Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.4% → 76.7%

Time: 11.7s

Alternatives: 19

Speedup: 1.2×

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: 66.4% 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: 76.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_1 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_2 := -0.25 \cdot t\_1\\ t_3 := \frac{t\_1}{d + d} \cdot t\_0\\ \mathbf{if}\;d \leq -4.6 \cdot 10^{-303}:\\ \;\;\;\;-\mathsf{fma}\left(\left(\frac{\left(t\_0 \cdot t\_2\right) \cdot h}{\ell \cdot d} \cdot \frac{-0.5}{d}\right) \cdot t\_0, t\_1, -1\right) \cdot \frac{-d}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;d \leq 4.8 \cdot 10^{-222}:\\ \;\;\;\;\frac{\left(d \cdot \sqrt{\frac{h}{d}}\right) \cdot \left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{d \cdot \ell}}\right)\right)}{h}\\ \mathbf{elif}\;d \leq 3 \cdot 10^{+156}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{t\_0}{d} \cdot t\_2}{d} \cdot \frac{\left(\left(t\_0 \cdot t\_1\right) \cdot -0.5\right) \cdot h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{d} \cdot \sqrt{\frac{d}{h \cdot \ell}}\right) \cdot \left(1 - \left(t\_3 \cdot \left(t\_3 \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (fmin (fabs M) (fabs D)))
        (t_1 (fmax (fabs M) (fabs D)))
        (t_2 (* -0.25 t_1))
        (t_3 (* (/ t_1 (+ d d)) t_0)))
   (if (<= d -4.6e-303)
     (-
      (*
       (fma (* (* (/ (* (* t_0 t_2) h) (* l d)) (/ -0.5 d)) t_0) t_1 -1.0)
       (/ (- d) (sqrt (* h l)))))
     (if (<= d 4.8e-222)
       (/ (* (* d (sqrt (/ h d))) (* -1.0 (* d (sqrt (/ 1.0 (* d l)))))) h)
       (if (<= d 3e+156)
         (*
          (* (sqrt (/ d l)) (sqrt (/ d h)))
          (- 1.0 (* (/ (* (/ t_0 d) t_2) d) (/ (* (* (* t_0 t_1) -0.5) h) l))))
         (*
          (* (sqrt d) (sqrt (/ d (* h l))))
          (- 1.0 (* (* t_3 (* t_3 0.5)) (/ h l)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fmin(fabs(M), fabs(D));
	double t_1 = fmax(fabs(M), fabs(D));
	double t_2 = -0.25 * t_1;
	double t_3 = (t_1 / (d + d)) * t_0;
	double tmp;
	if (d <= -4.6e-303) {
		tmp = -(fma((((((t_0 * t_2) * h) / (l * d)) * (-0.5 / d)) * t_0), t_1, -1.0) * (-d / sqrt((h * l))));
	} else if (d <= 4.8e-222) {
		tmp = ((d * sqrt((h / d))) * (-1.0 * (d * sqrt((1.0 / (d * l)))))) / h;
	} else if (d <= 3e+156) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((((t_0 / d) * t_2) / d) * ((((t_0 * t_1) * -0.5) * h) / l)));
	} else {
		tmp = (sqrt(d) * sqrt((d / (h * l)))) * (1.0 - ((t_3 * (t_3 * 0.5)) * (h / l)));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = fmin(abs(M), abs(D))
	t_1 = fmax(abs(M), abs(D))
	t_2 = Float64(-0.25 * t_1)
	t_3 = Float64(Float64(t_1 / Float64(d + d)) * t_0)
	tmp = 0.0
	if (d <= -4.6e-303)
		tmp = Float64(-Float64(fma(Float64(Float64(Float64(Float64(Float64(t_0 * t_2) * h) / Float64(l * d)) * Float64(-0.5 / d)) * t_0), t_1, -1.0) * Float64(Float64(-d) / sqrt(Float64(h * l)))));
	elseif (d <= 4.8e-222)
		tmp = Float64(Float64(Float64(d * sqrt(Float64(h / d))) * Float64(-1.0 * Float64(d * sqrt(Float64(1.0 / Float64(d * l)))))) / h);
	elseif (d <= 3e+156)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(Float64(Float64(t_0 / d) * t_2) / d) * Float64(Float64(Float64(Float64(t_0 * t_1) * -0.5) * h) / l))));
	else
		tmp = Float64(Float64(sqrt(d) * sqrt(Float64(d / Float64(h * l)))) * Float64(1.0 - Float64(Float64(t_3 * Float64(t_3 * 0.5)) * Float64(h / l))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Min[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(-0.25 * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$1 / N[(d + d), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[d, -4.6e-303], (-N[(N[(N[(N[(N[(N[(N[(t$95$0 * t$95$2), $MachinePrecision] * h), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(-0.5 / d), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$1 + -1.0), $MachinePrecision] * N[((-d) / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), If[LessEqual[d, 4.8e-222], N[(N[(N[(d * N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 * N[(d * N[Sqrt[N[(1.0 / N[(d * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[d, 3e+156], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(t$95$0 / d), $MachinePrecision] * t$95$2), $MachinePrecision] / d), $MachinePrecision] * N[(N[(N[(N[(t$95$0 * t$95$1), $MachinePrecision] * -0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] * N[Sqrt[N[(d / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(t$95$3 * N[(t$95$3 * 0.5), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -4.5999999999999999e-303

   1. Initial program 66.4%

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

      1. lift-*.f64 N/A

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

      2. *-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 66.4%

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

      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 66.4%

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

      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 66.4%

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

   3. Applied rewrites 66.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. distribute-frac-neg2 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. distribute-frac-neg2 N/A

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

      9. sqr-neg N/A

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

      10. lower-*.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. distribute-neg-frac2 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. distribute-lft-neg-in N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval N/A

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

      21. distribute-neg-frac N/A

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

      22. distribute-neg-frac2 N/A

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

      23. lower-/.f64 N/A

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

      24. lift-*.f64 N/A

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

      25. *-commutative N/A

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

      26. lower-*.f64 N/A

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

      27. lift-*.f64 N/A

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

      28. distribute-lft-neg-in N/A

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

   5. Applied rewrites 66.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 67.6%

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

   9. Applied rewrites 37.8%

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

   if -4.5999999999999999e-303 < d < 4.7999999999999999e-222

   1. Initial program 66.4%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

      2. 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 d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 31.1%

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

   7. Applied rewrites 31.1%

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

   8. Taylor expanded in d around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 21.8%

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

   10. Applied rewrites 21.8%

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

   if 4.7999999999999999e-222 < d < 3e156

   1. Initial program 66.4%

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

      1. lift-*.f64 N/A

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

      2. *-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 66.4%

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

      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 66.4%

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

      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 66.4%

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

   3. Applied rewrites 66.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. distribute-frac-neg2 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. distribute-frac-neg2 N/A

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

      9. sqr-neg N/A

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

      10. lower-*.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. distribute-neg-frac2 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. distribute-lft-neg-in N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval N/A

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

      21. distribute-neg-frac N/A

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

      22. distribute-neg-frac2 N/A

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

      23. lower-/.f64 N/A

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

      24. lift-*.f64 N/A

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

      25. *-commutative N/A

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

      26. lower-*.f64 N/A

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

      27. lift-*.f64 N/A

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

      28. distribute-lft-neg-in N/A

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

   5. Applied rewrites 66.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 67.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. associate-*l/ N/A

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

      15. times-frac N/A

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

   9. Applied rewrites 66.4%

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

   if 3e156 < d

   1. Initial program 66.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip N/A

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

      10. associate-*l* N/A

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

      11. sqrt-prod N/A

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

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

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

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

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

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

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

      15. lift-/.f64 N/A

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

      16. frac-2neg N/A

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

      17. frac-2neg N/A

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

      18. frac-times N/A

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

      19. remove-double-neg N/A

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

      20. *-lft-identity N/A

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

      21. remove-double-neg N/A

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

      22. remove-double-neg N/A

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

      23. remove-double-neg N/A

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

      24. lower-/.f64 N/A

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

   3. Applied rewrites 32.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-*.f64 N/A

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

      11. count-2-rev N/A

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

      12. lift-+.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 31.8%

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

      17. lift-/.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-/l* N/A

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

      20. lift-*.f64 N/A

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

      21. count-2-rev N/A

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

      22. lift-+.f64 N/A

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

      23. lift-/.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-*.f64 32.0%

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

   5. Applied rewrites 32.0%

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

Alternative 2: 75.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_1 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_2 := \frac{t\_0 \cdot t\_1}{-2 \cdot d}\\ \mathbf{if}\;\ell \leq 10^{-309}:\\ \;\;\;\;-\mathsf{fma}\left(\left(\frac{\left(t\_1 \cdot \left(-0.25 \cdot t\_0\right)\right) \cdot h}{\ell \cdot d} \cdot \frac{-0.5}{d}\right) \cdot t\_1, t\_0, -1\right) \cdot \frac{-d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(t\_2 \cdot t\_2\right)\right) \cdot \frac{h}{\ell}\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (fmax (fabs M) (fabs D)))
        (t_1 (fmin (fabs M) (fabs D)))
        (t_2 (/ (* t_0 t_1) (* -2.0 d))))
   (if (<= l 1e-309)
     (-
      (*
       (fma
        (* (* (/ (* (* t_1 (* -0.25 t_0)) h) (* l d)) (/ -0.5 d)) t_1)
        t_0
        -1.0)
       (/ (- d) (sqrt (* h l)))))
     (*
      (* (sqrt (/ d l)) (/ (sqrt d) (sqrt h)))
      (- 1.0 (* (* (/ 1.0 2.0) (* t_2 t_2)) (/ h l)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fmax(fabs(M), fabs(D));
	double t_1 = fmin(fabs(M), fabs(D));
	double t_2 = (t_0 * t_1) / (-2.0 * d);
	double tmp;
	if (l <= 1e-309) {
		tmp = -(fma((((((t_1 * (-0.25 * t_0)) * h) / (l * d)) * (-0.5 / d)) * t_1), t_0, -1.0) * (-d / sqrt((h * l))));
	} else {
		tmp = (sqrt((d / l)) * (sqrt(d) / sqrt(h))) * (1.0 - (((1.0 / 2.0) * (t_2 * t_2)) * (h / l)));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = fmax(abs(M), abs(D))
	t_1 = fmin(abs(M), abs(D))
	t_2 = Float64(Float64(t_0 * t_1) / Float64(-2.0 * d))
	tmp = 0.0
	if (l <= 1e-309)
		tmp = Float64(-Float64(fma(Float64(Float64(Float64(Float64(Float64(t_1 * Float64(-0.25 * t_0)) * h) / Float64(l * d)) * Float64(-0.5 / d)) * t_1), t_0, -1.0) * Float64(Float64(-d) / sqrt(Float64(h * l)))));
	else
		tmp = Float64(Float64(sqrt(Float64(d / l)) * Float64(sqrt(d) / sqrt(h))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * Float64(t_2 * t_2)) * Float64(h / l))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Max[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 * t$95$1), $MachinePrecision] / N[(-2.0 * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 1e-309], (-N[(N[(N[(N[(N[(N[(N[(t$95$1 * N[(-0.25 * t$95$0), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(-0.5 / d), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$0 + -1.0), $MachinePrecision] * N[((-d) / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if l < 1.0000000000000019e-309

