5 * (t_5 * 0.5)) * (h / l)))
	elif t_2 <= 0.0:
		tmp = t_3 * (1.0 - ((((M * D) * t_6) / (d + d)) * (h / l)))
	elif t_2 <= 5e+107:
		tmp = t_1 * t_0
	else:
		tmp = t_3 * (1.0 - ((((t_6 * t_4) * M) * h) / l))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / l))
	t_1 = sqrt(Float64(d / h))
	t_2 = 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_3 = Float64(abs(d) / sqrt(Float64(l * h)))
	t_4 = Float64(D / Float64(d + d))
	t_5 = Float64(t_4 * M)
	t_6 = Float64(0.25 * Float64(Float64(M * D) / d))
	tmp = 0.0
	if (t_2 <= -5e+42)
		tmp = Float64(Float64(t_0 * t_1) * Float64(1.0 - Float64(Float64(t_5 * Float64(t_5 * 0.5)) * Float64(h / l))));
	elseif (t_2 <= 0.0)
		tmp = Float64(t_3 * Float64(1.0 - Float64(Float64(Float64(Float64(M * D) * t_6) / Float64(d + d)) * Float64(h / l))));
	elseif (t_2 <= 5e+107)
		tmp = Float64(t_1 * t_0);
	else
		tmp = Float64(t_3 * Float64(1.0 - Float64(Float64(Float64(Float64(t_6 * t_4) * M) * h) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / l));
	t_1 = sqrt((d / h));
	t_2 = (((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_3 = abs(d) / sqrt((l * h));
	t_4 = D / (d + d);
	t_5 = t_4 * M;
	t_6 = 0.25 * ((M * D) / d);
	tmp = 0.0;
	if (t_2 <= -5e+42)
		tmp = (t_0 * t_1) * (1.0 - ((t_5 * (t_5 * 0.5)) * (h / l)));
	elseif (t_2 <= 0.0)
		tmp = t_3 * (1.0 - ((((M * D) * t_6) / (d + d)) * (h / l)));
	elseif (t_2 <= 5e+107)
		tmp = t_1 * t_0;
	else
		tmp = t_3 * (1.0 - ((((t_6 * t_4) * M) * h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = 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$3 = N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(D / N[(d + d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 * M), $MachinePrecision]}, Block[{t$95$6 = N[(0.25 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+42], N[(N[(t$95$0 * t$95$1), $MachinePrecision] * N[(1.0 - N[(N[(t$95$5 * N[(t$95$5 * 0.5), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(t$95$3 * N[(1.0 - N[(N[(N[(N[(M * D), $MachinePrecision] * t$95$6), $MachinePrecision] / N[(d + d), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+107], N[(t$95$1 * t$95$0), $MachinePrecision], N[(t$95$3 * N[(1.0 - N[(N[(N[(N[(t$95$6 * t$95$4), $MachinePrecision] * M), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5.00000000000000007e42

   1. Initial program 66.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lift-/.f64 N/A

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

      12. lower-/.f64 65.8

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

      13. lift-/.f64 N/A

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

      14. metadata-eval 65.8

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

   3. Applied rewrites 65.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-prod-down N/A

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

      9. unpow1/2 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. frac-times N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. sqrt-div N/A

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

      16. lower-unsound-sqrt.f32 N/A

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

      17. lower-sqrt.f32 N/A

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

      18. lift-*.f64 N/A

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

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

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

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

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

      21. lower-fabs.f64 N/A

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

      22. lower-unsound-sqrt.f64 70.2

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-*.f64 70.2

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

   5. Applied rewrites 70.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. associate-/r* N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. mult-flip N/A

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

      19. lift-/.f64 N/A

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

   7. Applied rewrites 70.3%

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

      1. lift-/.f64 N/A

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

      2. lift-fabs.f64 N/A

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

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

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. sqrt-undiv N/A

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

      9. lift-*.f64 N/A

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

      10. frac-times N/A

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

      11. sqrt-unprod N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-/.f64 66.1

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

   9. Applied rewrites 66.1%

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

   if -5.00000000000000007e42 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0

   1. Initial program 66.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lift-/.f64 N/A

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

      12. lower-/.f64 65.8

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

      13. lift-/.f64 N/A

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

      14. metadata-eval 65.8

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

   3. Applied rewrites 65.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-prod-down N/A

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

      9. unpow1/2 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. frac-times N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. sqrt-div N/A

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

      16. lower-unsound-sqrt.f32 N/A

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

      17. lower-sqrt.f32 N/A

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

      18. lift-*.f64 N/A

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

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

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

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

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

      21. lower-fabs.f64 N/A

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

      22. lower-unsound-sqrt.f64 70.2

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-*.f64 70.2

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

   5. Applied rewrites 70.2%

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

   6. Evaluated real constant 70.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. associate-/r* N/A

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

      10. count-2-rev N/A

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

      11. lift-+.f64 N/A

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

      12. mult-flip N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. pow2 N/A

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

      16. associate-*r* N/A

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

      17. lift-*.f64 N/A

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

   8. Applied rewrites 69.9%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000002e107

   1. Initial program 66.3%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 23.9

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

   4. Applied rewrites 23.9%

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

   5. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 31.5

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

   7. Applied rewrites 31.5%

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

   8. Taylor expanded in h around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 39.0

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

   10. Applied rewrites 39.0%

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

   if 5.0000000000000002e107 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lift-/.f64 N/A

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

      12. lower-/.f64 65.8

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

      13. lift-/.f64 N/A

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

      14. metadata-eval 65.8

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

   3. Applied rewrites 65.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-prod-down N/A

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

      9. unpow1/2 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. frac-times N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. sqrt-div N/A

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

      16. lower-unsound-sqrt.f32 N/A

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

      17. lower-sqrt.f32 N/A

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

      18. lift-*.f64 N/A

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

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

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

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

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

      21. lower-fabs.f64 N/A

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

      22. lower-unsound-sqrt.f64 70.2

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-*.f64 70.2

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

   5. Applied rewrites 70.2%

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

   6. Evaluated real constant 70.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

   8. Applied rewrites 73.3%

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

Alternative 4: 81.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ t_1 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_2 := \sqrt{\frac{d}{\ell}}\\ t_3 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_4 := t\_1 \cdot t\_3\\ t_5 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_4}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_6 := 0.25 \cdot \frac{t\_4}{d}\\ t_7 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;t\_5 \leq -5 \cdot 10^{+42}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(\left(t\_6 \cdot t\_1\right) \cdot \frac{t\_3}{d}\right) \cdot -0.5, \frac{h}{\ell}, 1\right) \cdot t\_2\right) \cdot t\_7\\ \mathbf{elif}\;t\_5 \leq 0:\\ \;\;\;\;t\_0 \cdot \left(1 - \frac{t\_4 \cdot t\_6}{d + d} \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;t\_5 \leq 5 \cdot 10^{+107}:\\ \;\;\;\;t\_7 \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(1 - \frac{\left(\left(t\_6 \cdot \frac{t\_3}{d + d}\right) \cdot t\_1\right) \cdot h}{\ell}\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ (fabs d) (sqrt (* l h))))
        (t_1 (fmin (fabs M) (fabs D)))
        (t_2 (sqrt (/ d l)))
        (t_3 (fmax (fabs M) (fabs D)))
        (t_4 (* t_1 t_3))
        (t_5
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ t_4 (* 2.0 d)) 2.0)) (/ h l)))))
        (t_6 (* 0.25 (/ t_4 d)))
        (t_7 (sqrt (/ d h))))
   (if (<= t_5 -5e+42)
     (* (* (fma (* (* (* t_6 t_1) (/ t_3 d)) -0.5) (/ h l) 1.0) t_2) t_7)
     (if (<= t_5 0.0)
       (* t_0 (- 1.0 (* (/ (* t_4 t_6) (+ d d)) (/ h l))))
       (if (<= t_5 5e+107)
         (* t_7 t_2)
         (* t_0 (- 1.0 (/ (* (* (* t_6 (/ t_3 (+ d d))) t_1) h) l))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fabs(d) / sqrt((l * h));
	double t_1 = fmin(fabs(M), fabs(D));
	double t_2 = sqrt((d / l));
	double t_3 = fmax(fabs(M), fabs(D));
	double t_4 = t_1 * t_3;
	double t_5 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow((t_4 / (2.0 * d)), 2.0)) * (h / l)));
	double t_6 = 0.25 * (t_4 / d);
	double t_7 = sqrt((d / h));
	double tmp;
	if (t_5 <= -5e+42) {
		tmp = (fma((((t_6 * t_1) * (t_3 / d)) * -0.5), (h / l), 1.0) * t_2) * t_7;
	} else if (t_5 <= 0.0) {
		tmp = t_0 * (1.0 - (((t_4 * t_6) / (d + d)) * (h / l)));
	} else if (t_5 <= 5e+107) {
		tmp = t_7 * t_2;
	} else {
		tmp = t_0 * (1.0 - ((((t_6 * (t_3 / (d + d))) * t_1) * h) / l));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(abs(d) / sqrt(Float64(l * h)))
	t_1 = fmin(abs(M), abs(D))
	t_2 = sqrt(Float64(d / l))
	t_3 = fmax(abs(M), abs(D))
	t_4 = Float64(t_1 * t_3)
	t_5 = 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(t_4 / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_6 = Float64(0.25 * Float64(t_4 / d))
	t_7 = sqrt(Float64(d / h))
	tmp = 0.0
	if (t_5 <= -5e+42)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(t_6 * t_1) * Float64(t_3 / d)) * -0.5), Float64(h / l), 1.0) * t_2) * t_7);
	elseif (t_5 <= 0.0)
		tmp = Float64(t_0 * Float64(1.0 - Float64(Float64(Float64(t_4 * t_6) / Float64(d + d)) * Float64(h / l))));
	elseif (t_5 <= 5e+107)
		tmp = Float64(t_7 * t_2);
	else
		tmp = Float64(t_0 * Float64(1.0 - Float64(Float64(Float64(Float64(t_6 * Float64(t_3 / Float64(d + d))) * t_1) * h) / l)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Max[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 * t$95$3), $MachinePrecision]}, Block[{t$95$5 = 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[(t$95$4 / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(0.25 * N[(t$95$4 / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$5, -5e+42], N[(N[(N[(N[(N[(N[(t$95$6 * t$95$1), $MachinePrecision] * N[(t$95$3 / d), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$7), $MachinePrecision], If[LessEqual[t$95$5, 0.0], N[(t$95$0 * N[(1.0 - N[(N[(N[(t$95$4 * t$95$6), $MachinePrecision] / N[(d + d), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 5e+107], N[(t$95$7 * t$95$2), $MachinePrecision], N[(t$95$0 * N[(1.0 - N[(N[(N[(N[(t$95$6 * N[(t$95$3 / N[(d + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5.00000000000000007e42

   1. Initial program 66.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lift-/.f64 N/A

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

      12. lower-/.f64 65.8

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

      13. lift-/.f64 N/A

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

      14. metadata-eval 65.8

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

   3. Applied rewrites 65.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-prod-down N/A

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

      9. unpow1/2 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. frac-times N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. sqrt-div N/A

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

      16. lower-unsound-sqrt.f32 N/A

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

      17. lower-sqrt.f32 N/A

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

      18. lift-*.f64 N/A

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

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

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

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

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

      21. lower-fabs.f64 N/A

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

      22. lower-unsound-sqrt.f64 70.2

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-*.f64 70.2

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

   5. Applied rewrites 70.2%

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

   6. Evaluated real constant 70.2%

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

   8. Applied rewrites 64.7%

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

   if -5.00000000000000007e42 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0

   1. Initial program 66.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lift-/.f64 N/A

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

      12. lower-/.f64 65.8

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

      13. lift-/.f64 N/A

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

      14. metadata-eval 65.8

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

   3. Applied rewrites 65.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-prod-down N/A

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

      9. unpow1/2 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. frac-times N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. sqrt-div N/A

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

      16. lower-unsound-sqrt.f32 N/A

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

      17. lower-sqrt.f32 N/A

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

      18. lift-*.f64 N/A

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

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

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

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

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

      21. lower-fabs.f64 N/A

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

      22. lower-unsound-sqrt.f64 70.2

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-*.f64 70.2

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

   5. Applied rewrites 70.2%

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

   6. Evaluated real constant 70.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. associate-/r* N/A

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

      10. count-2-rev N/A

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

      11. lift-+.f64 N/A

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

      12. mult-flip N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. pow2 N/A

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

      16. associate-*r* N/A

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

      17. lift-*.f64 N/A

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

   8. Applied rewrites 69.9%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000002e107

   1. Initial program 66.3%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 23.9

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

   4. Applied rewrites 23.9%

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

   5. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 31.5

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

   7. Applied rewrites 31.5%

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

   8. Taylor expanded in h around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 39.0

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

   10. Applied rewrites 39.0%

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

   if 5.0000000000000002e107 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lift-/.f64 N/A

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

      12. lower-/.f64 65.8

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

      13. lift-/.f64 N/A

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

      14. metadata-eval 65.8

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

   3. Applied rewrites 65.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-prod-down N/A

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

      9. unpow1/2 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. frac-times N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. sqrt-div N/A

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

      16. lower-unsound-sqrt.f32 N/A

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

      17. lower-sqrt.f32 N/A

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

      18. lift-*.f64 N/A

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

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

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

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

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

      21. lower-fabs.f64 N/A

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

      22. lower-unsound-sqrt.f64 70.2

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-*.f64 70.2

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

   5. Applied rewrites 70.2%

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

   6. Evaluated real constant 70.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

   8. Applied rewrites 73.3%

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

Alternative 5: 80.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|M\right|, D\right)\\ t_1 := \mathsf{max}\left(\left|M\right|, D\right)\\ t_2 := {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\\ t_3 := t\_0 \cdot t\_1\\ \mathbf{if}\;t\_2 \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_3}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+107}:\\ \;\;\;\;t\_2 \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(t\_1 \cdot \left(\frac{0.5}{d} \cdot t\_0\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{\left(\left(\left(0.25 \cdot \frac{t\_3}{d}\right) \cdot \frac{t\_1}{d + d}\right) \cdot t\_0\right) \cdot h}{\ell}\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (fmin (fabs M) D))
        (t_1 (fmax (fabs M) D))
        (t_2 (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))))
        (t_3 (* t_0 t_1)))
   (if (<=
        (* t_2 (- 1.0 (* (* (/ 1.0 2.0) (pow (/ t_3 (* 2.0 d)) 2.0)) (/ h l))))
        5e+107)
     (*
      t_2
      (- 1.0 (* (* (/ 1.0 2.0) (pow (* t_1 (* (/ 0.5 d) t_0)) 2.0)) (/ h l))))
     (*
      (/ (fabs d) (sqrt (* l h)))
      (- 1.0 (/ (* (* (* (* 0.25 (/ t_3 d)) (/ t_1 (+ d d))) t_0) h) l))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fmin(fabs(M), D);
	double t_1 = fmax(fabs(M), D);
	double t_2 = pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0));
	double t_3 = t_0 * t_1;
	double tmp;
	if ((t_2 * (1.0 - (((1.0 / 2.0) * pow((t_3 / (2.0 * d)), 2.0)) * (h / l)))) <= 5e+107) {
		tmp = t_2 * (1.0 - (((1.0 / 2.0) * pow((t_1 * ((0.5 / d) * t_0)), 2.0)) * (h / l)));
	} else {
		tmp = (fabs(d) / sqrt((l * h))) * (1.0 - (((((0.25 * (t_3 / d)) * (t_1 / (d + d))) * t_0) * 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) :: tmp
    t_0 = fmin(abs(m), d_1)
    t_1 = fmax(abs(m), d_1)
    t_2 = ((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))
    t_3 = t_0 * t_1
    if ((t_2 * (1.0d0 - (((1.0d0 / 2.0d0) * ((t_3 / (2.0d0 * d)) ** 2.0d0)) * (h / l)))) <= 5d+107) then
        tmp = t_2 * (1.0d0 - (((1.0d0 / 2.0d0) * ((t_1 * ((0.5d0 / d) * t_0)) ** 2.0d0)) * (h / l)))
    else
        tmp = (abs(d) / sqrt((l * h))) * (1.0d0 - (((((0.25d0 * (t_3 / d)) * (t_1 / (d + d))) * t_0) * 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 = fmin(Math.abs(M), D);
	double t_1 = fmax(Math.abs(M), D);
	double t_2 = Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0));
	double t_3 = t_0 * t_1;
	double tmp;
	if ((t_2 * (1.0 - (((1.0 / 2.0) * Math.pow((t_3 / (2.0 * d)), 2.0)) * (h / l)))) <= 5e+107) {
		tmp = t_2 * (1.0 - (((1.0 / 2.0) * Math.pow((t_1 * ((0.5 / d) * t_0)), 2.0)) * (h / l)));
	} else {
		tmp = (Math.abs(d) / Math.sqrt((l * h))) * (1.0 - (((((0.25 * (t_3 / d)) * (t_1 / (d + d))) * t_0) * h) / l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = fmin(math.fabs(M), D)
	t_1 = fmax(math.fabs(M), D)
	t_2 = math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))
	t_3 = t_0 * t_1
	tmp = 0
	if (t_2 * (1.0 - (((1.0 / 2.0) * math.pow((t_3 / (2.0 * d)), 2.0)) * (h / l)))) <= 5e+107:
		tmp = t_2 * (1.0 - (((1.0 / 2.0) * math.pow((t_1 * ((0.5 / d) * t_0)), 2.0)) * (h / l)))
	else:
		tmp = (math.fabs(d) / math.sqrt((l * h))) * (1.0 - (((((0.25 * (t_3 / d)) * (t_1 / (d + d))) * t_0) * h) / l))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = fmin(abs(M), D)
	t_1 = fmax(abs(M), D)
	t_2 = Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0)))
	t_3 = Float64(t_0 * t_1)
	tmp = 0.0
	if (Float64(t_2 * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(t_3 / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= 5e+107)
		tmp = Float64(t_2 * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(t_1 * Float64(Float64(0.5 / d) * t_0)) ^ 2.0)) * Float64(h / l))));
	else
		tmp = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(0.25 * Float64(t_3 / d)) * Float64(t_1 / Float64(d + d))) * t_0) * h) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = min(abs(M), D);
	t_1 = max(abs(M), D);
	t_2 = ((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0));
	t_3 = t_0 * t_1;
	tmp = 0.0;
	if ((t_2 * (1.0 - (((1.0 / 2.0) * ((t_3 / (2.0 * d)) ^ 2.0)) * (h / l)))) <= 5e+107)
		tmp = t_2 * (1.0 - (((1.0 / 2.0) * ((t_1 * ((0.5 / d) * t_0)) ^ 2.0)) * (h / l)));
	else
		tmp = (abs(d) / sqrt((l * h))) * (1.0 - (((((0.25 * (t_3 / d)) * (t_1 / (d + d))) * t_0) * h) / 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[Max[N[Abs[M], $MachinePrecision], D], $MachinePrecision]}, Block[{t$95$2 = 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]}, Block[{t$95$3 = N[(t$95$0 * t$95$1), $MachinePrecision]}, If[LessEqual[N[(t$95$2 * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(t$95$3 / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+107], N[(t$95$2 * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(t$95$1 * N[(N[(0.5 / d), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(0.25 * N[(t$95$3 / d), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / N[(d + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000002e107

   1. Initial program 66.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \color{blue}{\left(\frac{1}{2 \cdot d} \cdot M\right)}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{1}{\color{blue}{2 \cdot d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-/.f64 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. metadata-eval 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{0.5}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   if 5.0000000000000002e107 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. associate-/l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. mult-flip N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\left(D \cdot \frac{1}{2 \cdot d}\right)} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \color{blue}{\left(\frac{1}{2 \cdot d} \cdot M\right)}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{1}{\color{blue}{2 \cdot d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-/.f64 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. metadata-eval 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{0.5}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-unsound-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. lower-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. rem-sqrt-square-rev N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. lower-unsound-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. lower-fabs.f64 N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      22. lower-unsound-sqrt.f64 70.2

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      24. *-commutative N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      25. lower-*.f64 70.2

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 70.2%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Evaluated real constant 70.2%

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \color{blue}{\left({\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

   8. Applied rewrites 73.3%

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \color{blue}{\frac{\left(\left(\left(0.25 \cdot \frac{M \cdot D}{d}\right) \cdot \frac{D}{d + d}\right) \cdot M\right) \cdot h}{\ell}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 79.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|M\right|, D\right)\\ t_1 := \mathsf{max}\left(\left|M\right|, D\right)\\ t_2 := t\_0 \cdot t\_1\\ \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_2}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+107}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(t\_1 \cdot \left(\frac{0.5}{d} \cdot t\_0\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{\left(\left(\left(0.25 \cdot \frac{t\_2}{d}\right) \cdot \frac{t\_1}{d + d}\right) \cdot t\_0\right) \cdot h}{\ell}\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (fmin (fabs M) D)) (t_1 (fmax (fabs M) D)) (t_2 (* t_0 t_1)))
   (if (<=
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (- 1.0 (* (* (/ 1.0 2.0) (pow (/ t_2 (* 2.0 d)) 2.0)) (/ h l))))
        5e+107)
     (*
      (* (sqrt (/ d l)) (sqrt (/ d h)))
      (- 1.0 (* (* (/ 1.0 2.0) (pow (* t_1 (* (/ 0.5 d) t_0)) 2.0)) (/ h l))))
     (*
      (/ (fabs d) (sqrt (* l h)))
      (- 1.0 (/ (* (* (* (* 0.25 (/ t_2 d)) (/ t_1 (+ d d))) t_0) h) l))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fmin(fabs(M), D);
	double t_1 = fmax(fabs(M), D);
	double t_2 = t_0 * t_1;
	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_2 / (2.0 * d)), 2.0)) * (h / l)))) <= 5e+107) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (((1.0 / 2.0) * pow((t_1 * ((0.5 / d) * t_0)), 2.0)) * (h / l)));
	} else {
		tmp = (fabs(d) / sqrt((l * h))) * (1.0 - (((((0.25 * (t_2 / d)) * (t_1 / (d + d))) * t_0) * 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) :: tmp
    t_0 = fmin(abs(m), d_1)
    t_1 = fmax(abs(m), d_1)
    t_2 = t_0 * t_1
    if (((((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * ((t_2 / (2.0d0 * d)) ** 2.0d0)) * (h / l)))) <= 5d+107) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - (((1.0d0 / 2.0d0) * ((t_1 * ((0.5d0 / d) * t_0)) ** 2.0d0)) * (h / l)))
    else
        tmp = (abs(d) / sqrt((l * h))) * (1.0d0 - (((((0.25d0 * (t_2 / d)) * (t_1 / (d + d))) * t_0) * 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 = fmin(Math.abs(M), D);
	double t_1 = fmax(Math.abs(M), D);
	double t_2 = t_0 * t_1;
	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((t_2 / (2.0 * d)), 2.0)) * (h / l)))) <= 5e+107) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - (((1.0 / 2.0) * Math.pow((t_1 * ((0.5 / d) * t_0)), 2.0)) * (h / l)));
	} else {
		tmp = (Math.abs(d) / Math.sqrt((l * h))) * (1.0 - (((((0.25 * (t_2 / d)) * (t_1 / (d + d))) * t_0) * h) / l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = fmin(math.fabs(M), D)
	t_1 = fmax(math.fabs(M), D)
	t_2 = t_0 * t_1
	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((t_2 / (2.0 * d)), 2.0)) * (h / l)))) <= 5e+107:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - (((1.0 / 2.0) * math.pow((t_1 * ((0.5 / d) * t_0)), 2.0)) * (h / l)))
	else:
		tmp = (math.fabs(d) / math.sqrt((l * h))) * (1.0 - (((((0.25 * (t_2 / d)) * (t_1 / (d + d))) * t_0) * h) / l))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = fmin(abs(M), D)
	t_1 = fmax(abs(M), D)
	t_2 = Float64(t_0 * t_1)
	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(t_2 / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= 5e+107)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(t_1 * Float64(Float64(0.5 / d) * t_0)) ^ 2.0)) * Float64(h / l))));
	else
		tmp = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(0.25 * Float64(t_2 / d)) * Float64(t_1 / Float64(d + d))) * t_0) * h) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = min(abs(M), D);
	t_1 = max(abs(M), D);
	t_2 = t_0 * t_1;
	tmp = 0.0;
	if (((((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * ((t_2 / (2.0 * d)) ^ 2.0)) * (h / l)))) <= 5e+107)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (((1.0 / 2.0) * ((t_1 * ((0.5 / d) * t_0)) ^ 2.0)) * (h / l)));
	else
		tmp = (abs(d) / sqrt((l * h))) * (1.0 - (((((0.25 * (t_2 / d)) * (t_1 / (d + d))) * t_0) * h) / 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[Max[N[Abs[M], $MachinePrecision], D], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1), $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[(t$95$2 / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+107], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(t$95$1 * N[(N[(0.5 / d), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(0.25 * N[(t$95$2 / d), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / N[(d + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000002e107