   1. Initial program 66.4%

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

      1. lift-*.f64 N/A

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

      2. *-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 66.4%

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

      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 66.4%

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

      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 66.4%

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

   3. Applied rewrites 66.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. distribute-frac-neg2 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. distribute-frac-neg2 N/A

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

      9. sqr-neg N/A

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

      10. lower-*.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. distribute-neg-frac2 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. distribute-lft-neg-in N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval N/A

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

      21. distribute-neg-frac N/A

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

      22. distribute-neg-frac2 N/A

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

      23. lower-/.f64 N/A

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

      24. lift-*.f64 N/A

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

      25. *-commutative N/A

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

      26. lower-*.f64 N/A

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

      27. lift-*.f64 N/A

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

      28. distribute-lft-neg-in N/A

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

   5. Applied rewrites 66.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 67.6%

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

   9. Applied rewrites 37.8%

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

   if 1.0000000000000019e-309 < l

   1. Initial program 66.4%

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

      1. lift-*.f64 N/A

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

      2. *-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 66.4%

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

      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 66.4%

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

      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 66.4%

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

   3. Applied rewrites 66.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. distribute-frac-neg2 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. distribute-frac-neg2 N/A

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

      9. sqr-neg N/A

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

      10. lower-*.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. distribute-neg-frac2 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. distribute-lft-neg-in N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval N/A

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

      21. distribute-neg-frac N/A

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

      22. distribute-neg-frac2 N/A

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

      23. lower-/.f64 N/A

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

      24. lift-*.f64 N/A

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

      25. *-commutative N/A

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

      26. lower-*.f64 N/A

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

      27. lift-*.f64 N/A

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

      28. distribute-lft-neg-in N/A

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

   5. Applied rewrites 66.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

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

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

      5. lower-sqrt.f64 N/A

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

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

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

      7. lower-sqrt.f64 N/A

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

      8. lower-unsound-sqrt.f64 37.9%

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

   7. Applied rewrites 37.9%

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

Alternative 3: 74.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \mathsf{min}\left(M, D\right) \cdot \mathsf{max}\left(M, D\right)\\ t_2 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_1}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_3 := \sqrt{\frac{d}{\ell}}\\ t_4 := \left|\left(\frac{\frac{1}{d}}{\sqrt{h \cdot \ell}} \cdot d\right) \cdot d\right|\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{-132}:\\ \;\;\;\;\left(t\_3 \cdot t\_0\right) \cdot \left(1 - \frac{\left(t\_1 \cdot -0.5\right) \cdot \frac{\left(\mathsf{min}\left(M, D\right) \cdot \left(-0.25 \cdot \mathsf{max}\left(M, D\right)\right)\right) \cdot h}{\ell \cdot d}}{d}\right)\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq 10^{+281}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\mathsf{max}\left(M, D\right) \cdot \left(\left(\frac{\mathsf{max}\left(M, D\right)}{d} \cdot \mathsf{min}\left(M, D\right)\right) \cdot \mathsf{min}\left(M, D\right)\right)}{d} \cdot -0.125, \frac{h}{\ell}, 1\right) \cdot t\_0\right) \cdot t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h)))
        (t_1 (* (fmin M D) (fmax M 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_1 (* 2.0 d)) 2.0)) (/ h l)))))
        (t_3 (sqrt (/ d l)))
        (t_4 (fabs (* (* (/ (/ 1.0 d) (sqrt (* h l))) d) d))))
   (if (<= t_2 -4e-132)
     (*
      (* t_3 t_0)
      (-
       1.0
       (/
        (* (* t_1 -0.5) (/ (* (* (fmin M D) (* -0.25 (fmax M D))) h) (* l d)))
        d)))
     (if (<= t_2 0.0)
       t_4
       (if (<= t_2 1e+281)
         (*
          (*
           (fma
            (*
             (/
              (* (fmax M D) (* (* (/ (fmax M D) d) (fmin M D)) (fmin M D)))
              d)
             -0.125)
            (/ h l)
            1.0)
           t_0)
          t_3)
         t_4)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h));
	double t_1 = fmin(M, D) * fmax(M, 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_1 / (2.0 * d)), 2.0)) * (h / l)));
	double t_3 = sqrt((d / l));
	double t_4 = fabs(((((1.0 / d) / sqrt((h * l))) * d) * d));
	double tmp;
	if (t_2 <= -4e-132) {
		tmp = (t_3 * t_0) * (1.0 - (((t_1 * -0.5) * (((fmin(M, D) * (-0.25 * fmax(M, D))) * h) / (l * d))) / d));
	} else if (t_2 <= 0.0) {
		tmp = t_4;
	} else if (t_2 <= 1e+281) {
		tmp = (fma((((fmax(M, D) * (((fmax(M, D) / d) * fmin(M, D)) * fmin(M, D))) / d) * -0.125), (h / l), 1.0) * t_0) * t_3;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / h))
	t_1 = Float64(fmin(M, D) * fmax(M, 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_1 / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_3 = sqrt(Float64(d / l))
	t_4 = abs(Float64(Float64(Float64(Float64(1.0 / d) / sqrt(Float64(h * l))) * d) * d))
	tmp = 0.0
	if (t_2 <= -4e-132)
		tmp = Float64(Float64(t_3 * t_0) * Float64(1.0 - Float64(Float64(Float64(t_1 * -0.5) * Float64(Float64(Float64(fmin(M, D) * Float64(-0.25 * fmax(M, D))) * h) / Float64(l * d))) / d)));
	elseif (t_2 <= 0.0)
		tmp = t_4;
	elseif (t_2 <= 1e+281)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(fmax(M, D) * Float64(Float64(Float64(fmax(M, D) / d) * fmin(M, D)) * fmin(M, D))) / d) * -0.125), Float64(h / l), 1.0) * t_0) * t_3);
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Min[M, D], $MachinePrecision] * N[Max[M, 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$1 / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Abs[N[(N[(N[(N[(1.0 / d), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, -4e-132], N[(N[(t$95$3 * t$95$0), $MachinePrecision] * N[(1.0 - N[(N[(N[(t$95$1 * -0.5), $MachinePrecision] * N[(N[(N[(N[Min[M, D], $MachinePrecision] * N[(-0.25 * N[Max[M, D], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], t$95$4, If[LessEqual[t$95$2, 1e+281], N[(N[(N[(N[(N[(N[(N[Max[M, D], $MachinePrecision] * N[(N[(N[(N[Max[M, D], $MachinePrecision] / d), $MachinePrecision] * N[Min[M, D], $MachinePrecision]), $MachinePrecision] * N[Min[M, D], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$3), $MachinePrecision], t$95$4]]]]]]]]

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)))) < -3.9999999999999999e-132

   1. Initial program 66.4%

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

      1. lift-*.f64 N/A

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

      2. *-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 66.4%

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

      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 66.4%

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

      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 66.4%

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

   3. Applied rewrites 66.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. distribute-frac-neg2 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. distribute-frac-neg2 N/A

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

      9. sqr-neg N/A

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

      10. lower-*.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. distribute-neg-frac2 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. distribute-lft-neg-in N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval N/A

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

      21. distribute-neg-frac N/A

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

      22. distribute-neg-frac2 N/A

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

      23. lower-/.f64 N/A

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

      24. lift-*.f64 N/A

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

      25. *-commutative N/A

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

      26. lower-*.f64 N/A

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

      27. lift-*.f64 N/A

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

      28. distribute-lft-neg-in N/A

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

   5. Applied rewrites 66.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 67.6%

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

   9. Applied rewrites 66.1%

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

   if -3.9999999999999999e-132 < (*.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 1e281 < (*.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 66.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip N/A

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

      10. associate-*l* N/A

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

      11. sqrt-prod N/A

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

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

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

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

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

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

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

      15. lift-/.f64 N/A

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

      16. frac-2neg N/A

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

      17. frac-2neg N/A

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

      18. frac-times N/A

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

      19. remove-double-neg N/A

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

      20. *-lft-identity N/A

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

      21. remove-double-neg N/A

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

      22. remove-double-neg N/A

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

      23. remove-double-neg N/A

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

      24. lower-/.f64 N/A

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

   3. Applied rewrites 32.2%

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

   4. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 13.9%

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

   6. Applied rewrites 13.9%

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

   8. Applied rewrites 39.0%

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

   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)))) < 1e281

   1. Initial program 66.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. lift-/.f64 N/A

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

      17. mult-flip N/A

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

      18. associate-*l* N/A

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

   3. Applied rewrites 58.8%

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

   5. Applied rewrites 64.3%

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

Alternative 4: 74.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_1 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ \mathbf{if}\;\ell \leq 10^{-309}:\\ \;\;\;\;-\mathsf{fma}\left(\left(\frac{\left(t\_0 \cdot \left(-0.25 \cdot t\_1\right)\right) \cdot h}{\ell \cdot d} \cdot \frac{-0.5}{d}\right) \cdot t\_0, t\_1, -1\right) \cdot \frac{-d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right) \cdot \left(1 - \frac{\left(\left(0.5 \cdot \left(t\_1 \cdot -0.5\right)\right) \cdot \frac{t\_0}{d}\right) \cdot \left(\left(t\_0 \cdot t\_1\right) \cdot -0.5\right)}{d} \cdot \frac{h}{\ell}\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (fmin (fabs M) (fabs D))) (t_1 (fmax (fabs M) (fabs D))))
   (if (<= l 1e-309)
     (-
      (*
       (fma
        (* (* (/ (* (* t_0 (* -0.25 t_1)) h) (* l d)) (/ -0.5 d)) t_0)
        t_1
        -1.0)
       (/ (- d) (sqrt (* h l)))))
     (*
      (* (sqrt (/ d l)) (/ (sqrt d) (sqrt h)))
      (-
       1.0
       (*
        (/ (* (* (* 0.5 (* t_1 -0.5)) (/ t_0 d)) (* (* t_0 t_1) -0.5)) d)
        (/ h l)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fmin(fabs(M), fabs(D));
	double t_1 = fmax(fabs(M), fabs(D));
	double tmp;
	if (l <= 1e-309) {
		tmp = -(fma((((((t_0 * (-0.25 * t_1)) * h) / (l * d)) * (-0.5 / d)) * t_0), t_1, -1.0) * (-d / sqrt((h * l))));
	} else {
		tmp = (sqrt((d / l)) * (sqrt(d) / sqrt(h))) * (1.0 - (((((0.5 * (t_1 * -0.5)) * (t_0 / d)) * ((t_0 * t_1) * -0.5)) / d) * (h / l)));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = fmin(abs(M), abs(D))
	t_1 = fmax(abs(M), abs(D))
	tmp = 0.0
	if (l <= 1e-309)
		tmp = Float64(-Float64(fma(Float64(Float64(Float64(Float64(Float64(t_0 * Float64(-0.25 * t_1)) * h) / Float64(l * d)) * Float64(-0.5 / d)) * t_0), t_1, -1.0) * Float64(Float64(-d) / sqrt(Float64(h * l)))));
	else
		tmp = Float64(Float64(sqrt(Float64(d / l)) * Float64(sqrt(d) / sqrt(h))) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(0.5 * Float64(t_1 * -0.5)) * Float64(t_0 / d)) * Float64(Float64(t_0 * t_1) * -0.5)) / d) * Float64(h / l))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Min[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, 1e-309], (-N[(N[(N[(N[(N[(N[(N[(t$95$0 * N[(-0.25 * t$95$1), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(-0.5 / d), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$1 + -1.0), $MachinePrecision] * N[((-d) / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(0.5 * N[(t$95$1 * -0.5), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / d), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 * t$95$1), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if l < 1.0000000000000019e-309