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. associate-/l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. mult-flip N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\left(D \cdot \frac{1}{2 \cdot d}\right)} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \color{blue}{\left(\frac{1}{2 \cdot d} \cdot M\right)}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{1}{\color{blue}{2 \cdot d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-/.f64 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. metadata-eval 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{0.5}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.8

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. metadata-eval 65.8

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-sqrt.f64 65.8

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. metadata-eval 65.8

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-pow.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-sqrt.f64 65.8

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 65.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   if 5.0000000000000002e107 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. associate-/l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. mult-flip N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\left(D \cdot \frac{1}{2 \cdot d}\right)} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \color{blue}{\left(\frac{1}{2 \cdot d} \cdot M\right)}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{1}{\color{blue}{2 \cdot d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-/.f64 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. metadata-eval 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{0.5}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-unsound-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. lower-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. rem-sqrt-square-rev N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. lower-unsound-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. lower-fabs.f64 N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      22. lower-unsound-sqrt.f64 70.2

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      24. *-commutative N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      25. lower-*.f64 70.2

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 70.2%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Evaluated real constant 70.2%

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \color{blue}{\left({\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

   8. Applied rewrites 73.3%

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \color{blue}{\frac{\left(\left(\left(0.25 \cdot \frac{M \cdot D}{d}\right) \cdot \frac{D}{d + d}\right) \cdot M\right) \cdot h}{\ell}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 79.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\\ \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot t\_0\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+107}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(0.5 \cdot t\_0\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{\left(\left(\left(0.25 \cdot \frac{M \cdot D}{d}\right) \cdot \frac{D}{d + d}\right) \cdot M\right) \cdot h}{\ell}\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (pow (/ (* M D) (* 2.0 d)) 2.0)))
   (if (<=
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (- 1.0 (* (* (/ 1.0 2.0) t_0) (/ h l))))
        5e+107)
     (* (* (sqrt (/ d l)) (sqrt (/ d h))) (- 1.0 (* (* 0.5 t_0) (/ h l))))
     (*
      (/ (fabs d) (sqrt (* l h)))
      (- 1.0 (/ (* (* (* (* 0.25 (/ (* M D) d)) (/ D (+ d d))) M) h) l))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = pow(((M * D) / (2.0 * d)), 2.0);
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * t_0) * (h / l)))) <= 5e+107) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((0.5 * t_0) * (h / l)));
	} else {
		tmp = (fabs(d) / sqrt((l * h))) * (1.0 - (((((0.25 * ((M * D) / 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) :: tmp
    t_0 = ((m * d_1) / (2.0d0 * d)) ** 2.0d0
    if (((((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * t_0) * (h / l)))) <= 5d+107) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - ((0.5d0 * t_0) * (h / l)))
    else
        tmp = (abs(d) / sqrt((l * h))) * (1.0d0 - (((((0.25d0 * ((m * d_1) / 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 = Math.pow(((M * D) / (2.0 * d)), 2.0);
	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) * t_0) * (h / l)))) <= 5e+107) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - ((0.5 * t_0) * (h / l)));
	} else {
		tmp = (Math.abs(d) / Math.sqrt((l * h))) * (1.0 - (((((0.25 * ((M * D) / d)) * (D / (d + d))) * M) * h) / l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.pow(((M * D) / (2.0 * d)), 2.0)
	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) * t_0) * (h / l)))) <= 5e+107:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - ((0.5 * t_0) * (h / l)))
	else:
		tmp = (math.fabs(d) / math.sqrt((l * h))) * (1.0 - (((((0.25 * ((M * D) / d)) * (D / (d + d))) * M) * h) / l))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0
	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) * t_0) * Float64(h / l)))) <= 5e+107)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(0.5 * t_0) * Float64(h / l))));
	else
		tmp = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(0.25 * Float64(Float64(M * D) / d)) * Float64(D / Float64(d + d))) * M) * h) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = ((M * D) / (2.0 * d)) ^ 2.0;
	tmp = 0.0;
	if (((((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * t_0) * (h / l)))) <= 5e+107)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((0.5 * t_0) * (h / l)));
	else
		tmp = (abs(d) / sqrt((l * h))) * (1.0 - (((((0.25 * ((M * D) / d)) * (D / (d + d))) * M) * h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $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] * t$95$0), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+107], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(0.5 * t$95$0), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(0.25 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(D / N[(d + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000002e107

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 66.3

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      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 66.3

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      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 66.3

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 66.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Evaluated real constant 66.3%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   if 5.0000000000000002e107 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. associate-/l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. mult-flip N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\left(D \cdot \frac{1}{2 \cdot d}\right)} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \color{blue}{\left(\frac{1}{2 \cdot d} \cdot M\right)}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{1}{\color{blue}{2 \cdot d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-/.f64 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. metadata-eval 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{0.5}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-unsound-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. lower-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. rem-sqrt-square-rev N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. lower-unsound-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. lower-fabs.f64 N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      22. lower-unsound-sqrt.f64 70.2

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      24. *-commutative N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      25. lower-*.f64 70.2

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 70.2%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Evaluated real constant 70.2%

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \color{blue}{\left({\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

   8. Applied rewrites 73.3%

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \color{blue}{\frac{\left(\left(\left(0.25 \cdot \frac{M \cdot D}{d}\right) \cdot \frac{D}{d + d}\right) \cdot M\right) \cdot h}{\ell}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 79.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := 0.25 \cdot \frac{M \cdot D}{d}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_0 \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot t\_2}{d + d} \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+107}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(1 - \frac{\left(\left(t\_2 \cdot \frac{D}{d + d}\right) \cdot M\right) \cdot h}{\ell}\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ (fabs d) (sqrt (* l h))))
        (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 (* 0.25 (/ (* M D) d))))
   (if (<= t_1 0.0)
     (* t_0 (- 1.0 (* (/ (* (* M D) t_2) (+ d d)) (/ h l))))
     (if (<= t_1 5e+107)
       (* (sqrt (/ d h)) (sqrt (/ d l)))
       (* t_0 (- 1.0 (/ (* (* (* t_2 (/ D (+ d d))) M) h) l)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fabs(d) / sqrt((l * h));
	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 = 0.25 * ((M * D) / d);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0 * (1.0 - ((((M * D) * t_2) / (d + d)) * (h / l)));
	} else if (t_1 <= 5e+107) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else {
		tmp = t_0 * (1.0 - ((((t_2 * (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) :: t_2
    real(8) :: tmp
    t_0 = abs(d) / sqrt((l * h))
    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 = 0.25d0 * ((m * d_1) / d)
    if (t_1 <= 0.0d0) then
        tmp = t_0 * (1.0d0 - ((((m * d_1) * t_2) / (d + d)) * (h / l)))
    else if (t_1 <= 5d+107) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else
        tmp = t_0 * (1.0d0 - ((((t_2 * (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 = Math.abs(d) / Math.sqrt((l * h));
	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 = 0.25 * ((M * D) / d);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0 * (1.0 - ((((M * D) * t_2) / (d + d)) * (h / l)));
	} else if (t_1 <= 5e+107) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else {
		tmp = t_0 * (1.0 - ((((t_2 * (D / (d + d))) * M) * h) / l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.fabs(d) / math.sqrt((l * h))
	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 = 0.25 * ((M * D) / d)
	tmp = 0
	if t_1 <= 0.0:
		tmp = t_0 * (1.0 - ((((M * D) * t_2) / (d + d)) * (h / l)))
	elif t_1 <= 5e+107:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	else:
		tmp = t_0 * (1.0 - ((((t_2 * (D / (d + d))) * M) * h) / l))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(abs(d) / sqrt(Float64(l * h)))
	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(0.25 * Float64(Float64(M * D) / d))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(t_0 * Float64(1.0 - Float64(Float64(Float64(Float64(M * D) * t_2) / Float64(d + d)) * Float64(h / l))));
	elseif (t_1 <= 5e+107)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	else
		tmp = Float64(t_0 * Float64(1.0 - Float64(Float64(Float64(Float64(t_2 * Float64(D / Float64(d + d))) * M) * h) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = abs(d) / sqrt((l * h));
	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 = 0.25 * ((M * D) / d);
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = t_0 * (1.0 - ((((M * D) * t_2) / (d + d)) * (h / l)));
	elseif (t_1 <= 5e+107)
		tmp = sqrt((d / h)) * sqrt((d / l));
	else
		tmp = t_0 * (1.0 - ((((t_2 * (D / (d + d))) * M) * h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.25 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(t$95$0 * N[(1.0 - N[(N[(N[(N[(M * D), $MachinePrecision] * t$95$2), $MachinePrecision] / N[(d + d), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+107], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 - N[(N[(N[(N[(t$95$2 * N[(D / N[(d + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision] * h), $MachinePrecision] / l), $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)))) < 0.0

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. associate-/l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. mult-flip N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\left(D \cdot \frac{1}{2 \cdot d}\right)} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \color{blue}{\left(\frac{1}{2 \cdot d} \cdot M\right)}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{1}{\color{blue}{2 \cdot d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-/.f64 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. metadata-eval 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{0.5}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-unsound-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. lower-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. rem-sqrt-square-rev N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. lower-unsound-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. lower-fabs.f64 N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      22. lower-unsound-sqrt.f64 70.2

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      24. *-commutative N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      25. lower-*.f64 70.2

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 70.2%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Evaluated real constant 70.2%

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \color{blue}{\left({\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left(\color{blue}{{\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left({\color{blue}{\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left({\left(D \cdot \color{blue}{\left(\frac{\frac{1}{2}}{d} \cdot M\right)}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left({\color{blue}{\left(\left(D \cdot \frac{\frac{1}{2}}{d}\right) \cdot M\right)}}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left({\left(\left(D \cdot \color{blue}{\frac{\frac{1}{2}}{d}}\right) \cdot M\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left({\left(\left(D \cdot \frac{\color{blue}{\frac{1}{2}}}{d}\right) \cdot M\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. associate-/r* N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left({\left(\left(D \cdot \color{blue}{\frac{1}{2 \cdot d}}\right) \cdot M\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. count-2-rev N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left({\left(\left(D \cdot \frac{1}{\color{blue}{d + d}}\right) \cdot M\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left({\left(\left(D \cdot \frac{1}{\color{blue}{d + d}}\right) \cdot M\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. mult-flip N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left({\left(\color{blue}{\frac{D}{d + d}} \cdot M\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left({\left(\color{blue}{\frac{D}{d + d}} \cdot M\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left({\color{blue}{\left(\frac{D}{d + d} \cdot M\right)}}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. pow2 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left(\color{blue}{\left(\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right)} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. associate-*r* N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot M\right) \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left(\left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{D}{d + d} \cdot M\right) \cdot \frac{1}{2}\right)}\right) \cdot \frac{h}{\ell}\right) \]

   8. Applied rewrites 69.9%

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \color{blue}{\frac{\left(M \cdot D\right) \cdot \left(0.25 \cdot \frac{M \cdot D}{d}\right)}{d + d}} \cdot \frac{h}{\ell}\right) \]

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000002e107

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.9

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.9%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      6. lower-*.f64 31.5

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   7. Applied rewrites 31.5%

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   8. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      3. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      5. lower-/.f64 39.0

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

   10. Applied rewrites 39.0%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

   if 5.0000000000000002e107 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. associate-/l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. mult-flip N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\left(D \cdot \frac{1}{2 \cdot d}\right)} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \color{blue}{\left(\frac{1}{2 \cdot d} \cdot M\right)}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{1}{\color{blue}{2 \cdot d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-/.f64 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. metadata-eval 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{0.5}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-unsound-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. lower-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. rem-sqrt-square-rev N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. lower-unsound-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. lower-fabs.f64 N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      22. lower-unsound-sqrt.f64 70.2

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      24. *-commutative N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      25. lower-*.f64 70.2

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 70.2%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Evaluated real constant 70.2%

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \color{blue}{\left({\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

   8. Applied rewrites 73.3%

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \color{blue}{\frac{\left(\left(\left(0.25 \cdot \frac{M \cdot D}{d}\right) \cdot \frac{D}{d + d}\right) \cdot M\right) \cdot h}{\ell}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 79.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ t_1 := \mathsf{min}\left(\left|M\right|, D\right)\\ t_2 := \mathsf{max}\left(\left|M\right|, D\right)\\ t_3 := t\_1 \cdot t\_2\\ t_4 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_3}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_5 := 0.25 \cdot \frac{t\_3}{d}\\ \mathbf{if}\;t\_4 \leq 0:\\ \;\;\;\;t\_0 \cdot \left(1 - \frac{t\_2 \cdot \left(t\_5 \cdot t\_1\right)}{d + d} \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{+107}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(1 - \frac{\left(\left(t\_5 \cdot \frac{t\_2}{d + d}\right) \cdot t\_1\right) \cdot h}{\ell}\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ (fabs d) (sqrt (* l h))))
        (t_1 (fmin (fabs M) D))
        (t_2 (fmax (fabs M) D))
        (t_3 (* t_1 t_2))
        (t_4
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ t_3 (* 2.0 d)) 2.0)) (/ h l)))))
        (t_5 (* 0.25 (/ t_3 d))))
   (if (<= t_4 0.0)
     (* t_0 (- 1.0 (* (/ (* t_2 (* t_5 t_1)) (+ d d)) (/ h l))))
     (if (<= t_4 5e+107)
       (* (sqrt (/ d h)) (sqrt (/ d l)))
       (* t_0 (- 1.0 (/ (* (* (* t_5 (/ t_2 (+ d d))) t_1) h) l)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fabs(d) / sqrt((l * h));
	double t_1 = fmin(fabs(M), D);
	double t_2 = fmax(fabs(M), D);
	double t_3 = t_1 * t_2;
	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_3 / (2.0 * d)), 2.0)) * (h / l)));
	double t_5 = 0.25 * (t_3 / d);
	double tmp;
	if (t_4 <= 0.0) {
		tmp = t_0 * (1.0 - (((t_2 * (t_5 * t_1)) / (d + d)) * (h / l)));
	} else if (t_4 <= 5e+107) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else {
		tmp = t_0 * (1.0 - ((((t_5 * (t_2 / (d + d))) * t_1) * 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) :: t_5
    real(8) :: tmp
    t_0 = abs(d) / sqrt((l * h))
    t_1 = fmin(abs(m), d_1)
    t_2 = fmax(abs(m), d_1)
    t_3 = t_1 * t_2
    t_4 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * ((t_3 / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    t_5 = 0.25d0 * (t_3 / d)
    if (t_4 <= 0.0d0) then
        tmp = t_0 * (1.0d0 - (((t_2 * (t_5 * t_1)) / (d + d)) * (h / l)))
    else if (t_4 <= 5d+107) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else
        tmp = t_0 * (1.0d0 - ((((t_5 * (t_2 / (d + d))) * t_1) * 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 = Math.abs(d) / Math.sqrt((l * h));
	double t_1 = fmin(Math.abs(M), D);
	double t_2 = fmax(Math.abs(M), D);
	double t_3 = t_1 * t_2;
	double t_4 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow((t_3 / (2.0 * d)), 2.0)) * (h / l)));
	double t_5 = 0.25 * (t_3 / d);
	double tmp;
	if (t_4 <= 0.0) {
		tmp = t_0 * (1.0 - (((t_2 * (t_5 * t_1)) / (d + d)) * (h / l)));
	} else if (t_4 <= 5e+107) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else {
		tmp = t_0 * (1.0 - ((((t_5 * (t_2 / (d + d))) * t_1) * h) / l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.fabs(d) / math.sqrt((l * h))
	t_1 = fmin(math.fabs(M), D)
	t_2 = fmax(math.fabs(M), D)
	t_3 = t_1 * t_2
	t_4 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow((t_3 / (2.0 * d)), 2.0)) * (h / l)))
	t_5 = 0.25 * (t_3 / d)
	tmp = 0
	if t_4 <= 0.0:
		tmp = t_0 * (1.0 - (((t_2 * (t_5 * t_1)) / (d + d)) * (h / l)))
	elif t_4 <= 5e+107:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	else:
		tmp = t_0 * (1.0 - ((((t_5 * (t_2 / (d + d))) * t_1) * h) / l))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(abs(d) / sqrt(Float64(l * h)))
	t_1 = fmin(abs(M), D)
	t_2 = fmax(abs(M), D)
	t_3 = Float64(t_1 * t_2)
	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(t_3 / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_5 = Float64(0.25 * Float64(t_3 / d))
	tmp = 0.0
	if (t_4 <= 0.0)
		tmp = Float64(t_0 * Float64(1.0 - Float64(Float64(Float64(t_2 * Float64(t_5 * t_1)) / Float64(d + d)) * Float64(h / l))));
	elseif (t_4 <= 5e+107)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	else
		tmp = Float64(t_0 * Float64(1.0 - Float64(Float64(Float64(Float64(t_5 * Float64(t_2 / Float64(d + d))) * t_1) * h) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = abs(d) / sqrt((l * h));
	t_1 = min(abs(M), D);
	t_2 = max(abs(M), D);
	t_3 = t_1 * t_2;
	t_4 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * ((t_3 / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_5 = 0.25 * (t_3 / d);
	tmp = 0.0;
	if (t_4 <= 0.0)
		tmp = t_0 * (1.0 - (((t_2 * (t_5 * t_1)) / (d + d)) * (h / l)));
	elseif (t_4 <= 5e+107)
		tmp = sqrt((d / h)) * sqrt((d / l));
	else
		tmp = t_0 * (1.0 - ((((t_5 * (t_2 / (d + d))) * t_1) * h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[M], $MachinePrecision], D], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Abs[M], $MachinePrecision], D], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * t$95$2), $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[(t$95$3 / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(0.25 * N[(t$95$3 / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[(t$95$0 * N[(1.0 - N[(N[(N[(t$95$2 * N[(t$95$5 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(d + d), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 5e+107], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 - N[(N[(N[(N[(t$95$5 * N[(t$95$2 / N[(d + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * h), $MachinePrecision] / l), $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)))) < 0.0

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. associate-/l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. mult-flip N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\left(D \cdot \frac{1}{2 \cdot d}\right)} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \color{blue}{\left(\frac{1}{2 \cdot d} \cdot M\right)}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{1}{\color{blue}{2 \cdot d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-/.f64 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. metadata-eval 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{0.5}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-unsound-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. lower-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. rem-sqrt-square-rev N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. lower-unsound-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. lower-fabs.f64 N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      22. lower-unsound-sqrt.f64 70.2

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      24. *-commutative N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      25. lower-*.f64 70.2

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 70.2%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Evaluated real constant 70.2%

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \color{blue}{\left({\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left(\color{blue}{{\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left({\color{blue}{\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left({\left(D \cdot \color{blue}{\left(\frac{\frac{1}{2}}{d} \cdot M\right)}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left({\color{blue}{\left(\left(D \cdot \frac{\frac{1}{2}}{d}\right) \cdot M\right)}}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left({\left(\left(D \cdot \color{blue}{\frac{\frac{1}{2}}{d}}\right) \cdot M\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left({\left(\left(D \cdot \frac{\color{blue}{\frac{1}{2}}}{d}\right) \cdot M\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. associate-/r* N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left({\left(\left(D \cdot \color{blue}{\frac{1}{2 \cdot d}}\right) \cdot M\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. count-2-rev N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left({\left(\left(D \cdot \frac{1}{\color{blue}{d + d}}\right) \cdot M\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left({\left(\left(D \cdot \frac{1}{\color{blue}{d + d}}\right) \cdot M\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. mult-flip N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left({\left(\color{blue}{\frac{D}{d + d}} \cdot M\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left({\left(\color{blue}{\frac{D}{d + d}} \cdot M\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left({\color{blue}{\left(\frac{D}{d + d} \cdot M\right)}}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. pow2 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left(\color{blue}{\left(\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right)} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. associate-*r* N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot M\right) \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left(\left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{D}{d + d} \cdot M\right) \cdot \frac{1}{2}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left(\color{blue}{\left(\frac{D}{d + d} \cdot M\right)} \cdot \left(\left(\frac{D}{d + d} \cdot M\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      19. associate-*l* N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot \left(M \cdot \left(\left(\frac{D}{d + d} \cdot M\right) \cdot \frac{1}{2}\right)\right)\right)} \cdot \frac{h}{\ell}\right) \]

   8. Applied rewrites 69.1%

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \color{blue}{\frac{D \cdot \left(\left(0.25 \cdot \frac{M \cdot D}{d}\right) \cdot M\right)}{d + d}} \cdot \frac{h}{\ell}\right) \]

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000002e107

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.9

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.9%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      6. lower-*.f64 31.5

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   7. Applied rewrites 31.5%

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   8. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      3. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      5. lower-/.f64 39.0

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

   10. Applied rewrites 39.0%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

   if 5.0000000000000002e107 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. associate-/l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. mult-flip N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\left(D \cdot \frac{1}{2 \cdot d}\right)} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \color{blue}{\left(\frac{1}{2 \cdot d} \cdot M\right)}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{1}{\color{blue}{2 \cdot d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-/.f64 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. metadata-eval 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{0.5}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-unsound-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. lower-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. rem-sqrt-square-rev N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. lower-unsound-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. lower-fabs.f64 N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      22. lower-unsound-sqrt.f64 70.2

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      24. *-commutative N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      25. lower-*.f64 70.2

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 70.2%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Evaluated real constant 70.2%

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \color{blue}{\left({\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

   8. Applied rewrites 73.3%

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \color{blue}{\frac{\left(\left(\left(0.25 \cdot \frac{M \cdot D}{d}\right) \cdot \frac{D}{d + d}\right) \cdot M\right) \cdot h}{\ell}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 78.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(M, D\right) \cdot \mathsf{max}\left(M, D\right)\\ t_1 := 0.25 \cdot \frac{t\_0}{d}\\ t_2 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_0}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(t\_1 \cdot \mathsf{min}\left(M, D\right)\right) \cdot \frac{\mathsf{max}\left(M, D\right)}{d}\right) \cdot -0.5, \frac{h}{\ell}, 1\right) \cdot \left|d\right|}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+107}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{\left(\left(t\_1 \cdot \frac{\mathsf{max}\left(M, D\right)}{d + d}\right) \cdot \mathsf{min}\left(M, D\right)\right) \cdot h}{\ell}\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (fmin M D) (fmax M D)))
        (t_1 (* 0.25 (/ t_0 d)))
        (t_2
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ t_0 (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_2 0.0)
     (/
      (*
       (fma (* (* (* t_1 (fmin M D)) (/ (fmax M D) d)) -0.5) (/ h l) 1.0)
       (fabs d))
      (sqrt (* h l)))
     (if (<= t_2 5e+107)
       (* (sqrt (/ d h)) (sqrt (/ d l)))
       (*
        (/ (fabs d) (sqrt (* l h)))
        (- 1.0 (/ (* (* (* t_1 (/ (fmax M D) (+ d d))) (fmin M D)) h) l)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fmin(M, D) * fmax(M, D);
	double t_1 = 0.25 * (t_0 / d);
	double t_2 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow((t_0 / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = (fma((((t_1 * fmin(M, D)) * (fmax(M, D) / d)) * -0.5), (h / l), 1.0) * fabs(d)) / sqrt((h * l));
	} else if (t_2 <= 5e+107) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else {
		tmp = (fabs(d) / sqrt((l * h))) * (1.0 - ((((t_1 * (fmax(M, D) / (d + d))) * fmin(M, D)) * h) / l));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(fmin(M, D) * fmax(M, D))
	t_1 = Float64(0.25 * Float64(t_0 / d))
	t_2 = 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(t_0 / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(t_1 * fmin(M, D)) * Float64(fmax(M, D) / d)) * -0.5), Float64(h / l), 1.0) * abs(d)) / sqrt(Float64(h * l)));
	elseif (t_2 <= 5e+107)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	else
		tmp = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * Float64(1.0 - Float64(Float64(Float64(Float64(t_1 * Float64(fmax(M, D) / Float64(d + d))) * fmin(M, D)) * h) / l)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Min[M, D], $MachinePrecision] * N[Max[M, D], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.25 * N[(t$95$0 / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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[(t$95$0 / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(N[(N[(N[(N[(N[(t$95$1 * N[Min[M, D], $MachinePrecision]), $MachinePrecision] * N[(N[Max[M, D], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Abs[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+107], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(t$95$1 * N[(N[Max[M, D], $MachinePrecision] / N[(d + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Min[M, D], $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / l), $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)))) < 0.0