   1. Initial program 66.4%

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

      1. lift-*.f64 N/A

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

      2. *-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 66.4%

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

      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 66.4%

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

      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 66.4%

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

   3. Applied rewrites 66.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. distribute-frac-neg2 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. distribute-frac-neg2 N/A

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

      9. sqr-neg N/A

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

      10. lower-*.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. distribute-neg-frac2 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. distribute-lft-neg-in N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval N/A

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

      21. distribute-neg-frac N/A

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

      22. distribute-neg-frac2 N/A

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

      23. lower-/.f64 N/A

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

      24. lift-*.f64 N/A

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

      25. *-commutative N/A

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

      26. lower-*.f64 N/A

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

      27. lift-*.f64 N/A

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

      28. distribute-lft-neg-in N/A

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

   5. Applied rewrites 66.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 67.6%

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

   9. Applied rewrites 37.8%

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

   if 1.0000000000000019e-309 < l

   1. Initial program 66.4%

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

      1. lift-*.f64 N/A

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

      2. *-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 66.4%

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

      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 66.4%

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

      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 66.4%

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

   3. Applied rewrites 66.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. distribute-frac-neg2 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. distribute-frac-neg2 N/A

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

      9. sqr-neg N/A

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

      10. lower-*.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. distribute-neg-frac2 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. distribute-lft-neg-in N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval N/A

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

      21. distribute-neg-frac N/A

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

      22. distribute-neg-frac2 N/A

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

      23. lower-/.f64 N/A

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

      24. lift-*.f64 N/A

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

      25. *-commutative N/A

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

      26. lower-*.f64 N/A

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

      27. lift-*.f64 N/A

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

      28. distribute-lft-neg-in N/A

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

   5. Applied rewrites 66.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   7. Applied rewrites 65.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

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

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

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

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

      6. lower-unsound-sqrt.f64 37.1%

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

   9. Applied rewrites 37.1%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}}\right) \cdot \left(1 - \frac{\left(\left(0.5 \cdot \left(D \cdot -0.5\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot -0.5\right)}{d} \cdot \frac{h}{\ell}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 74.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_1 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_0 \cdot t\_1}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 10^{+281}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \left(\left(0.5 \cdot \left(t\_1 \cdot -0.5\right)\right) \cdot \frac{t\_0}{d}\right)\right) \cdot \left(\left(t\_0 \cdot \frac{-0.5}{d}\right) \cdot t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{\frac{1}{d}}{\sqrt{h \cdot \ell}} \cdot d\right) \cdot d\right|\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (fmin (fabs M) (fabs D))) (t_1 (fmax (fabs M) (fabs D))))
   (if (<=
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (-
          1.0
          (* (* (/ 1.0 2.0) (pow (/ (* t_0 t_1) (* 2.0 d)) 2.0)) (/ h l))))
        1e+281)
     (*
      (* (sqrt (/ d l)) (sqrt (/ d h)))
      (-
       1.0
       (*
        (* (/ h l) (* (* 0.5 (* t_1 -0.5)) (/ t_0 d)))
        (* (* t_0 (/ -0.5 d)) t_1))))
     (fabs (* (* (/ (/ 1.0 d) (sqrt (* h l))) d) d)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fmin(fabs(M), fabs(D));
	double t_1 = fmax(fabs(M), fabs(D));
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((t_0 * t_1) / (2.0 * d)), 2.0)) * (h / l)))) <= 1e+281) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (((h / l) * ((0.5 * (t_1 * -0.5)) * (t_0 / d))) * ((t_0 * (-0.5 / d)) * t_1)));
	} else {
		tmp = fabs(((((1.0 / d) / sqrt((h * l))) * d) * d));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = fmin(abs(m), abs(d_1))
    t_1 = fmax(abs(m), abs(d_1))
    if (((((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((t_0 * t_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))) <= 1d+281) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - (((h / l) * ((0.5d0 * (t_1 * (-0.5d0))) * (t_0 / d))) * ((t_0 * ((-0.5d0) / d)) * t_1)))
    else
        tmp = abs(((((1.0d0 / d) / sqrt((h * l))) * d) * d))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = fmin(Math.abs(M), Math.abs(D));
	double t_1 = fmax(Math.abs(M), Math.abs(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(((t_0 * t_1) / (2.0 * d)), 2.0)) * (h / l)))) <= 1e+281) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - (((h / l) * ((0.5 * (t_1 * -0.5)) * (t_0 / d))) * ((t_0 * (-0.5 / d)) * t_1)));
	} else {
		tmp = Math.abs(((((1.0 / d) / Math.sqrt((h * l))) * d) * d));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = fmin(math.fabs(M), math.fabs(D))
	t_1 = fmax(math.fabs(M), math.fabs(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(((t_0 * t_1) / (2.0 * d)), 2.0)) * (h / l)))) <= 1e+281:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - (((h / l) * ((0.5 * (t_1 * -0.5)) * (t_0 / d))) * ((t_0 * (-0.5 / d)) * t_1)))
	else:
		tmp = math.fabs(((((1.0 / d) / math.sqrt((h * l))) * d) * d))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = fmin(abs(M), abs(D))
	t_1 = fmax(abs(M), abs(D))
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(t_0 * t_1) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= 1e+281)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(Float64(h / l) * Float64(Float64(0.5 * Float64(t_1 * -0.5)) * Float64(t_0 / d))) * Float64(Float64(t_0 * Float64(-0.5 / d)) * t_1))));
	else
		tmp = abs(Float64(Float64(Float64(Float64(1.0 / d) / sqrt(Float64(h * l))) * d) * d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = min(abs(M), abs(D));
	t_1 = max(abs(M), abs(D));
	tmp = 0.0;
	if (((((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((t_0 * t_1) / (2.0 * d)) ^ 2.0)) * (h / l)))) <= 1e+281)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (((h / l) * ((0.5 * (t_1 * -0.5)) * (t_0 / d))) * ((t_0 * (-0.5 / d)) * t_1)));
	else
		tmp = abs(((((1.0 / d) / sqrt((h * l))) * d) * d));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Min[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(t$95$0 * t$95$1), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+281], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(h / l), $MachinePrecision] * N[(N[(0.5 * N[(t$95$1 * -0.5), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 * N[(-0.5 / d), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[N[(N[(N[(N[(1.0 / d), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision] * d), $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)))) < 1e281

   1. Initial program 66.4%

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

      1. lift-*.f64 N/A

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

      2. *-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 66.4%

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

      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 66.4%

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

      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 66.4%

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

   3. Applied rewrites 66.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. distribute-frac-neg2 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. distribute-frac-neg2 N/A

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

      9. sqr-neg N/A

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

      10. lower-*.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. distribute-neg-frac2 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. distribute-lft-neg-in N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval N/A

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

      21. distribute-neg-frac N/A

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

      22. distribute-neg-frac2 N/A

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

      23. lower-/.f64 N/A

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

      24. lift-*.f64 N/A

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

      25. *-commutative N/A

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

      26. lower-*.f64 N/A

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

      27. lift-*.f64 N/A

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

      28. distribute-lft-neg-in N/A

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

   5. Applied rewrites 66.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 67.6%

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

   if 1e281 < (*.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 66.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip N/A

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

      10. associate-*l* N/A

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

      11. sqrt-prod N/A

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

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

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

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

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

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

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

      15. lift-/.f64 N/A

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

      16. frac-2neg N/A

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

      17. frac-2neg N/A

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

      18. frac-times N/A

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

      19. remove-double-neg N/A

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

      20. *-lft-identity N/A

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

      21. remove-double-neg N/A

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

      22. remove-double-neg N/A

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

      23. remove-double-neg N/A

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

      24. lower-/.f64 N/A

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

   3. Applied rewrites 32.2%

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

   4. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 13.9%

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

   6. Applied rewrites 13.9%

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

   8. Applied rewrites 39.0%

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

Alternative 6: 74.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{\mathsf{max}\left(M, D\right)}{d + d} \cdot \mathsf{min}\left(M, D\right)\\ \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\mathsf{min}\left(M, D\right) \cdot \mathsf{max}\left(M, D\right)}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 10^{+281}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(t\_0 \cdot \left(t\_0 \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{\frac{1}{d}}{\sqrt{h \cdot \ell}} \cdot d\right) \cdot d\right|\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ (fmax M D) (+ d d)) (fmin M D))))
   (if (<=
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (-
          1.0
          (*
           (* (/ 1.0 2.0) (pow (/ (* (fmin M D) (fmax M D)) (* 2.0 d)) 2.0))
           (/ h l))))
        1e+281)
     (*
      (* (sqrt (/ d l)) (sqrt (/ d h)))
      (- 1.0 (* (* t_0 (* t_0 0.5)) (/ h l))))
     (fabs (* (* (/ (/ 1.0 d) (sqrt (* h l))) d) d)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (fmax(M, D) / (d + d)) * fmin(M, 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(((fmin(M, D) * fmax(M, D)) / (2.0 * d)), 2.0)) * (h / l)))) <= 1e+281) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((t_0 * (t_0 * 0.5)) * (h / l)));
	} else {
		tmp = fabs(((((1.0 / d) / sqrt((h * l))) * d) * d));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (fmax(m, d_1) / (d + d)) * fmin(m, d_1)
    if (((((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((fmin(m, d_1) * fmax(m, d_1)) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))) <= 1d+281) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - ((t_0 * (t_0 * 0.5d0)) * (h / l)))
    else
        tmp = abs(((((1.0d0 / d) / sqrt((h * l))) * d) * d))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (fmax(M, D) / (d + d)) * fmin(M, 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(((fmin(M, D) * fmax(M, D)) / (2.0 * d)), 2.0)) * (h / l)))) <= 1e+281) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - ((t_0 * (t_0 * 0.5)) * (h / l)));
	} else {
		tmp = Math.abs(((((1.0 / d) / Math.sqrt((h * l))) * d) * d));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (fmax(M, D) / (d + d)) * fmin(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(((fmin(M, D) * fmax(M, D)) / (2.0 * d)), 2.0)) * (h / l)))) <= 1e+281:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - ((t_0 * (t_0 * 0.5)) * (h / l)))
	else:
		tmp = math.fabs(((((1.0 / d) / math.sqrt((h * l))) * d) * d))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(fmax(M, D) / Float64(d + d)) * fmin(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(fmin(M, D) * fmax(M, D)) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= 1e+281)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(t_0 * Float64(t_0 * 0.5)) * Float64(h / l))));
	else
		tmp = abs(Float64(Float64(Float64(Float64(1.0 / d) / sqrt(Float64(h * l))) * d) * d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (max(M, D) / (d + d)) * min(M, D);
	tmp = 0.0;
	if (((((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((min(M, D) * max(M, D)) / (2.0 * d)) ^ 2.0)) * (h / l)))) <= 1e+281)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((t_0 * (t_0 * 0.5)) * (h / l)));
	else
		tmp = abs(((((1.0 / d) / sqrt((h * l))) * d) * d));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Max[M, D], $MachinePrecision] / N[(d + d), $MachinePrecision]), $MachinePrecision] * N[Min[M, D], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(N[Min[M, D], $MachinePrecision] * N[Max[M, D], $MachinePrecision]), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+281], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(t$95$0 * N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[N[(N[(N[(N[(1.0 / d), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision] * d), $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)))) < 1e281