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. associate-/l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. mult-flip N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\left(D \cdot \frac{1}{2 \cdot d}\right)} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \color{blue}{\left(\frac{1}{2 \cdot d} \cdot M\right)}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{1}{\color{blue}{2 \cdot d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-/.f64 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. metadata-eval 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{0.5}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-unsound-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. lower-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. rem-sqrt-square-rev N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. lower-unsound-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. lower-fabs.f64 N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      22. lower-unsound-sqrt.f64 70.2

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      24. *-commutative N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      25. lower-*.f64 70.2

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 70.2%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Evaluated real constant 70.2%

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   8. Applied rewrites 69.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\left(0.25 \cdot \frac{M \cdot D}{d}\right) \cdot M\right) \cdot \frac{D}{d}\right) \cdot -0.5, \frac{h}{\ell}, 1\right) \cdot \left|d\right|}{\sqrt{h \cdot \ell}}} \]

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000002e107

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.9

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.9%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      6. lower-*.f64 31.5

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   7. Applied rewrites 31.5%

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   8. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      3. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      5. lower-/.f64 39.0

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

   10. Applied rewrites 39.0%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

   if 5.0000000000000002e107 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. associate-/l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. mult-flip N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\left(D \cdot \frac{1}{2 \cdot d}\right)} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \color{blue}{\left(\frac{1}{2 \cdot d} \cdot M\right)}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{1}{\color{blue}{2 \cdot d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-/.f64 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. metadata-eval 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{0.5}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-unsound-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. lower-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. rem-sqrt-square-rev N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. lower-unsound-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. lower-fabs.f64 N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      22. lower-unsound-sqrt.f64 70.2

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      24. *-commutative N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      25. lower-*.f64 70.2

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 70.2%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Evaluated real constant 70.2%

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \color{blue}{\left({\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

   8. Applied rewrites 73.3%

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \color{blue}{\frac{\left(\left(\left(0.25 \cdot \frac{M \cdot D}{d}\right) \cdot \frac{D}{d + d}\right) \cdot M\right) \cdot h}{\ell}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 78.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(M, \left|D\right|\right)\\ t_1 := \mathsf{max}\left(M, \left|D\right|\right)\\ t_2 := t\_0 \cdot t\_1\\ t_3 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_2}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(\left(0.25 \cdot \frac{t\_2}{d}\right) \cdot t\_0\right) \cdot \frac{t\_1}{d}\right) \cdot -0.5, \frac{h}{\ell}, 1\right) \cdot \left|d\right|}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+259}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{\left(t\_0 \cdot \left(\frac{t\_1 \cdot \left(t\_1 \cdot t\_0\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot 0.5}{\ell}\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (fmin M (fabs D)))
        (t_1 (fmax M (fabs D)))
        (t_2 (* t_0 t_1))
        (t_3
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ t_2 (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_3 0.0)
     (/
      (*
       (fma (* (* (* (* 0.25 (/ t_2 d)) t_0) (/ t_1 d)) -0.5) (/ h l) 1.0)
       (fabs d))
      (sqrt (* h l)))
     (if (<= t_3 2e+259)
       (* (sqrt (/ d h)) (sqrt (/ d l)))
       (*
        (/ (fabs d) (sqrt (* l h)))
        (-
         1.0
         (/
          (* (* t_0 (* (/ (* t_1 (* t_1 t_0)) (* 4.0 (* d d))) h)) 0.5)
          l)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fmin(M, fabs(D));
	double t_1 = fmax(M, fabs(D));
	double t_2 = t_0 * t_1;
	double t_3 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow((t_2 / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = (fma(((((0.25 * (t_2 / d)) * t_0) * (t_1 / d)) * -0.5), (h / l), 1.0) * fabs(d)) / sqrt((h * l));
	} else if (t_3 <= 2e+259) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else {
		tmp = (fabs(d) / sqrt((l * h))) * (1.0 - (((t_0 * (((t_1 * (t_1 * t_0)) / (4.0 * (d * d))) * h)) * 0.5) / l));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = fmin(M, abs(D))
	t_1 = fmax(M, abs(D))
	t_2 = Float64(t_0 * t_1)
	t_3 = 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(t_2 / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(Float64(0.25 * Float64(t_2 / d)) * t_0) * Float64(t_1 / d)) * -0.5), Float64(h / l), 1.0) * abs(d)) / sqrt(Float64(h * l)));
	elseif (t_3 <= 2e+259)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	else
		tmp = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * Float64(1.0 - Float64(Float64(Float64(t_0 * Float64(Float64(Float64(t_1 * Float64(t_1 * t_0)) / Float64(4.0 * Float64(d * d))) * h)) * 0.5) / l)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Min[M, N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Max[M, N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1), $MachinePrecision]}, Block[{t$95$3 = 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[(t$95$2 / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[(N[(N[(N[(N[(N[(N[(0.25 * N[(t$95$2 / d), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(t$95$1 / d), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Abs[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+259], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(t$95$0 * N[(N[(N[(t$95$1 * N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(4.0 * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] / l), $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)))) < 0.0

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. associate-/l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. mult-flip N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\left(D \cdot \frac{1}{2 \cdot d}\right)} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \color{blue}{\left(\frac{1}{2 \cdot d} \cdot M\right)}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{1}{\color{blue}{2 \cdot d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-/.f64 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. metadata-eval 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{0.5}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-unsound-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. lower-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. rem-sqrt-square-rev N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. lower-unsound-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. lower-fabs.f64 N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      22. lower-unsound-sqrt.f64 70.2

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      24. *-commutative N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      25. lower-*.f64 70.2

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 70.2%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Evaluated real constant 70.2%

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   8. Applied rewrites 69.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\left(0.25 \cdot \frac{M \cdot D}{d}\right) \cdot M\right) \cdot \frac{D}{d}\right) \cdot -0.5, \frac{h}{\ell}, 1\right) \cdot \left|d\right|}{\sqrt{h \cdot \ell}}} \]

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2e259

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.9

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.9%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      6. lower-*.f64 31.5

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   7. Applied rewrites 31.5%

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   8. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      3. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      5. lower-/.f64 39.0

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

   10. Applied rewrites 39.0%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

   if 2e259 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{h}{\ell}}\right) \cdot \frac{1}{2}\right) \]

      6. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell}} \cdot \frac{1}{2}\right) \]

      7. associate-*l/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right) \cdot \frac{1}{2}}{\ell}}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right) \cdot \frac{1}{2}}{\ell}}\right) \]

   3. Applied rewrites 54.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot 0.5}{\ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      9. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      12. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      15. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      16. lower-unsound-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      17. lower-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      19. rem-sqrt-square-rev N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      20. lower-unsound-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      21. lower-fabs.f64 N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      22. lower-unsound-sqrt.f64 60.9

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot 0.5}{\ell}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      24. *-commutative N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      25. lower-*.f64 60.9

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot 0.5}{\ell}\right) \]

   5. Applied rewrites 60.9%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot 0.5}{\ell}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 75.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(M, D\right) \cdot \mathsf{max}\left(M, D\right)\\ t_1 := \frac{\mathsf{fma}\left(\left(\left(\left(0.25 \cdot \frac{t\_0}{d}\right) \cdot \mathsf{min}\left(M, D\right)\right) \cdot \frac{\mathsf{max}\left(M, D\right)}{d}\right) \cdot -0.5, \frac{h}{\ell}, 1\right) \cdot \left|d\right|}{\sqrt{h \cdot \ell}}\\ t_2 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_0}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+107}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (fmin M D) (fmax M D)))
        (t_1
         (/
          (*
           (fma
            (* (* (* (* 0.25 (/ t_0 d)) (fmin M D)) (/ (fmax M D) d)) -0.5)
            (/ h l)
            1.0)
           (fabs d))
          (sqrt (* h l))))
        (t_2
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ t_0 (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_2 0.0)
     t_1
     (if (<= t_2 5e+107) (* (sqrt (/ d h)) (sqrt (/ d l))) t_1))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fmin(M, D) * fmax(M, D);
	double t_1 = (fma(((((0.25 * (t_0 / d)) * fmin(M, D)) * (fmax(M, D) / d)) * -0.5), (h / l), 1.0) * fabs(d)) / sqrt((h * l));
	double t_2 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow((t_0 / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = t_1;
	} else if (t_2 <= 5e+107) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(fmin(M, D) * fmax(M, D))
	t_1 = Float64(Float64(fma(Float64(Float64(Float64(Float64(0.25 * Float64(t_0 / d)) * fmin(M, D)) * Float64(fmax(M, D) / d)) * -0.5), Float64(h / l), 1.0) * abs(d)) / sqrt(Float64(h * l)))
	t_2 = 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(t_0 / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = t_1;
	elseif (t_2 <= 5e+107)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Min[M, D], $MachinePrecision] * N[Max[M, D], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(N[(N[(0.25 * N[(t$95$0 / d), $MachinePrecision]), $MachinePrecision] * N[Min[M, D], $MachinePrecision]), $MachinePrecision] * N[(N[Max[M, D], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Abs[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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[(t$95$0 / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], t$95$1, If[LessEqual[t$95$2, 5e+107], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 5.0000000000000002e107 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. associate-/l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. mult-flip N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\left(D \cdot \frac{1}{2 \cdot d}\right)} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \color{blue}{\left(\frac{1}{2 \cdot d} \cdot M\right)}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{1}{\color{blue}{2 \cdot d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-/.f64 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. metadata-eval 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{0.5}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-unsound-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. lower-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. rem-sqrt-square-rev N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. lower-unsound-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. lower-fabs.f64 N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      22. lower-unsound-sqrt.f64 70.2

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      24. *-commutative N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      25. lower-*.f64 70.2

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 70.2%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Evaluated real constant 70.2%

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   8. Applied rewrites 69.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\left(0.25 \cdot \frac{M \cdot D}{d}\right) \cdot M\right) \cdot \frac{D}{d}\right) \cdot -0.5, \frac{h}{\ell}, 1\right) \cdot \left|d\right|}{\sqrt{h \cdot \ell}}} \]

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000002e107

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.9

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.9%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      6. lower-*.f64 31.5

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   7. Applied rewrites 31.5%

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   8. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      3. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      5. lower-/.f64 39.0

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

   10. Applied rewrites 39.0%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 71.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ t_2 := t\_1 \cdot 1\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+113}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{0.25}{d \cdot d}\right) \cdot -0.5, \frac{h}{\ell}, 1\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+107}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(M, \frac{\frac{\left(D \cdot D\right) \cdot \left(M \cdot h\right)}{d \cdot d} \cdot 0.125}{\ell}, -1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (/ (fabs d) (sqrt (* l h))))
        (t_2 (* t_1 1.0)))
   (if (<= t_0 -1e+113)
     (*
      (fma (* (* (* (* M D) (* M D)) (/ 0.25 (* d d))) -0.5) (/ h l) 1.0)
      t_1)
     (if (<= t_0 0.0)
       t_2
       (if (<= t_0 5e+107)
         (* (sqrt (/ d h)) (sqrt (/ d l)))
         (if (<= t_0 INFINITY)
           t_2
           (-
            (*
             (fma M (/ (* (/ (* (* D D) (* M h)) (* d d)) 0.125) l) -1.0)
             (/ (fabs d) (sqrt (* h l)))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = fabs(d) / sqrt((l * h));
	double t_2 = t_1 * 1.0;
	double tmp;
	if (t_0 <= -1e+113) {
		tmp = fma(((((M * D) * (M * D)) * (0.25 / (d * d))) * -0.5), (h / l), 1.0) * t_1;
	} else if (t_0 <= 0.0) {
		tmp = t_2;
	} else if (t_0 <= 5e+107) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = -(fma(M, (((((D * D) * (M * h)) / (d * d)) * 0.125) / l), -1.0) * (fabs(d) / sqrt((h * l))));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(abs(d) / sqrt(Float64(l * h)))
	t_2 = Float64(t_1 * 1.0)
	tmp = 0.0
	if (t_0 <= -1e+113)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) * Float64(0.25 / Float64(d * d))) * -0.5), Float64(h / l), 1.0) * t_1);
	elseif (t_0 <= 0.0)
		tmp = t_2;
	elseif (t_0 <= 5e+107)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (t_0 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(-Float64(fma(M, Float64(Float64(Float64(Float64(Float64(D * D) * Float64(M * h)) / Float64(d * d)) * 0.125) / l), -1.0) * Float64(abs(d) / sqrt(Float64(h * l)))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+113], N[(N[(N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * N[(0.25 / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 0.0], t$95$2, If[LessEqual[t$95$0, 5e+107], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], t$95$2, (-N[(N[(M * N[(N[(N[(N[(N[(D * D), $MachinePrecision] * N[(M * h), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] / l), $MachinePrecision] + -1.0), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1e113