   1. Initial program 66.4%

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

      1. lift-*.f64 N/A

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

      2. *-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 66.4%

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

      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 66.4%

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

      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 66.4%

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

   3. Applied rewrites 66.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-*.f64 N/A

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

      11. count-2-rev N/A

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

      12. lift-+.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 65.5%

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

      17. lift-/.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-/l* N/A

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

      20. lift-*.f64 N/A

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

      21. count-2-rev N/A

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

      22. lift-+.f64 N/A

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

      23. lift-/.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-*.f64 65.9%

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

   5. Applied rewrites 65.9%

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

   if 1e281 < (*.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 66.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip N/A

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

      10. associate-*l* N/A

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

      11. sqrt-prod N/A

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

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

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

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

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

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

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

      15. lift-/.f64 N/A

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

      16. frac-2neg N/A

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

      17. frac-2neg N/A

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

      18. frac-times N/A

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

      19. remove-double-neg N/A

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

      20. *-lft-identity N/A

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

      21. remove-double-neg N/A

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

      22. remove-double-neg N/A

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

      23. remove-double-neg N/A

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

      24. lower-/.f64 N/A

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

   3. Applied rewrites 32.2%

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

   4. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 13.9%

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

   6. Applied rewrites 13.9%

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

   8. Applied rewrites 39.0%

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

Alternative 7: 74.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\mathsf{min}\left(M, D\right) \cdot \mathsf{max}\left(M, D\right)}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 10^{+281}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(h \cdot \frac{\mathsf{min}\left(M, D\right)}{d}\right) \cdot \left(-0.25 \cdot \mathsf{max}\left(M, D\right)\right)}{\ell} \cdot \left(\left(\mathsf{min}\left(M, D\right) \cdot \frac{-0.5}{d}\right) \cdot \mathsf{max}\left(M, D\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{\frac{1}{d}}{\sqrt{h \cdot \ell}} \cdot d\right) \cdot d\right|\\ \end{array} \]

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 (/ (* (fmin M D) (fmax M D)) (* 2.0 d)) 2.0))
         (/ h l))))
      1e+281)
   (*
    (* (sqrt (/ d l)) (sqrt (/ d h)))
    (-
     1.0
     (*
      (/ (* (* h (/ (fmin M D) d)) (* -0.25 (fmax M D))) l)
      (* (* (fmin M D) (/ -0.5 d)) (fmax M D)))))
   (fabs (* (* (/ (/ 1.0 d) (sqrt (* h l))) d) d))))

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(((fmin(M, D) * fmax(M, D)) / (2.0 * d)), 2.0)) * (h / l)))) <= 1e+281) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((((h * (fmin(M, D) / d)) * (-0.25 * fmax(M, D))) / l) * ((fmin(M, D) * (-0.5 / d)) * fmax(M, D))));
	} else {
		tmp = fabs(((((1.0 / d) / sqrt((h * l))) * d) * d));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((fmin(m, d_1) * fmax(m, d_1)) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))) <= 1d+281) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - ((((h * (fmin(m, d_1) / d)) * ((-0.25d0) * fmax(m, d_1))) / l) * ((fmin(m, d_1) * ((-0.5d0) / d)) * fmax(m, d_1))))
    else
        tmp = abs(((((1.0d0 / d) / sqrt((h * l))) * d) * d))
    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(((fmin(M, D) * fmax(M, D)) / (2.0 * d)), 2.0)) * (h / l)))) <= 1e+281) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - ((((h * (fmin(M, D) / d)) * (-0.25 * fmax(M, D))) / l) * ((fmin(M, D) * (-0.5 / d)) * fmax(M, D))));
	} else {
		tmp = Math.abs(((((1.0 / d) / Math.sqrt((h * l))) * d) * d));
	}
	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(((fmin(M, D) * fmax(M, D)) / (2.0 * d)), 2.0)) * (h / l)))) <= 1e+281:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - ((((h * (fmin(M, D) / d)) * (-0.25 * fmax(M, D))) / l) * ((fmin(M, D) * (-0.5 / d)) * fmax(M, D))))
	else:
		tmp = math.fabs(((((1.0 / d) / math.sqrt((h * l))) * d) * d))
	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(fmin(M, D) * fmax(M, D)) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= 1e+281)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(Float64(Float64(h * Float64(fmin(M, D) / d)) * Float64(-0.25 * fmax(M, D))) / l) * Float64(Float64(fmin(M, D) * Float64(-0.5 / d)) * fmax(M, D)))));
	else
		tmp = abs(Float64(Float64(Float64(Float64(1.0 / d) / sqrt(Float64(h * l))) * d) * d));
	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) * (((min(M, D) * max(M, D)) / (2.0 * d)) ^ 2.0)) * (h / l)))) <= 1e+281)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((((h * (min(M, D) / d)) * (-0.25 * max(M, D))) / l) * ((min(M, D) * (-0.5 / d)) * max(M, D))));
	else
		tmp = abs(((((1.0 / d) / sqrt((h * l))) * d) * d));
	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[(N[Min[M, D], $MachinePrecision] * N[Max[M, D], $MachinePrecision]), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+281], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(h * N[(N[Min[M, D], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(-0.25 * N[Max[M, D], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Min[M, D], $MachinePrecision] * N[(-0.5 / d), $MachinePrecision]), $MachinePrecision] * N[Max[M, D], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[N[(N[(N[(N[(1.0 / d), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision] * d), $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)))) < 1e281

   1. Initial program 66.4%

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

      1. lift-*.f64 N/A

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

      2. *-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 66.4%

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

      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 66.4%

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

      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 66.4%

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

   3. Applied rewrites 66.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. distribute-frac-neg2 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. distribute-frac-neg2 N/A

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

      9. sqr-neg N/A

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

      10. lower-*.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. distribute-neg-frac2 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. distribute-lft-neg-in N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval N/A

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

      21. distribute-neg-frac N/A

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

      22. distribute-neg-frac2 N/A

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

      23. lower-/.f64 N/A

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

      24. lift-*.f64 N/A

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

      25. *-commutative N/A

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

      26. lower-*.f64 N/A

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

      27. lift-*.f64 N/A

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

      28. distribute-lft-neg-in N/A

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

   5. Applied rewrites 66.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 67.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 68.6%

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 68.6%

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

   9. Applied rewrites 68.6%

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

   if 1e281 < (*.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 66.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip N/A

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

      10. associate-*l* N/A

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

      11. sqrt-prod N/A

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

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

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

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

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

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

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

      15. lift-/.f64 N/A

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

      16. frac-2neg N/A

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

      17. frac-2neg N/A

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

      18. frac-times N/A

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

      19. remove-double-neg N/A

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

      20. *-lft-identity N/A

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

      21. remove-double-neg N/A

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

      22. remove-double-neg N/A

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

      23. remove-double-neg N/A

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

      24. lower-/.f64 N/A

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

   3. Applied rewrites 32.2%

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

   4. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 13.9%

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

   6. Applied rewrites 13.9%

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

   8. Applied rewrites 39.0%

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

Alternative 8: 73.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_1 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_2 := t\_0 \cdot t\_1\\ \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_2}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 10^{+281}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\left(t\_2 \cdot \left(\frac{t\_0}{d} \cdot \left(-0.25 \cdot t\_1\right)\right)\right) \cdot 0.5}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{\frac{1}{d}}{\sqrt{h \cdot \ell}} \cdot d\right) \cdot d\right|\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (fmin (fabs M) (fabs D)))
        (t_1 (fmax (fabs M) (fabs D)))
        (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))))
        1e+281)
     (*
      (*
       (fma (/ (* (* t_2 (* (/ t_0 d) (* -0.25 t_1))) 0.5) d) (/ h l) 1.0)
       (sqrt (/ d l)))
      (sqrt (/ d h)))
     (fabs (* (* (/ (/ 1.0 d) (sqrt (* h l))) d) d)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fmin(fabs(M), fabs(D));
	double t_1 = fmax(fabs(M), fabs(D));
	double 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)))) <= 1e+281) {
		tmp = (fma((((t_2 * ((t_0 / d) * (-0.25 * t_1))) * 0.5) / d), (h / l), 1.0) * sqrt((d / l))) * sqrt((d / h));
	} else {
		tmp = fabs(((((1.0 / d) / sqrt((h * l))) * d) * d));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = fmin(abs(M), abs(D))
	t_1 = fmax(abs(M), abs(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)))) <= 1e+281)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(t_2 * Float64(Float64(t_0 / d) * Float64(-0.25 * t_1))) * 0.5) / d), Float64(h / l), 1.0) * sqrt(Float64(d / l))) * sqrt(Float64(d / h)));
	else
		tmp = abs(Float64(Float64(Float64(Float64(1.0 / d) / sqrt(Float64(h * l))) * d) * d));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Min[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, 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], 1e+281], N[(N[(N[(N[(N[(N[(t$95$2 * N[(N[(t$95$0 / d), $MachinePrecision] * N[(-0.25 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] / d), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(N[(N[(N[(1.0 / d), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision] * d), $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)))) < 1e281

   1. Initial program 66.4%

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

      1. lift-*.f64 N/A

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

      2. *-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 66.4%

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

      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 66.4%

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

      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 66.4%

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

   3. Applied rewrites 66.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. distribute-frac-neg2 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. distribute-frac-neg2 N/A

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

      9. sqr-neg N/A

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

      10. lower-*.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. distribute-neg-frac2 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. distribute-lft-neg-in N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval N/A

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

      21. distribute-neg-frac N/A

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

      22. distribute-neg-frac2 N/A

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

      23. lower-/.f64 N/A

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

      24. lift-*.f64 N/A

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

      25. *-commutative N/A

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

      26. lower-*.f64 N/A

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

      27. lift-*.f64 N/A

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

      28. distribute-lft-neg-in N/A

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

   5. Applied rewrites 66.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 67.6%

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

   9. Applied rewrites 64.9%

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

   if 1e281 < (*.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 66.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip N/A

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

      10. associate-*l* N/A

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

      11. sqrt-prod N/A

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

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

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

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

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

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

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

      15. lift-/.f64 N/A

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

      16. frac-2neg N/A

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

      17. frac-2neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{1}{h} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. frac-times N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\color{blue}{\frac{1 \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)\right)}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. remove-double-neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{1 \cdot \color{blue}{d}}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. *-lft-identity N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{\color{blue}{d}}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. remove-double-neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{d}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(h\right)\right)\right)\right)} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      22. remove-double-neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{d}{\color{blue}{h} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      23. remove-double-neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{d}{h \cdot \color{blue}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      24. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\color{blue}{\frac{d}{h \cdot \ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 32.2%

      \[\leadsto \color{blue}{\left(\sqrt{d} \cdot \sqrt{\frac{d}{h \cdot \ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \color{blue}{{d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {d}^{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{d}}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \color{blue}{\sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\color{blue}{\frac{1}{d \cdot \left(h \cdot \ell\right)}}}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{\color{blue}{1}}{d \cdot \left(h \cdot \ell\right)}}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right) \]

      9. lower-*.f64 13.9%

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right) \]

   6. Applied rewrites 13.9%

      \[\leadsto \color{blue}{{d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right)} \]

   8. Applied rewrites 39.0%

      \[\leadsto \left|\left(\frac{\frac{1}{d}}{\sqrt{h \cdot \ell}} \cdot d\right) \cdot d\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 71.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\mathsf{min}\left(M, D\right) \cdot \mathsf{max}\left(M, D\right)}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 10^{+281}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\mathsf{max}\left(M, D\right) \cdot \left(\left(\frac{\mathsf{max}\left(M, D\right)}{d} \cdot \mathsf{min}\left(M, D\right)\right) \cdot \mathsf{min}\left(M, D\right)\right)}{d} \cdot -0.125, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{\frac{1}{d}}{\sqrt{h \cdot \ell}} \cdot d\right) \cdot d\right|\\ \end{array} \]

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 (/ (* (fmin M D) (fmax M D)) (* 2.0 d)) 2.0))
         (/ h l))))
      1e+281)
   (*
    (*
     (fma
      (*
       (/ (* (fmax M D) (* (* (/ (fmax M D) d) (fmin M D)) (fmin M D))) d)
       -0.125)
      (/ h l)
      1.0)
     (sqrt (/ d h)))
    (sqrt (/ d l)))
   (fabs (* (* (/ (/ 1.0 d) (sqrt (* h l))) d) d))))

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(((fmin(M, D) * fmax(M, D)) / (2.0 * d)), 2.0)) * (h / l)))) <= 1e+281) {
		tmp = (fma((((fmax(M, D) * (((fmax(M, D) / d) * fmin(M, D)) * fmin(M, D))) / d) * -0.125), (h / l), 1.0) * sqrt((d / h))) * sqrt((d / l));
	} else {
		tmp = fabs(((((1.0 / d) / sqrt((h * l))) * d) * d));
	}
	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(fmin(M, D) * fmax(M, D)) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= 1e+281)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(fmax(M, D) * Float64(Float64(Float64(fmax(M, D) / d) * fmin(M, D)) * fmin(M, D))) / d) * -0.125), Float64(h / l), 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	else
		tmp = abs(Float64(Float64(Float64(Float64(1.0 / d) / sqrt(Float64(h * l))) * d) * d));
	end
	return 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[(N[Min[M, D], $MachinePrecision] * N[Max[M, D], $MachinePrecision]), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+281], N[(N[(N[(N[(N[(N[(N[Max[M, D], $MachinePrecision] * N[(N[(N[(N[Max[M, D], $MachinePrecision] / d), $MachinePrecision] * N[Min[M, D], $MachinePrecision]), $MachinePrecision] * N[Min[M, D], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(N[(N[(N[(1.0 / d), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision] * d), $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)))) < 1e281

   1. Initial program 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      8. associate-/l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      9. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      10. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot \left(M \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right)\right)} \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{2 \cdot d} \cdot \left(M \cdot \left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \frac{1}{2}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{2 \cdot d} \cdot \left(M \cdot \left(\frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \frac{1}{2}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      13. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{2 \cdot d} \cdot \left(M \cdot \left(\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \frac{1}{2}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      14. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{2 \cdot d} \cdot \left(M \cdot \left(\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \frac{1}{2}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      15. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{2 \cdot d} \cdot \left(M \cdot \color{blue}{\left(\frac{D}{2 \cdot d} \cdot \left(M \cdot \frac{1}{2}\right)\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{2 \cdot d} \cdot \left(M \cdot \left(\frac{D}{2 \cdot d} \cdot \left(M \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      17. mult-flip N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{2 \cdot d} \cdot \left(M \cdot \left(\frac{D}{2 \cdot d} \cdot \color{blue}{\frac{M}{2}}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{2 \cdot d} \cdot \color{blue}{\left(\left(M \cdot \frac{D}{2 \cdot d}\right) \cdot \frac{M}{2}\right)}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 58.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right)\right)} \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 64.3%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{D \cdot \left(\left(\frac{D}{d} \cdot M\right) \cdot M\right)}{d} \cdot -0.125, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]

   if 1e281 < (*.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 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. mult-flip N/A

         \[\leadsto \sqrt{\color{blue}{\left(d \cdot \frac{1}{h}\right)} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. associate-*l* N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \left(\frac{1}{h} \cdot \frac{d}{\ell}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. sqrt-prod N/A

          \[\leadsto \color{blue}{\left(\sqrt{d} \cdot \sqrt{\frac{1}{h} \cdot \frac{d}{\ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-unsound-*.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{d} \cdot \sqrt{\frac{1}{h} \cdot \frac{d}{\ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-unsound-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{d}} \cdot \sqrt{\frac{1}{h} \cdot \frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-unsound-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{d} \cdot \color{blue}{\sqrt{\frac{1}{h} \cdot \frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{1}{h} \cdot \color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. frac-2neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{1}{h} \cdot \color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. frac-2neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{1}{h} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. frac-times N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\color{blue}{\frac{1 \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)\right)}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. remove-double-neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{1 \cdot \color{blue}{d}}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. *-lft-identity N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{\color{blue}{d}}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. remove-double-neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{d}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(h\right)\right)\right)\right)} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      22. remove-double-neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{d}{\color{blue}{h} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      23. remove-double-neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{d}{h \cdot \color{blue}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      24. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\color{blue}{\frac{d}{h \cdot \ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 32.2%

      \[\leadsto \color{blue}{\left(\sqrt{d} \cdot \sqrt{\frac{d}{h \cdot \ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \color{blue}{{d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {d}^{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{d}}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \color{blue}{\sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\color{blue}{\frac{1}{d \cdot \left(h \cdot \ell\right)}}}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{\color{blue}{1}}{d \cdot \left(h \cdot \ell\right)}}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right) \]

      9. lower-*.f64 13.9%

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right) \]

   6. Applied rewrites 13.9%

      \[\leadsto \color{blue}{{d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right)} \]

   8. Applied rewrites 39.0%

      \[\leadsto \left|\left(\frac{\frac{1}{d}}{\sqrt{h \cdot \ell}} \cdot d\right) \cdot d\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 70.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left|\left(\frac{\frac{1}{d}}{\sqrt{h \cdot \ell}} \cdot d\right) \cdot d\right|\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\mathsf{min}\left(M, D\right) \cdot \mathsf{max}\left(M, D\right)}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-132}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{max}\left(M, D\right) \cdot \left(\left(\frac{\mathsf{max}\left(M, D\right)}{d} \cdot \mathsf{min}\left(M, D\right)\right) \cdot \mathsf{min}\left(M, D\right)\right)}{d} \cdot -0.125, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h \cdot \ell} \cdot d}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+281}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (fabs (* (* (/ (/ 1.0 d) (sqrt (* h l))) d) d)))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (*
            (* (/ 1.0 2.0) (pow (/ (* (fmin M D) (fmax M D)) (* 2.0 d)) 2.0))
            (/ h l))))))
   (if (<= t_1 -4e-132)
     (*
      (fma
       (*
        (/ (* (fmax M D) (* (* (/ (fmax M D) d) (fmin M D)) (fmin M D))) d)
        -0.125)
       (/ h l)
       1.0)
      (sqrt (* (/ d (* h l)) d)))
     (if (<= t_1 0.0)
       t_0
       (if (<= t_1 1e+281) (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0) t_0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fabs(((((1.0 / d) / sqrt((h * l))) * d) * d));
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((fmin(M, D) * fmax(M, D)) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= -4e-132) {
		tmp = fma((((fmax(M, D) * (((fmax(M, D) / d) * fmin(M, D)) * fmin(M, D))) / d) * -0.125), (h / l), 1.0) * sqrt(((d / (h * l)) * d));
	} else if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 1e+281) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = abs(Float64(Float64(Float64(Float64(1.0 / d) / sqrt(Float64(h * l))) * d) * d))
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(fmin(M, D) * fmax(M, D)) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_1 <= -4e-132)
		tmp = Float64(fma(Float64(Float64(Float64(fmax(M, D) * Float64(Float64(Float64(fmax(M, D) / d) * fmin(M, D)) * fmin(M, D))) / d) * -0.125), Float64(h / l), 1.0) * sqrt(Float64(Float64(d / Float64(h * l)) * d)));
	elseif (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 1e+281)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Abs[N[(N[(N[(N[(1.0 / d), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(N[Min[M, D], $MachinePrecision] * N[Max[M, D], $MachinePrecision]), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-132], N[(N[(N[(N[(N[(N[Max[M, D], $MachinePrecision] * N[(N[(N[(N[Max[M, D], $MachinePrecision] / d), $MachinePrecision] * N[Min[M, D], $MachinePrecision]), $MachinePrecision] * N[Min[M, D], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(d / N[(h * l), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], t$95$0, If[LessEqual[t$95$1, 1e+281], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $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)))) < -3.9999999999999999e-132