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. associate-/l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. mult-flip N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\left(D \cdot \frac{1}{2 \cdot d}\right)} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \color{blue}{\left(\frac{1}{2 \cdot d} \cdot M\right)}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{1}{\color{blue}{2 \cdot d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-/.f64 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. metadata-eval 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{0.5}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 56.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{0.25}{d \cdot d}\right) \cdot -0.5, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \]

   if -1e113 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 5.0000000000000002e107 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. associate-/l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. mult-flip N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\left(D \cdot \frac{1}{2 \cdot d}\right)} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \color{blue}{\left(\frac{1}{2 \cdot d} \cdot M\right)}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{1}{\color{blue}{2 \cdot d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-/.f64 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. metadata-eval 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{0.5}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-unsound-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. lower-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. rem-sqrt-square-rev N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. lower-unsound-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. lower-fabs.f64 N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      22. lower-unsound-sqrt.f64 70.2

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      24. *-commutative N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      25. lower-*.f64 70.2

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 70.2%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 43.1%

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \color{blue}{1} \]

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000002e107

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.9

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.9%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      6. lower-*.f64 31.5

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   7. Applied rewrites 31.5%

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   8. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      3. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      5. lower-/.f64 39.0

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

   10. Applied rewrites 39.0%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{h}{\ell}}\right) \cdot \frac{1}{2}\right) \]

      6. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell}} \cdot \frac{1}{2}\right) \]

      7. associate-*l/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right) \cdot \frac{1}{2}}{\ell}}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right) \cdot \frac{1}{2}}{\ell}}\right) \]

   3. Applied rewrites 54.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot 0.5}{\ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      9. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      12. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      14. associate-*r/ N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \sqrt{d \cdot \color{blue}{\frac{d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      17. lift-sqrt.f64 43.5

          \[\leadsto \color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot 0.5}{\ell}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      19. *-commutative N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h \cdot \ell} \cdot d}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      20. lower-*.f64 43.5

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h \cdot \ell} \cdot d}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot 0.5}{\ell}\right) \]

      21. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d}{\color{blue}{h \cdot \ell}} \cdot d} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      22. *-commutative N/A

          \[\leadsto \sqrt{\frac{d}{\color{blue}{\ell \cdot h}} \cdot d} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      23. lower-*.f64 43.5

          \[\leadsto \sqrt{\frac{d}{\color{blue}{\ell \cdot h}} \cdot d} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot 0.5}{\ell}\right) \]

   5. Applied rewrites 43.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell \cdot h} \cdot d}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot 0.5}{\ell}\right) \]

   7. Applied rewrites 56.0%

      \[\leadsto \color{blue}{-\mathsf{fma}\left(M, \frac{\frac{\left(D \cdot D\right) \cdot \left(M \cdot h\right)}{d \cdot d} \cdot 0.125}{\ell}, -1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 14: 70.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ t_2 := t\_1 \cdot 1\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{\left(D \cdot D\right) \cdot M}{d \cdot d} \cdot -0.125\right) \cdot M, \frac{h}{\ell}, 1\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+107}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(M, \frac{\frac{\left(D \cdot D\right) \cdot \left(M \cdot h\right)}{d \cdot d} \cdot 0.125}{\ell}, -1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (/ (fabs d) (sqrt (* l h))))
        (t_2 (* t_1 1.0)))
   (if (<= t_0 -5e+42)
     (* (fma (* (* (/ (* (* D D) M) (* d d)) -0.125) M) (/ h l) 1.0) t_1)
     (if (<= t_0 0.0)
       t_2
       (if (<= t_0 5e+107)
         (* (sqrt (/ d h)) (sqrt (/ d l)))
         (if (<= t_0 INFINITY)
           t_2
           (-
            (*
             (fma M (/ (* (/ (* (* D D) (* M h)) (* d d)) 0.125) l) -1.0)
             (/ (fabs d) (sqrt (* h l)))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = fabs(d) / sqrt((l * h));
	double t_2 = t_1 * 1.0;
	double tmp;
	if (t_0 <= -5e+42) {
		tmp = fma((((((D * D) * M) / (d * d)) * -0.125) * M), (h / l), 1.0) * t_1;
	} else if (t_0 <= 0.0) {
		tmp = t_2;
	} else if (t_0 <= 5e+107) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = -(fma(M, (((((D * D) * (M * h)) / (d * d)) * 0.125) / l), -1.0) * (fabs(d) / sqrt((h * l))));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(abs(d) / sqrt(Float64(l * h)))
	t_2 = Float64(t_1 * 1.0)
	tmp = 0.0
	if (t_0 <= -5e+42)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(Float64(D * D) * M) / Float64(d * d)) * -0.125) * M), Float64(h / l), 1.0) * t_1);
	elseif (t_0 <= 0.0)
		tmp = t_2;
	elseif (t_0 <= 5e+107)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (t_0 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(-Float64(fma(M, Float64(Float64(Float64(Float64(Float64(D * D) * Float64(M * h)) / Float64(d * d)) * 0.125) / l), -1.0) * Float64(abs(d) / sqrt(Float64(h * l)))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+42], N[(N[(N[(N[(N[(N[(N[(D * D), $MachinePrecision] * M), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * M), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 0.0], t$95$2, If[LessEqual[t$95$0, 5e+107], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], t$95$2, (-N[(N[(M * N[(N[(N[(N[(N[(D * D), $MachinePrecision] * N[(M * h), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] / l), $MachinePrecision] + -1.0), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5.00000000000000007e42

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 29.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{h}{\ell}, \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot -0.5\right)\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}, \sqrt{d \cdot \frac{d}{h \cdot \ell}}\right)} \]
   4. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\frac{h}{\ell} \cdot \left(\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{-1}{2}\right)\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}\right) + \sqrt{d \cdot \frac{d}{h \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{h}{\ell} \cdot \color{blue}{\left(\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{-1}{2}\right)\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}\right)} + \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{h}{\ell} \cdot \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}} + \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      4. distribute-lft1-in N/A

         \[\leadsto \color{blue}{\left(\frac{h}{\ell} \cdot \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{-1}{2}\right)\right) + 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{h}{\ell} \cdot \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{-1}{2}\right)\right) + 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}} \]

   5. Applied rewrites 52.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{\left(D \cdot D\right) \cdot M}{d \cdot d} \cdot -0.125\right) \cdot M, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \]

   if -5.00000000000000007e42 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 5.0000000000000002e107 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. associate-/l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. mult-flip N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\left(D \cdot \frac{1}{2 \cdot d}\right)} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \color{blue}{\left(\frac{1}{2 \cdot d} \cdot M\right)}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{1}{\color{blue}{2 \cdot d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-/.f64 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. metadata-eval 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{0.5}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-unsound-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. lower-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. rem-sqrt-square-rev N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. lower-unsound-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. lower-fabs.f64 N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      22. lower-unsound-sqrt.f64 70.2

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      24. *-commutative N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      25. lower-*.f64 70.2

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 70.2%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 43.1%

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \color{blue}{1} \]

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000002e107

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.9

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.9%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      6. lower-*.f64 31.5

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   7. Applied rewrites 31.5%

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   8. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      3. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      5. lower-/.f64 39.0

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

   10. Applied rewrites 39.0%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{h}{\ell}}\right) \cdot \frac{1}{2}\right) \]

      6. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell}} \cdot \frac{1}{2}\right) \]

      7. associate-*l/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right) \cdot \frac{1}{2}}{\ell}}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right) \cdot \frac{1}{2}}{\ell}}\right) \]

   3. Applied rewrites 54.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot 0.5}{\ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      9. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      12. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      14. associate-*r/ N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \sqrt{d \cdot \color{blue}{\frac{d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      17. lift-sqrt.f64 43.5

          \[\leadsto \color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot 0.5}{\ell}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      19. *-commutative N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h \cdot \ell} \cdot d}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      20. lower-*.f64 43.5

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h \cdot \ell} \cdot d}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot 0.5}{\ell}\right) \]

      21. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d}{\color{blue}{h \cdot \ell}} \cdot d} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      22. *-commutative N/A

          \[\leadsto \sqrt{\frac{d}{\color{blue}{\ell \cdot h}} \cdot d} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      23. lower-*.f64 43.5

          \[\leadsto \sqrt{\frac{d}{\color{blue}{\ell \cdot h}} \cdot d} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot 0.5}{\ell}\right) \]

   5. Applied rewrites 43.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell \cdot h} \cdot d}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot 0.5}{\ell}\right) \]

   7. Applied rewrites 56.0%

      \[\leadsto \color{blue}{-\mathsf{fma}\left(M, \frac{\frac{\left(D \cdot D\right) \cdot \left(M \cdot h\right)}{d \cdot d} \cdot 0.125}{\ell}, -1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 15: 70.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ t_2 := t\_1 \cdot 1\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{\left(D \cdot D\right) \cdot M}{d \cdot d} \cdot -0.125\right) \cdot M, \frac{h}{\ell}, 1\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+107}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(\ell - \left(\frac{\left(D \cdot D\right) \cdot \left(M \cdot h\right)}{d \cdot d} \cdot 0.125\right) \cdot M\right)}{\ell}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (/ (fabs d) (sqrt (* l h))))
        (t_2 (* t_1 1.0)))
   (if (<= t_0 -5e+42)
     (* (fma (* (* (/ (* (* D D) M) (* d d)) -0.125) M) (/ h l) 1.0) t_1)
     (if (<= t_0 0.0)
       t_2
       (if (<= t_0 5e+107)
         (* (sqrt (/ d h)) (sqrt (/ d l)))
         (if (<= t_0 INFINITY)
           t_2
           (/
            (*
             (/ (fabs d) (sqrt (* h l)))
             (- l (* (* (/ (* (* D D) (* M h)) (* d d)) 0.125) M)))
            l)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = fabs(d) / sqrt((l * h));
	double t_2 = t_1 * 1.0;
	double tmp;
	if (t_0 <= -5e+42) {
		tmp = fma((((((D * D) * M) / (d * d)) * -0.125) * M), (h / l), 1.0) * t_1;
	} else if (t_0 <= 0.0) {
		tmp = t_2;
	} else if (t_0 <= 5e+107) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = ((fabs(d) / sqrt((h * l))) * (l - (((((D * D) * (M * h)) / (d * d)) * 0.125) * M))) / l;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(abs(d) / sqrt(Float64(l * h)))
	t_2 = Float64(t_1 * 1.0)
	tmp = 0.0
	if (t_0 <= -5e+42)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(Float64(D * D) * M) / Float64(d * d)) * -0.125) * M), Float64(h / l), 1.0) * t_1);
	elseif (t_0 <= 0.0)
		tmp = t_2;
	elseif (t_0 <= 5e+107)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (t_0 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(Float64(Float64(abs(d) / sqrt(Float64(h * l))) * Float64(l - Float64(Float64(Float64(Float64(Float64(D * D) * Float64(M * h)) / Float64(d * d)) * 0.125) * M))) / l);
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+42], N[(N[(N[(N[(N[(N[(N[(D * D), $MachinePrecision] * M), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * M), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 0.0], t$95$2, If[LessEqual[t$95$0, 5e+107], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], t$95$2, N[(N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(l - N[(N[(N[(N[(N[(D * D), $MachinePrecision] * N[(M * h), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5.00000000000000007e42

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 29.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{h}{\ell}, \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot -0.5\right)\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}, \sqrt{d \cdot \frac{d}{h \cdot \ell}}\right)} \]
   4. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\frac{h}{\ell} \cdot \left(\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{-1}{2}\right)\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}\right) + \sqrt{d \cdot \frac{d}{h \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{h}{\ell} \cdot \color{blue}{\left(\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{-1}{2}\right)\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}\right)} + \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{h}{\ell} \cdot \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}} + \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      4. distribute-lft1-in N/A

         \[\leadsto \color{blue}{\left(\frac{h}{\ell} \cdot \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{-1}{2}\right)\right) + 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{h}{\ell} \cdot \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{-1}{2}\right)\right) + 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}} \]

   5. Applied rewrites 52.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{\left(D \cdot D\right) \cdot M}{d \cdot d} \cdot -0.125\right) \cdot M, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \]

   if -5.00000000000000007e42 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 5.0000000000000002e107 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. associate-/l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. mult-flip N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\left(D \cdot \frac{1}{2 \cdot d}\right)} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \color{blue}{\left(\frac{1}{2 \cdot d} \cdot M\right)}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{1}{\color{blue}{2 \cdot d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-/.f64 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. metadata-eval 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{0.5}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-unsound-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. lower-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. rem-sqrt-square-rev N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. lower-unsound-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. lower-fabs.f64 N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      22. lower-unsound-sqrt.f64 70.2