   1. Initial program 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      8. associate-/l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      9. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      10. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot \left(M \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right)\right)} \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{2 \cdot d} \cdot \left(M \cdot \left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \frac{1}{2}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{2 \cdot d} \cdot \left(M \cdot \left(\frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \frac{1}{2}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      13. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{2 \cdot d} \cdot \left(M \cdot \left(\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \frac{1}{2}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      14. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{2 \cdot d} \cdot \left(M \cdot \left(\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \frac{1}{2}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      15. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{2 \cdot d} \cdot \left(M \cdot \color{blue}{\left(\frac{D}{2 \cdot d} \cdot \left(M \cdot \frac{1}{2}\right)\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{2 \cdot d} \cdot \left(M \cdot \left(\frac{D}{2 \cdot d} \cdot \left(M \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      17. mult-flip N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{2 \cdot d} \cdot \left(M \cdot \left(\frac{D}{2 \cdot d} \cdot \color{blue}{\frac{M}{2}}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{2 \cdot d} \cdot \color{blue}{\left(\left(M \cdot \frac{D}{2 \cdot d}\right) \cdot \frac{M}{2}\right)}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 58.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right)\right)} \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 49.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{D \cdot \left(\left(\frac{D}{d} \cdot M\right) \cdot M\right)}{d} \cdot -0.125, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h \cdot \ell} \cdot d}} \]

   if -3.9999999999999999e-132 < (*.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 1e281 < (*.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 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. mult-flip N/A

         \[\leadsto \sqrt{\color{blue}{\left(d \cdot \frac{1}{h}\right)} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. associate-*l* N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \left(\frac{1}{h} \cdot \frac{d}{\ell}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. sqrt-prod N/A

          \[\leadsto \color{blue}{\left(\sqrt{d} \cdot \sqrt{\frac{1}{h} \cdot \frac{d}{\ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-unsound-*.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{d} \cdot \sqrt{\frac{1}{h} \cdot \frac{d}{\ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-unsound-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{d}} \cdot \sqrt{\frac{1}{h} \cdot \frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-unsound-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{d} \cdot \color{blue}{\sqrt{\frac{1}{h} \cdot \frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{1}{h} \cdot \color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. frac-2neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{1}{h} \cdot \color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. frac-2neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{1}{h} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. frac-times N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\color{blue}{\frac{1 \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)\right)}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. remove-double-neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{1 \cdot \color{blue}{d}}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. *-lft-identity N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{\color{blue}{d}}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. remove-double-neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{d}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(h\right)\right)\right)\right)} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      22. remove-double-neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{d}{\color{blue}{h} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      23. remove-double-neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{d}{h \cdot \color{blue}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      24. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\color{blue}{\frac{d}{h \cdot \ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 32.2%

      \[\leadsto \color{blue}{\left(\sqrt{d} \cdot \sqrt{\frac{d}{h \cdot \ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \color{blue}{{d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {d}^{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{d}}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \color{blue}{\sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\color{blue}{\frac{1}{d \cdot \left(h \cdot \ell\right)}}}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{\color{blue}{1}}{d \cdot \left(h \cdot \ell\right)}}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right) \]

      9. lower-*.f64 13.9%

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right) \]

   6. Applied rewrites 13.9%

      \[\leadsto \color{blue}{{d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right)} \]

   8. Applied rewrites 39.0%

      \[\leadsto \left|\left(\frac{\frac{1}{d}}{\sqrt{h \cdot \ell}} \cdot d\right) \cdot d\right| \]

   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)))) < 1e281

   1. Initial program 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-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 66.4%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      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 66.4%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      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 66.4%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 66.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 39.1%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 65.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left|\left(\frac{\frac{1}{d}}{\sqrt{h \cdot \ell}} \cdot d\right) \cdot d\right|\\ t_1 := \mathsf{min}\left(\left|M\right|, D\right)\\ t_2 := \mathsf{max}\left(\left|M\right|, D\right)\\ t_3 := t\_1 \cdot t\_2\\ t_4 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_3}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_4 \leq -2 \cdot 10^{-71}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{t\_3 \cdot t\_2}{d \cdot d} \cdot -0.125\right) \cdot t\_1, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h \cdot \ell} \cdot d}\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_4 \leq 10^{+281}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (fabs (* (* (/ (/ 1.0 d) (sqrt (* h l))) d) d)))
        (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))))))
   (if (<= t_4 -2e-71)
     (*
      (fma (* (* (/ (* t_3 t_2) (* d d)) -0.125) t_1) (/ h l) 1.0)
      (sqrt (* (/ d (* h l)) d)))
     (if (<= t_4 0.0)
       t_0
       (if (<= t_4 1e+281) (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0) t_0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fabs(((((1.0 / d) / sqrt((h * l))) * d) * d));
	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 tmp;
	if (t_4 <= -2e-71) {
		tmp = fma(((((t_3 * t_2) / (d * d)) * -0.125) * t_1), (h / l), 1.0) * sqrt(((d / (h * l)) * d));
	} else if (t_4 <= 0.0) {
		tmp = t_0;
	} else if (t_4 <= 1e+281) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = abs(Float64(Float64(Float64(Float64(1.0 / d) / sqrt(Float64(h * l))) * d) * d))
	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))))
	tmp = 0.0
	if (t_4 <= -2e-71)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(t_3 * t_2) / Float64(d * d)) * -0.125) * t_1), Float64(h / l), 1.0) * sqrt(Float64(Float64(d / Float64(h * l)) * d)));
	elseif (t_4 <= 0.0)
		tmp = t_0;
	elseif (t_4 <= 1e+281)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Abs[N[(N[(N[(N[(1.0 / d), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision] * d), $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]}, If[LessEqual[t$95$4, -2e-71], N[(N[(N[(N[(N[(N[(t$95$3 * t$95$2), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(d / N[(h * l), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.0], t$95$0, If[LessEqual[t$95$4, 1e+281], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $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)))) < -1.9999999999999998e-71

   1. Initial program 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-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 66.4%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      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 66.4%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      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 66.4%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 66.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      4. frac-2neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{\mathsf{neg}\left(M \cdot D\right)}{\mathsf{neg}\left(2 \cdot d\right)}} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      5. distribute-frac-neg2 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(M \cdot D\right)}{2 \cdot d}\right)\right)} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(M \cdot D\right)}{2 \cdot d}\right)\right) \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      7. frac-2neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(M \cdot D\right)}{2 \cdot d}\right)\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(M \cdot D\right)}{\mathsf{neg}\left(2 \cdot d\right)}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      8. distribute-frac-neg2 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(M \cdot D\right)}{2 \cdot d}\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(M \cdot D\right)}{2 \cdot d}\right)\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      9. sqr-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{\mathsf{neg}\left(M \cdot D\right)}{2 \cdot d} \cdot \frac{\mathsf{neg}\left(M \cdot D\right)}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{\mathsf{neg}\left(M \cdot D\right)}{2 \cdot d} \cdot \frac{\mathsf{neg}\left(M \cdot D\right)}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      11. distribute-neg-frac N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right)} \cdot \frac{\mathsf{neg}\left(M \cdot D\right)}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      12. distribute-neg-frac2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}} \cdot \frac{\mathsf{neg}\left(M \cdot D\right)}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}} \cdot \frac{\mathsf{neg}\left(M \cdot D\right)}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{\color{blue}{M \cdot D}}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{\mathsf{neg}\left(M \cdot D\right)}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      15. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{\mathsf{neg}\left(M \cdot D\right)}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{\mathsf{neg}\left(M \cdot D\right)}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\mathsf{neg}\left(\color{blue}{2 \cdot d}\right)} \cdot \frac{\mathsf{neg}\left(M \cdot D\right)}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      18. distribute-lft-neg-in N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}} \cdot \frac{\mathsf{neg}\left(M \cdot D\right)}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}} \cdot \frac{\mathsf{neg}\left(M \cdot D\right)}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      20. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\color{blue}{-2} \cdot d} \cdot \frac{\mathsf{neg}\left(M \cdot D\right)}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      21. distribute-neg-frac N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      22. distribute-neg-frac2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \color{blue}{\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      23. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \color{blue}{\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      24. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{\color{blue}{M \cdot D}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      25. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      26. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      27. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\mathsf{neg}\left(\color{blue}{2 \cdot d}\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      28. distribute-lft-neg-in N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}}\right)\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 66.4%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{-2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

   7. Applied rewrites 40.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{\left(M \cdot D\right) \cdot D}{d \cdot d} \cdot -0.125\right) \cdot M, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h \cdot \ell} \cdot d}} \]

   if -1.9999999999999998e-71 < (*.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 1e281 < (*.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 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. mult-flip N/A

         \[\leadsto \sqrt{\color{blue}{\left(d \cdot \frac{1}{h}\right)} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. associate-*l* N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \left(\frac{1}{h} \cdot \frac{d}{\ell}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. sqrt-prod N/A

          \[\leadsto \color{blue}{\left(\sqrt{d} \cdot \sqrt{\frac{1}{h} \cdot \frac{d}{\ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-unsound-*.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{d} \cdot \sqrt{\frac{1}{h} \cdot \frac{d}{\ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-unsound-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{d}} \cdot \sqrt{\frac{1}{h} \cdot \frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-unsound-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{d} \cdot \color{blue}{\sqrt{\frac{1}{h} \cdot \frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{1}{h} \cdot \color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. frac-2neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{1}{h} \cdot \color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. frac-2neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{1}{h} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. frac-times N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\color{blue}{\frac{1 \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)\right)}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. remove-double-neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{1 \cdot \color{blue}{d}}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. *-lft-identity N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{\color{blue}{d}}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. remove-double-neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{d}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(h\right)\right)\right)\right)} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      22. remove-double-neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{d}{\color{blue}{h} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      23. remove-double-neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{d}{h \cdot \color{blue}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      24. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\color{blue}{\frac{d}{h \cdot \ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 32.2%

      \[\leadsto \color{blue}{\left(\sqrt{d} \cdot \sqrt{\frac{d}{h \cdot \ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \color{blue}{{d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {d}^{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{d}}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \color{blue}{\sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\color{blue}{\frac{1}{d \cdot \left(h \cdot \ell\right)}}}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{\color{blue}{1}}{d \cdot \left(h \cdot \ell\right)}}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right) \]

      9. lower-*.f64 13.9%

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right) \]

   6. Applied rewrites 13.9%

      \[\leadsto \color{blue}{{d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right)} \]

   8. Applied rewrites 39.0%

      \[\leadsto \left|\left(\frac{\frac{1}{d}}{\sqrt{h \cdot \ell}} \cdot d\right) \cdot d\right| \]

   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)))) < 1e281

   1. Initial program 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-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 66.4%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      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 66.4%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      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 66.4%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 66.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 39.1%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 55.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \left|\left(\frac{\frac{1}{d}}{\sqrt{h \cdot \ell}} \cdot d\right) \cdot d\right|\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-132}:\\ \;\;\;\;-1 \cdot \sqrt{\frac{{d}^{2}}{h \cdot \ell}}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+281}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

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 (* (* (/ (/ 1.0 d) (sqrt (* h l))) d) d))))
   (if (<= t_0 -4e-132)
     (* -1.0 (sqrt (/ (pow d 2.0) (* h l))))
     (if (<= t_0 0.0)
       t_1
       (if (<= t_0 1e+281) (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0) t_1)))))