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      24. *-commutative N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      25. lower-*.f64 70.2

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 70.2%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 43.1%

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \color{blue}{1} \]

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000002e107

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.9

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.9%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      6. lower-*.f64 31.5

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   7. Applied rewrites 31.5%

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   8. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      3. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      5. lower-/.f64 39.0

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

   10. Applied rewrites 39.0%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{h}{\ell}}\right) \cdot \frac{1}{2}\right) \]

      6. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell}} \cdot \frac{1}{2}\right) \]

      7. associate-*l/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right) \cdot \frac{1}{2}}{\ell}}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right) \cdot \frac{1}{2}}{\ell}}\right) \]

   3. Applied rewrites 54.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot 0.5}{\ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      9. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      12. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      14. associate-*r/ N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \sqrt{d \cdot \color{blue}{\frac{d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      17. lift-sqrt.f64 43.5

          \[\leadsto \color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot 0.5}{\ell}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      19. *-commutative N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h \cdot \ell} \cdot d}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      20. lower-*.f64 43.5

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h \cdot \ell} \cdot d}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot 0.5}{\ell}\right) \]

      21. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d}{\color{blue}{h \cdot \ell}} \cdot d} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      22. *-commutative N/A

          \[\leadsto \sqrt{\frac{d}{\color{blue}{\ell \cdot h}} \cdot d} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot \frac{1}{2}}{\ell}\right) \]

      23. lower-*.f64 43.5

          \[\leadsto \sqrt{\frac{d}{\color{blue}{\ell \cdot h}} \cdot d} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot 0.5}{\ell}\right) \]

   5. Applied rewrites 43.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell \cdot h} \cdot d}} \cdot \left(1 - \frac{\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot h\right)\right) \cdot 0.5}{\ell}\right) \]

   7. Applied rewrites 52.1%

      \[\leadsto \color{blue}{\frac{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(\ell - \left(\frac{\left(D \cdot D\right) \cdot \left(M \cdot h\right)}{d \cdot d} \cdot 0.125\right) \cdot M\right)}{\ell}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 16: 66.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ t_2 := t\_1 \cdot 1\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{\left(D \cdot D\right) \cdot M}{d \cdot d} \cdot -0.125\right) \cdot M, \frac{h}{\ell}, 1\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+107}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (/ (fabs d) (sqrt (* l h))))
        (t_2 (* t_1 1.0)))
   (if (<= t_0 -5e+42)
     (* (fma (* (* (/ (* (* D D) M) (* d d)) -0.125) M) (/ h l) 1.0) t_1)
     (if (<= t_0 0.0)
       t_2
       (if (<= t_0 5e+107) (* (sqrt (/ d h)) (sqrt (/ d l))) t_2)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = fabs(d) / sqrt((l * h));
	double t_2 = t_1 * 1.0;
	double tmp;
	if (t_0 <= -5e+42) {
		tmp = fma((((((D * D) * M) / (d * d)) * -0.125) * M), (h / l), 1.0) * t_1;
	} else if (t_0 <= 0.0) {
		tmp = t_2;
	} else if (t_0 <= 5e+107) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(abs(d) / sqrt(Float64(l * h)))
	t_2 = Float64(t_1 * 1.0)
	tmp = 0.0
	if (t_0 <= -5e+42)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(Float64(D * D) * M) / Float64(d * d)) * -0.125) * M), Float64(h / l), 1.0) * t_1);
	elseif (t_0 <= 0.0)
		tmp = t_2;
	elseif (t_0 <= 5e+107)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+42], N[(N[(N[(N[(N[(N[(N[(D * D), $MachinePrecision] * M), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * M), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 0.0], t$95$2, If[LessEqual[t$95$0, 5e+107], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

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)))) < -5.00000000000000007e42

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 29.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{h}{\ell}, \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot -0.5\right)\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}, \sqrt{d \cdot \frac{d}{h \cdot \ell}}\right)} \]
   4. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\frac{h}{\ell} \cdot \left(\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{-1}{2}\right)\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}\right) + \sqrt{d \cdot \frac{d}{h \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{h}{\ell} \cdot \color{blue}{\left(\left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{-1}{2}\right)\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}\right)} + \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{h}{\ell} \cdot \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}} + \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      4. distribute-lft1-in N/A

         \[\leadsto \color{blue}{\left(\frac{h}{\ell} \cdot \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{-1}{2}\right)\right) + 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{h}{\ell} \cdot \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{-1}{2}\right)\right) + 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}} \]

   5. Applied rewrites 52.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{\left(D \cdot D\right) \cdot M}{d \cdot d} \cdot -0.125\right) \cdot M, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \]

   if -5.00000000000000007e42 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 5.0000000000000002e107 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. associate-/l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. mult-flip N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\left(D \cdot \frac{1}{2 \cdot d}\right)} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \color{blue}{\left(\frac{1}{2 \cdot d} \cdot M\right)}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{1}{\color{blue}{2 \cdot d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-/.f64 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. metadata-eval 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{0.5}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-unsound-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. lower-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. rem-sqrt-square-rev N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. lower-unsound-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. lower-fabs.f64 N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      22. lower-unsound-sqrt.f64 70.2

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      24. *-commutative N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      25. lower-*.f64 70.2

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 70.2%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 43.1%

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \color{blue}{1} \]

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000002e107

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.9

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.9%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      6. lower-*.f64 31.5

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   7. Applied rewrites 31.5%

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   8. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      3. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      5. lower-/.f64 39.0

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

   10. Applied rewrites 39.0%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 55.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{h \cdot \ell}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{\sqrt{t\_0 \cdot t\_0}} \cdot \left(-d\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+107}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* h l)))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_1 0.0)
     (* (sqrt (sqrt (* t_0 t_0))) (- d))
     (if (<= t_1 5e+107)
       (* (sqrt (/ d h)) (sqrt (/ d l)))
       (* (/ (fabs d) (sqrt (* l h))) 1.0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 / (h * l);
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = sqrt(sqrt((t_0 * t_0))) * -d;
	} else if (t_1 <= 5e+107) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 / (h * l)
    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)))
    if (t_1 <= 0.0d0) then
        tmp = sqrt(sqrt((t_0 * t_0))) * -d
    else if (t_1 <= 5d+107) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    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 t_0 = 1.0 / (h * l);
	double t_1 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = Math.sqrt(Math.sqrt((t_0 * t_0))) * -d;
	} else if (t_1 <= 5e+107) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else {
		tmp = (Math.abs(d) / Math.sqrt((l * h))) * 1.0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = 1.0 / (h * l)
	t_1 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_1 <= 0.0:
		tmp = math.sqrt(math.sqrt((t_0 * t_0))) * -d
	elif t_1 <= 5e+107:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	else:
		tmp = (math.fabs(d) / math.sqrt((l * h))) * 1.0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(1.0 / Float64(h * l))
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(sqrt(sqrt(Float64(t_0 * t_0))) * Float64(-d));
	elseif (t_1 <= 5e+107)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	else
		tmp = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = 1.0 / (h * l);
	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)));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = sqrt(sqrt((t_0 * t_0))) * -d;
	elseif (t_1 <= 5e+107)
		tmp = sqrt((d / h)) * sqrt((d / l));
	else
		tmp = (abs(d) / sqrt((l * h))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[Sqrt[N[Sqrt[N[(t$95$0 * t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], If[LessEqual[t$95$1, 5e+107], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 29.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{h}{\ell}, \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot -0.5\right)\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}, \sqrt{d \cdot \frac{d}{h \cdot \ell}}\right)} \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. lower-*.f64 26.2

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   6. Applied rewrites 26.2%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      7. lower-neg.f64 26.2

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right) \]

   8. Applied rewrites 26.2%

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]
   9. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \sqrt{\sqrt{\frac{1}{h \cdot \ell}} \cdot \sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(-d\right) \]

      2. sqrt-unprod N/A

         \[\leadsto \sqrt{\sqrt{\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}}} \cdot \left(-d\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\sqrt{\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}}} \cdot \left(-d\right) \]

      4. lower-*.f64 22.5

         \[\leadsto \sqrt{\sqrt{\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}}} \cdot \left(-d\right) \]

   10. Applied rewrites 22.5%

       \[\leadsto \sqrt{\sqrt{\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}}} \cdot \left(-d\right) \]

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000002e107

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.9

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.9%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      6. lower-*.f64 31.5

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   7. Applied rewrites 31.5%

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   8. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      3. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      5. lower-/.f64 39.0

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

   10. Applied rewrites 39.0%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

   if 5.0000000000000002e107 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. associate-/l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. mult-flip N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\left(D \cdot \frac{1}{2 \cdot d}\right)} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \color{blue}{\left(\frac{1}{2 \cdot d} \cdot M\right)}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{1}{\color{blue}{2 \cdot d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-/.f64 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. metadata-eval 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{0.5}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-unsound-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. lower-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. rem-sqrt-square-rev N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. lower-unsound-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. lower-fabs.f64 N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      22. lower-unsound-sqrt.f64 70.2

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      24. *-commutative N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      25. lower-*.f64 70.2

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 70.2%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 43.1%

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 54.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-250}:\\ \;\;\;\;-1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+107}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ (fabs d) (sqrt (* l h))) 1.0))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_1 -2e-250)
     (* -1.0 (/ (* d (sqrt (/ h l))) h))
     (if (<= t_1 0.0)
       t_0
       (if (<= t_1 5e+107) (* (sqrt (/ d h)) (sqrt (/ d l))) t_0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (fabs(d) / sqrt((l * h))) * 1.0;
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= -2e-250) {
		tmp = -1.0 * ((d * sqrt((h / l))) / h);
	} else if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 5e+107) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else {
		tmp = 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) :: tmp
    t_0 = (abs(d) / sqrt((l * h))) * 1.0d0
    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)))
    if (t_1 <= (-2d-250)) then
        tmp = (-1.0d0) * ((d * sqrt((h / l))) / h)
    else if (t_1 <= 0.0d0) then
        tmp = t_0
    else if (t_1 <= 5d+107) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else
        tmp = 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 = (Math.abs(d) / Math.sqrt((l * h))) * 1.0;
	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 tmp;
	if (t_1 <= -2e-250) {
		tmp = -1.0 * ((d * Math.sqrt((h / l))) / h);
	} else if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 5e+107) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.fabs(d) / math.sqrt((l * h))) * 1.0
	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)))
	tmp = 0
	if t_1 <= -2e-250:
		tmp = -1.0 * ((d * math.sqrt((h / l))) / h)
	elif t_1 <= 0.0:
		tmp = t_0
	elif t_1 <= 5e+107:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	else:
		tmp = t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * 1.0)
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_1 <= -2e-250)
		tmp = Float64(-1.0 * Float64(Float64(d * sqrt(Float64(h / l))) / h));
	elseif (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 5e+107)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (abs(d) / sqrt((l * h))) * 1.0;
	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)));
	tmp = 0.0;
	if (t_1 <= -2e-250)
		tmp = -1.0 * ((d * sqrt((h / l))) / h);
	elseif (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 5e+107)
		tmp = sqrt((d / h)) * sqrt((d / l));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-250], N[(-1.0 * N[(N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], t$95$0, If[LessEqual[t$95$1, 5e+107], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -2.0000000000000001e-250

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 29.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{h}{\ell}, \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot -0.5\right)\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}, \sqrt{d \cdot \frac{d}{h \cdot \ell}}\right)} \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. lower-*.f64 26.2

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   6. Applied rewrites 26.2%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   7. Taylor expanded in h around 0

      \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      4. lower-/.f64 14.1

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   9. Applied rewrites 14.1%

      \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

   if -2.0000000000000001e-250 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 5.0000000000000002e107 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. associate-/l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. mult-flip N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\left(D \cdot \frac{1}{2 \cdot d}\right)} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \color{blue}{\left(\frac{1}{2 \cdot d} \cdot M\right)}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{1}{\color{blue}{2 \cdot d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-/.f64 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. metadata-eval 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{0.5}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-unsound-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. lower-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. rem-sqrt-square-rev N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. lower-unsound-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. lower-fabs.f64 N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      22. lower-unsound-sqrt.f64 70.2

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      24. *-commutative N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      25. lower-*.f64 70.2

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 70.2%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 43.1%

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \color{blue}{1} \]

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000002e107

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.9

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.9%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      6. lower-*.f64 31.5

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   7. Applied rewrites 31.5%

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   8. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      3. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      5. lower-/.f64 39.0

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

   10. Applied rewrites 39.0%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 52.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-250}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}}}{h} \cdot \left(-d\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+107}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ (fabs d) (sqrt (* l h))) 1.0))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_1 -2e-250)
     (* (/ (sqrt (/ h l)) h) (- d))
     (if (<= t_1 0.0)
       t_0
       (if (<= t_1 5e+107) (* (sqrt (/ d h)) (sqrt (/ d l))) t_0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (fabs(d) / sqrt((l * h))) * 1.0;
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= -2e-250) {
		tmp = (sqrt((h / l)) / h) * -d;
	} else if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 5e+107) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else {
		tmp = 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) :: tmp
    t_0 = (abs(d) / sqrt((l * h))) * 1.0d0
    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)))
    if (t_1 <= (-2d-250)) then
        tmp = (sqrt((h / l)) / h) * -d
    else if (t_1 <= 0.0d0) then
        tmp = t_0
    else if (t_1 <= 5d+107) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else
        tmp = 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 = (Math.abs(d) / Math.sqrt((l * h))) * 1.0;
	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 tmp;
	if (t_1 <= -2e-250) {
		tmp = (Math.sqrt((h / l)) / h) * -d;
	} else if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 5e+107) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.fabs(d) / math.sqrt((l * h))) * 1.0
	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)))
	tmp = 0
	if t_1 <= -2e-250:
		tmp = (math.sqrt((h / l)) / h) * -d
	elif t_1 <= 0.0:
		tmp = t_0
	elif t_1 <= 5e+107:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	else:
		tmp = t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * 1.0)
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_1 <= -2e-250)
		tmp = Float64(Float64(sqrt(Float64(h / l)) / h) * Float64(-d));
	elseif (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 5e+107)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (abs(d) / sqrt((l * h))) * 1.0;
	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)));
	tmp = 0.0;
	if (t_1 <= -2e-250)
		tmp = (sqrt((h / l)) / h) * -d;
	elseif (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 5e+107)
		tmp = sqrt((d / h)) * sqrt((d / l));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-250], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] / h), $MachinePrecision] * (-d)), $MachinePrecision], If[LessEqual[t$95$1, 0.0], t$95$0, If[LessEqual[t$95$1, 5e+107], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -2.0000000000000001e-250

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 29.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{h}{\ell}, \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot -0.5\right)\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}, \sqrt{d \cdot \frac{d}{h \cdot \ell}}\right)} \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. lower-*.f64 26.2

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   6. Applied rewrites 26.2%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      7. lower-neg.f64 26.2

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right) \]

   8. Applied rewrites 26.2%

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   9. Taylor expanded in h around 0

      \[\leadsto \frac{\sqrt{\frac{h}{\ell}}}{h} \cdot \left(-\color{blue}{d}\right) \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{h}{\ell}}}{h} \cdot \left(-d\right) \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{h}{\ell}}}{h} \cdot \left(-d\right) \]

       3. lower-/.f64 13.6

          \[\leadsto \frac{\sqrt{\frac{h}{\ell}}}{h} \cdot \left(-d\right) \]

   11. Applied rewrites 13.6%

       \[\leadsto \frac{\sqrt{\frac{h}{\ell}}}{h} \cdot \left(-\color{blue}{d}\right) \]

   if -2.0000000000000001e-250 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 5.0000000000000002e107 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. associate-/l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. mult-flip N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\left(D \cdot \frac{1}{2 \cdot d}\right)} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \color{blue}{\left(\frac{1}{2 \cdot d} \cdot M\right)}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{1}{\color{blue}{2 \cdot d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-/.f64 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. metadata-eval 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{0.5}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-unsound-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. lower-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. rem-sqrt-square-rev N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. lower-unsound-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. lower-fabs.f64 N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      22. lower-unsound-sqrt.f64 70.2

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      24. *-commutative N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      25. lower-*.f64 70.2

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 70.2%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 43.1%

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \color{blue}{1} \]

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000002e107

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.9

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.9%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      6. lower-*.f64 31.5

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   7. Applied rewrites 31.5%

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   8. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      3. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      5. lower-/.f64 39.0

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

   10. Applied rewrites 39.0%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 50.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -2 \cdot 10^{-250}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}}}{h} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1\\ \end{array} \]

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))))
      -2e-250)
   (* (/ (sqrt (/ h l)) h) (- d))
   (* (/ (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)))) <= -2e-250) {
		tmp = (sqrt((h / l)) / h) * -d;
	} 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)))) <= (-2d-250)) then
        tmp = (sqrt((h / l)) / h) * -d
    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)))) <= -2e-250) {
		tmp = (Math.sqrt((h / l)) / h) * -d;
	} 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)))) <= -2e-250:
		tmp = (math.sqrt((h / l)) / h) * -d
	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)))) <= -2e-250)
		tmp = Float64(Float64(sqrt(Float64(h / l)) / h) * Float64(-d));
	else
		tmp = Float64(Float64(abs(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)))) <= -2e-250)
		tmp = (sqrt((h / l)) / h) * -d;
	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], -2e-250], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] / h), $MachinePrecision] * (-d)), $MachinePrecision], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $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)))) < -2.0000000000000001e-250

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 29.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{h}{\ell}, \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot -0.5\right)\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}, \sqrt{d \cdot \frac{d}{h \cdot \ell}}\right)} \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. lower-*.f64 26.2

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   6. Applied rewrites 26.2%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      7. lower-neg.f64 26.2

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right) \]

   8. Applied rewrites 26.2%

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   9. Taylor expanded in h around 0

      \[\leadsto \frac{\sqrt{\frac{h}{\ell}}}{h} \cdot \left(-\color{blue}{d}\right) \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{h}{\ell}}}{h} \cdot \left(-d\right) \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{h}{\ell}}}{h} \cdot \left(-d\right) \]

       3. lower-/.f64 13.6

          \[\leadsto \frac{\sqrt{\frac{h}{\ell}}}{h} \cdot \left(-d\right) \]

   11. Applied rewrites 13.6%

       \[\leadsto \frac{\sqrt{\frac{h}{\ell}}}{h} \cdot \left(-\color{blue}{d}\right) \]

   if -2.0000000000000001e-250 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. associate-/l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. mult-flip N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\left(D \cdot \frac{1}{2 \cdot d}\right)} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \color{blue}{\left(\frac{1}{2 \cdot d} \cdot M\right)}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{1}{\color{blue}{2 \cdot d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-/.f64 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. metadata-eval 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{0.5}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-unsound-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. lower-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. rem-sqrt-square-rev N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. lower-unsound-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. lower-fabs.f64 N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      22. lower-unsound-sqrt.f64 70.2

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      24. *-commutative N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      25. lower-*.f64 70.2

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 70.2%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 43.1%

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 46.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-156}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1\\ \end{array} \]

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))))
      -5e-156)
   (* d (sqrt (/ 1.0 (* h l))))
   (* (/ (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)))) <= -5e-156) {
		tmp = d * sqrt((1.0 / (h * l)));
	} 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)))) <= (-5d-156)) then
        tmp = d * sqrt((1.0d0 / (h * l)))
    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)))) <= -5e-156) {
		tmp = d * Math.sqrt((1.0 / (h * l)));
	} 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)))) <= -5e-156:
		tmp = d * math.sqrt((1.0 / (h * l)))
	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)))) <= -5e-156)
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(h * l))));
	else
		tmp = Float64(Float64(abs(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)))) <= -5e-156)
		tmp = d * sqrt((1.0 / (h * l)));
	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], -5e-156], N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $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)))) < -5.00000000000000007e-156

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 29.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{h}{\ell}, \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot -0.5\right)\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}, \sqrt{d \cdot \frac{d}{h \cdot \ell}}\right)} \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. lower-*.f64 26.2

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   6. Applied rewrites 26.2%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      7. lower-neg.f64 26.2

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right) \]

   8. Applied rewrites 26.2%

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   9. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

       2. lower-sqrt.f64 N/A

          \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       3. lower-/.f64 N/A

          \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       4. lower-*.f64 27.0

          \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   11. Applied rewrites 27.0%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   if -5.00000000000000007e-156 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. associate-/l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. mult-flip N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\left(D \cdot \frac{1}{2 \cdot d}\right)} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{2 \cdot d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \color{blue}{\left(\frac{1}{2 \cdot d} \cdot M\right)}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{1}{\color{blue}{2 \cdot d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-/.f64 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. metadata-eval 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\color{blue}{0.5}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-unsound-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. lower-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. rem-sqrt-square-rev N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. lower-unsound-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. lower-fabs.f64 N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      22. lower-unsound-sqrt.f64 70.2

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      24. *-commutative N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      25. lower-*.f64 70.2

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 70.2%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 43.1%

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 41.6% accurate, 5.5× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{if}\;d \leq 1.65 \cdot 10^{-127}:\\ \;\;\;\;t\_0 \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot t\_0\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (* h l)))))
   (if (<= d 1.65e-127) (* t_0 (- d)) (* d t_0))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((1.0 / (h * l)));
	double tmp;
	if (d <= 1.65e-127) {
		tmp = t_0 * -d;
	} else {
		tmp = d * 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) :: tmp
    t_0 = sqrt((1.0d0 / (h * l)))
    if (d <= 1.65d-127) then
        tmp = t_0 * -d
    else
        tmp = d * 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 = Math.sqrt((1.0 / (h * l)));
	double tmp;
	if (d <= 1.65e-127) {
		tmp = t_0 * -d;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((1.0 / (h * l)))
	tmp = 0
	if d <= 1.65e-127:
		tmp = t_0 * -d
	else:
		tmp = d * t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(1.0 / Float64(h * l)))
	tmp = 0.0
	if (d <= 1.65e-127)
		tmp = Float64(t_0 * Float64(-d));
	else
		tmp = Float64(d * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((1.0 / (h * l)));
	tmp = 0.0;
	if (d <= 1.65e-127)
		tmp = t_0 * -d;
	else
		tmp = d * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, 1.65e-127], N[(t$95$0 * (-d)), $MachinePrecision], N[(d * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < 1.6499999999999999e-127

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 29.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{h}{\ell}, \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot -0.5\right)\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}, \sqrt{d \cdot \frac{d}{h \cdot \ell}}\right)} \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. lower-*.f64 26.2

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   6. Applied rewrites 26.2%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      7. lower-neg.f64 26.2

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right) \]

   8. Applied rewrites 26.2%

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   if 1.6499999999999999e-127 < d

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 29.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{h}{\ell}, \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot -0.5\right)\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}, \sqrt{d \cdot \frac{d}{h \cdot \ell}}\right)} \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. lower-*.f64 26.2

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   6. Applied rewrites 26.2%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      7. lower-neg.f64 26.2

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right) \]

   8. Applied rewrites 26.2%

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   9. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

       2. lower-sqrt.f64 N/A

          \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       3. lower-/.f64 N/A

          \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       4. lower-*.f64 27.0

          \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   11. Applied rewrites 27.0%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 27.0% accurate, 7.7× speedup

Math

\[d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

FPCore

(FPCore (d h l M D) :precision binary64 (* d (sqrt (/ 1.0 (* h l)))))

C

double code(double d, double h, double l, double M, double D) {
	return d * sqrt((1.0 / (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 * sqrt((1.0d0 / (h * l)))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return d * Math.sqrt((1.0 / (h * l)));
}

Python

def code(d, h, l, M, D):
	return d * math.sqrt((1.0 / (h * l)))

Julia

function code(d, h, l, M, D)
	return Float64(d * sqrt(Float64(1.0 / Float64(h * l))))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = d * sqrt((1.0 / (h * l)));
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.3%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Applied rewrites 29.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{h}{\ell}, \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot -0.5\right)\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}, \sqrt{d \cdot \frac{d}{h \cdot \ell}}\right)} \]

4. Taylor expanded in d around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
5. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto -1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \]

   3. lower-sqrt.f64 N/A

      \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   4. lower-/.f64 N/A

      \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   5. lower-*.f64 26.2

      \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

6. Applied rewrites 26.2%

   \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right) \]

   5. distribute-rgt-neg-in N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

   7. lower-neg.f64 26.2

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right) \]

8. Applied rewrites 26.2%

   \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

9. Taylor expanded in d around inf

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
10. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

    2. lower-sqrt.f64 N/A

       \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

    3. lower-/.f64 N/A

       \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

    4. lower-*.f64 27.0

       \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

11. Applied rewrites 27.0%

    \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025175 
(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)))))