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(((((1.0 / d) / sqrt((h * l))) * d) * d));
	double tmp;
	if (t_0 <= -4e-132) {
		tmp = -1.0 * sqrt((pow(d, 2.0) / (h * l)));
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 1e+281) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((d / 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_1 = abs(((((1.0d0 / d) / sqrt((h * l))) * d) * d))
    if (t_0 <= (-4d-132)) then
        tmp = (-1.0d0) * sqrt(((d ** 2.0d0) / (h * l)))
    else if (t_0 <= 0.0d0) then
        tmp = t_1
    else if (t_0 <= 1d+281) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0d0
    else
        tmp = t_1
    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))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = Math.abs(((((1.0 / d) / Math.sqrt((h * l))) * d) * d));
	double tmp;
	if (t_0 <= -4e-132) {
		tmp = -1.0 * Math.sqrt((Math.pow(d, 2.0) / (h * l)));
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 1e+281) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * 1.0;
	} else {
		tmp = t_1;
	}
	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))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_1 = math.fabs(((((1.0 / d) / math.sqrt((h * l))) * d) * d))
	tmp = 0
	if t_0 <= -4e-132:
		tmp = -1.0 * math.sqrt((math.pow(d, 2.0) / (h * l)))
	elif t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 1e+281:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * 1.0
	else:
		tmp = t_1
	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 = abs(Float64(Float64(Float64(Float64(1.0 / d) / sqrt(Float64(h * l))) * d) * d))
	tmp = 0.0
	if (t_0 <= -4e-132)
		tmp = Float64(-1.0 * sqrt(Float64((d ^ 2.0) / Float64(h * l))));
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 1e+281)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	else
		tmp = t_1;
	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))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_1 = abs(((((1.0 / d) / sqrt((h * l))) * d) * d));
	tmp = 0.0;
	if (t_0 <= -4e-132)
		tmp = -1.0 * sqrt(((d ^ 2.0) / (h * l)));
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 1e+281)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = 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[Abs[N[(N[(N[(N[(1.0 / d), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -4e-132], N[(-1.0 * N[Sqrt[N[(N[Power[d, 2.0], $MachinePrecision] / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 1e+281], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 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)))) < -3.9999999999999999e-132

   1. Initial program 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. 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. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. sqrt-unprod N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      6. lower-*.f64 21.6%

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      8. *-commutative N/A

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

      9. lower-*.f64 21.6%

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   6. Applied rewrites 21.6%

      \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{\color{blue}{h}} \]

   7. Taylor expanded in h around -inf

      \[\leadsto -1 \cdot \color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]

      3. lower-/.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]

      4. lower-pow.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]

      5. lower-*.f64 11.8%

         \[\leadsto -1 \cdot \sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]

   9. Applied rewrites 11.8%

      \[\leadsto -1 \cdot \color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell}}} \]

   if -3.9999999999999999e-132 < (*.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 1e281 < (*.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 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. mult-flip N/A

         \[\leadsto \sqrt{\color{blue}{\left(d \cdot \frac{1}{h}\right)} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. associate-*l* N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \left(\frac{1}{h} \cdot \frac{d}{\ell}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. sqrt-prod N/A

          \[\leadsto \color{blue}{\left(\sqrt{d} \cdot \sqrt{\frac{1}{h} \cdot \frac{d}{\ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-unsound-*.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{d} \cdot \sqrt{\frac{1}{h} \cdot \frac{d}{\ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-unsound-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{d}} \cdot \sqrt{\frac{1}{h} \cdot \frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-unsound-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{d} \cdot \color{blue}{\sqrt{\frac{1}{h} \cdot \frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{1}{h} \cdot \color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. frac-2neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{1}{h} \cdot \color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. frac-2neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{1}{h} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. frac-times N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\color{blue}{\frac{1 \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)\right)}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. remove-double-neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{1 \cdot \color{blue}{d}}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. *-lft-identity N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{\color{blue}{d}}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. remove-double-neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{d}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(h\right)\right)\right)\right)} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      22. remove-double-neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{d}{\color{blue}{h} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      23. remove-double-neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{d}{h \cdot \color{blue}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      24. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\color{blue}{\frac{d}{h \cdot \ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 32.2%

      \[\leadsto \color{blue}{\left(\sqrt{d} \cdot \sqrt{\frac{d}{h \cdot \ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \color{blue}{{d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {d}^{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{d}}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \color{blue}{\sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\color{blue}{\frac{1}{d \cdot \left(h \cdot \ell\right)}}}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{\color{blue}{1}}{d \cdot \left(h \cdot \ell\right)}}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right) \]

      9. lower-*.f64 13.9%

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right) \]

   6. Applied rewrites 13.9%

      \[\leadsto \color{blue}{{d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right)} \]

   8. Applied rewrites 39.0%

      \[\leadsto \left|\left(\frac{\frac{1}{d}}{\sqrt{h \cdot \ell}} \cdot d\right) \cdot d\right| \]

   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)))) < 1e281

   1. Initial program 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-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 66.4%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      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 66.4%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      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 66.4%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 66.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 39.1%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 55.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \left|\left(\frac{\frac{1}{d}}{\sqrt{h \cdot \ell}} \cdot d\right) \cdot d\right|\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-71}:\\ \;\;\;\;\frac{\sqrt{d \cdot h} \cdot \left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{d \cdot \ell}}\right)\right)}{h}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+281}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

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 (* (* (/ (/ 1.0 d) (sqrt (* h l))) d) d))))
   (if (<= t_0 -2e-71)
     (/ (* (sqrt (* d h)) (* -1.0 (* d (sqrt (/ 1.0 (* d l)))))) h)
     (if (<= t_0 0.0)
       t_1
       (if (<= t_0 1e+281) (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0) t_1)))))

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(((((1.0 / d) / sqrt((h * l))) * d) * d));
	double tmp;
	if (t_0 <= -2e-71) {
		tmp = (sqrt((d * h)) * (-1.0 * (d * sqrt((1.0 / (d * l)))))) / h;
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 1e+281) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((d / 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_1 = abs(((((1.0d0 / d) / sqrt((h * l))) * d) * d))
    if (t_0 <= (-2d-71)) then
        tmp = (sqrt((d * h)) * ((-1.0d0) * (d * sqrt((1.0d0 / (d * l)))))) / h
    else if (t_0 <= 0.0d0) then
        tmp = t_1
    else if (t_0 <= 1d+281) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0d0
    else
        tmp = t_1
    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))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = Math.abs(((((1.0 / d) / Math.sqrt((h * l))) * d) * d));
	double tmp;
	if (t_0 <= -2e-71) {
		tmp = (Math.sqrt((d * h)) * (-1.0 * (d * Math.sqrt((1.0 / (d * l)))))) / h;
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 1e+281) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * 1.0;
	} else {
		tmp = t_1;
	}
	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))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_1 = math.fabs(((((1.0 / d) / math.sqrt((h * l))) * d) * d))
	tmp = 0
	if t_0 <= -2e-71:
		tmp = (math.sqrt((d * h)) * (-1.0 * (d * math.sqrt((1.0 / (d * l)))))) / h
	elif t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 1e+281:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * 1.0
	else:
		tmp = t_1
	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 = abs(Float64(Float64(Float64(Float64(1.0 / d) / sqrt(Float64(h * l))) * d) * d))
	tmp = 0.0
	if (t_0 <= -2e-71)
		tmp = Float64(Float64(sqrt(Float64(d * h)) * Float64(-1.0 * Float64(d * sqrt(Float64(1.0 / Float64(d * l)))))) / h);
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 1e+281)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	else
		tmp = t_1;
	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))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_1 = abs(((((1.0 / d) / sqrt((h * l))) * d) * d));
	tmp = 0.0;
	if (t_0 <= -2e-71)
		tmp = (sqrt((d * h)) * (-1.0 * (d * sqrt((1.0 / (d * l)))))) / h;
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 1e+281)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = 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[Abs[N[(N[(N[(N[(1.0 / d), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -2e-71], N[(N[(N[Sqrt[N[(d * h), $MachinePrecision]], $MachinePrecision] * N[(-1.0 * N[(d * N[Sqrt[N[(1.0 / N[(d * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 1e+281], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.9999999999999998e-71

   1. Initial program 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. 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 d around -inf

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{d \cdot \ell}}\right)\right)}{h} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{d \cdot \ell}}\right)\right)}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{d \cdot \ell}}\right)\right)}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{d \cdot \ell}}\right)\right)}{h} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{d \cdot \ell}}\right)\right)}{h} \]

      5. lower-*.f64 14.3%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{d \cdot \ell}}\right)\right)}{h} \]

   7. Applied rewrites 14.3%

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{d \cdot \ell}}\right)\right)}{h} \]

   if -1.9999999999999998e-71 < (*.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 1e281 < (*.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 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. mult-flip N/A

         \[\leadsto \sqrt{\color{blue}{\left(d \cdot \frac{1}{h}\right)} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. associate-*l* N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \left(\frac{1}{h} \cdot \frac{d}{\ell}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. sqrt-prod N/A

          \[\leadsto \color{blue}{\left(\sqrt{d} \cdot \sqrt{\frac{1}{h} \cdot \frac{d}{\ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-unsound-*.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{d} \cdot \sqrt{\frac{1}{h} \cdot \frac{d}{\ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-unsound-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{d}} \cdot \sqrt{\frac{1}{h} \cdot \frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-unsound-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{d} \cdot \color{blue}{\sqrt{\frac{1}{h} \cdot \frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{1}{h} \cdot \color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. frac-2neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{1}{h} \cdot \color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. frac-2neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{1}{h} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. frac-times N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\color{blue}{\frac{1 \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)\right)}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. remove-double-neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{1 \cdot \color{blue}{d}}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. *-lft-identity N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{\color{blue}{d}}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. remove-double-neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{d}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(h\right)\right)\right)\right)} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      22. remove-double-neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{d}{\color{blue}{h} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      23. remove-double-neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{d}{h \cdot \color{blue}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      24. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\color{blue}{\frac{d}{h \cdot \ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 32.2%

      \[\leadsto \color{blue}{\left(\sqrt{d} \cdot \sqrt{\frac{d}{h \cdot \ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \color{blue}{{d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {d}^{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{d}}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \color{blue}{\sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\color{blue}{\frac{1}{d \cdot \left(h \cdot \ell\right)}}}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{\color{blue}{1}}{d \cdot \left(h \cdot \ell\right)}}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right) \]

      9. lower-*.f64 13.9%

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right) \]

   6. Applied rewrites 13.9%

      \[\leadsto \color{blue}{{d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right)} \]

   8. Applied rewrites 39.0%

      \[\leadsto \left|\left(\frac{\frac{1}{d}}{\sqrt{h \cdot \ell}} \cdot d\right) \cdot d\right| \]

   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)))) < 1e281

   1. Initial program 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-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 66.4%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      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 66.4%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      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 66.4%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 66.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 39.1%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 54.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left|\left(\frac{\frac{1}{d}}{\sqrt{h \cdot \ell}} \cdot d\right) \cdot d\right|\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-132}:\\ \;\;\;\;\frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+281}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (fabs (* (* (/ (/ 1.0 d) (sqrt (* h l))) d) d)))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_1 -4e-132)
     (/ (* -1.0 (* d (sqrt (/ h l)))) h)
     (if (<= t_1 0.0)
       t_0
       (if (<= t_1 1e+281) (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0) t_0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fabs(((((1.0 / d) / sqrt((h * l))) * d) * d));
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= -4e-132) {
		tmp = (-1.0 * (d * sqrt((h / l)))) / h;
	} else if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 1e+281) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	} 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(((((1.0d0 / d) / sqrt((h * l))) * d) * d))
    t_1 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_1 <= (-4d-132)) then
        tmp = ((-1.0d0) * (d * sqrt((h / l)))) / h
    else if (t_1 <= 0.0d0) then
        tmp = t_0
    else if (t_1 <= 1d+281) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0d0
    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(((((1.0 / d) / Math.sqrt((h * l))) * d) * d));
	double t_1 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= -4e-132) {
		tmp = (-1.0 * (d * Math.sqrt((h / l)))) / h;
	} else if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 1e+281) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.fabs(((((1.0 / d) / math.sqrt((h * l))) * d) * d))
	t_1 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_1 <= -4e-132:
		tmp = (-1.0 * (d * math.sqrt((h / l)))) / h
	elif t_1 <= 0.0:
		tmp = t_0
	elif t_1 <= 1e+281:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = abs(Float64(Float64(Float64(Float64(1.0 / d) / sqrt(Float64(h * l))) * d) * d))
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_1 <= -4e-132)
		tmp = Float64(Float64(-1.0 * Float64(d * sqrt(Float64(h / l)))) / h);
	elseif (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 1e+281)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = abs(((((1.0 / d) / sqrt((h * l))) * d) * d));
	t_1 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_1 <= -4e-132)
		tmp = (-1.0 * (d * sqrt((h / l)))) / h;
	elseif (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 1e+281)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Abs[N[(N[(N[(N[(1.0 / d), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-132], N[(N[(-1.0 * N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[t$95$1, 0.0], t$95$0, If[LessEqual[t$95$1, 1e+281], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $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)))) < -3.9999999999999999e-132

   1. Initial program 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. 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. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. sqrt-unprod N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      6. lower-*.f64 21.6%

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      8. *-commutative N/A

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

      9. lower-*.f64 21.6%

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   6. Applied rewrites 21.6%

      \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{\color{blue}{h}} \]

   7. Taylor expanded in d around -inf

      \[\leadsto \frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]

      4. lower-/.f64 13.1%

         \[\leadsto \frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]

   9. Applied rewrites 13.1%

      \[\leadsto \frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]

   if -3.9999999999999999e-132 < (*.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 1e281 < (*.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 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. mult-flip N/A

         \[\leadsto \sqrt{\color{blue}{\left(d \cdot \frac{1}{h}\right)} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. associate-*l* N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \left(\frac{1}{h} \cdot \frac{d}{\ell}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. sqrt-prod N/A

          \[\leadsto \color{blue}{\left(\sqrt{d} \cdot \sqrt{\frac{1}{h} \cdot \frac{d}{\ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-unsound-*.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{d} \cdot \sqrt{\frac{1}{h} \cdot \frac{d}{\ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-unsound-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{d}} \cdot \sqrt{\frac{1}{h} \cdot \frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-unsound-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{d} \cdot \color{blue}{\sqrt{\frac{1}{h} \cdot \frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{1}{h} \cdot \color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. frac-2neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{1}{h} \cdot \color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. frac-2neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{1}{h} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. frac-times N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\color{blue}{\frac{1 \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)\right)}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. remove-double-neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{1 \cdot \color{blue}{d}}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. *-lft-identity N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{\color{blue}{d}}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. remove-double-neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{d}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(h\right)\right)\right)\right)} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      22. remove-double-neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{d}{\color{blue}{h} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      23. remove-double-neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{d}{h \cdot \color{blue}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      24. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\color{blue}{\frac{d}{h \cdot \ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 32.2%

      \[\leadsto \color{blue}{\left(\sqrt{d} \cdot \sqrt{\frac{d}{h \cdot \ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \color{blue}{{d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {d}^{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{d}}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \color{blue}{\sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\color{blue}{\frac{1}{d \cdot \left(h \cdot \ell\right)}}}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{\color{blue}{1}}{d \cdot \left(h \cdot \ell\right)}}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right) \]

      9. lower-*.f64 13.9%

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right) \]

   6. Applied rewrites 13.9%

      \[\leadsto \color{blue}{{d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right)} \]

   8. Applied rewrites 39.0%

      \[\leadsto \left|\left(\frac{\frac{1}{d}}{\sqrt{h \cdot \ell}} \cdot d\right) \cdot d\right| \]

   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)))) < 1e281

   1. Initial program 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-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 66.4%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      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 66.4%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      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 66.4%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 66.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 39.1%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 43.8% accurate, 4.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\ell \leq -7.5 \cdot 10^{-282}:\\ \;\;\;\;-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{-194}:\\ \;\;\;\;\frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -7.5e-282)
   (* -1.0 (* d (sqrt (/ 1.0 (* h l)))))
   (if (<= l 1.25e-194)
     (/ (* -1.0 (* d (sqrt (/ h l)))) h)
     (/ (* d (/ (sqrt h) (sqrt l))) h))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -7.5e-282) {
		tmp = -1.0 * (d * sqrt((1.0 / (h * l))));
	} else if (l <= 1.25e-194) {
		tmp = (-1.0 * (d * sqrt((h / l)))) / h;
	} else {
		tmp = (d * (sqrt(h) / sqrt(l))) / h;
	}
	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 (l <= (-7.5d-282)) then
        tmp = (-1.0d0) * (d * sqrt((1.0d0 / (h * l))))
    else if (l <= 1.25d-194) then
        tmp = ((-1.0d0) * (d * sqrt((h / l)))) / h
    else
        tmp = (d * (sqrt(h) / sqrt(l))) / h
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -7.5e-282) {
		tmp = -1.0 * (d * Math.sqrt((1.0 / (h * l))));
	} else if (l <= 1.25e-194) {
		tmp = (-1.0 * (d * Math.sqrt((h / l)))) / h;
	} else {
		tmp = (d * (Math.sqrt(h) / Math.sqrt(l))) / h;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -7.5e-282:
		tmp = -1.0 * (d * math.sqrt((1.0 / (h * l))))
	elif l <= 1.25e-194:
		tmp = (-1.0 * (d * math.sqrt((h / l)))) / h
	else:
		tmp = (d * (math.sqrt(h) / math.sqrt(l))) / h
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -7.5e-282)
		tmp = Float64(-1.0 * Float64(d * sqrt(Float64(1.0 / Float64(h * l)))));
	elseif (l <= 1.25e-194)
		tmp = Float64(Float64(-1.0 * Float64(d * sqrt(Float64(h / l)))) / h);
	else
		tmp = Float64(Float64(d * Float64(sqrt(h) / sqrt(l))) / h);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -7.5e-282)
		tmp = -1.0 * (d * sqrt((1.0 / (h * l))));
	elseif (l <= 1.25e-194)
		tmp = (-1.0 * (d * sqrt((h / l)))) / h;
	else
		tmp = (d * (sqrt(h) / sqrt(l))) / h;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -7.5e-282], N[(-1.0 * N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.25e-194], N[(N[(-1.0 * N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(N[(d * N[(N[Sqrt[h], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -7.4999999999999994e-282

   1. Initial program 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 29.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{h}{\ell}, \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot -0.5\right)\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}, \sqrt{d \cdot \frac{d}{h \cdot \ell}}\right)} \]

   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 25.9%

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   6. Applied rewrites 25.9%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   if -7.4999999999999994e-282 < l < 1.2500000000000001e-194

   1. Initial program 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. 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. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. sqrt-unprod N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      6. lower-*.f64 21.6%

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      8. *-commutative N/A

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

      9. lower-*.f64 21.6%

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   6. Applied rewrites 21.6%

      \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{\color{blue}{h}} \]

   7. Taylor expanded in d around -inf

      \[\leadsto \frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]

      4. lower-/.f64 13.1%

         \[\leadsto \frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]

   9. Applied rewrites 13.1%

      \[\leadsto \frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]

   if 1.2500000000000001e-194 < l

   1. Initial program 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. 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. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. sqrt-unprod N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      6. lower-*.f64 21.6%

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      8. *-commutative N/A

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

      9. lower-*.f64 21.6%

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   6. Applied rewrites 21.6%

      \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{\color{blue}{h}} \]

   7. Taylor expanded in d around 0

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      3. lower-/.f64 37.1%

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   9. Applied rewrites 37.1%

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   10. Step-by-step derivation

       1. lift-sqrt.f64 N/A

          \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

       3. sqrt-div N/A

          \[\leadsto \frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h} \]

       4. lower-unsound-/.f64 N/A

          \[\leadsto \frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h} \]

       5. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h} \]

       6. lower-unsound-sqrt.f64 22.3%

          \[\leadsto \frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h} \]

   11. Applied rewrites 22.3%

       \[\leadsto \frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 43.3% accurate, 4.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\ell \leq 1.8 \cdot 10^{-258}:\\ \;\;\;\;-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l 1.8e-258)
   (* -1.0 (* d (sqrt (/ 1.0 (* h l)))))
   (/ (* d (/ (sqrt h) (sqrt l))) h)))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 1.8e-258) {
		tmp = -1.0 * (d * sqrt((1.0 / (h * l))));
	} else {
		tmp = (d * (sqrt(h) / sqrt(l))) / h;
	}
	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 (l <= 1.8d-258) then
        tmp = (-1.0d0) * (d * sqrt((1.0d0 / (h * l))))
    else
        tmp = (d * (sqrt(h) / sqrt(l))) / h
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 1.8e-258) {
		tmp = -1.0 * (d * Math.sqrt((1.0 / (h * l))));
	} else {
		tmp = (d * (Math.sqrt(h) / Math.sqrt(l))) / h;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= 1.8e-258:
		tmp = -1.0 * (d * math.sqrt((1.0 / (h * l))))
	else:
		tmp = (d * (math.sqrt(h) / math.sqrt(l))) / h
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 1.8e-258)
		tmp = Float64(-1.0 * Float64(d * sqrt(Float64(1.0 / Float64(h * l)))));
	else
		tmp = Float64(Float64(d * Float64(sqrt(h) / sqrt(l))) / h);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= 1.8e-258)
		tmp = -1.0 * (d * sqrt((1.0 / (h * l))));
	else
		tmp = (d * (sqrt(h) / sqrt(l))) / h;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, 1.8e-258], N[(-1.0 * N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d * N[(N[Sqrt[h], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.7999999999999999e-258

   1. Initial program 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 29.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{h}{\ell}, \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot -0.5\right)\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}, \sqrt{d \cdot \frac{d}{h \cdot \ell}}\right)} \]

   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 25.9%

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   6. Applied rewrites 25.9%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   if 1.7999999999999999e-258 < l

   1. Initial program 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. 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. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. sqrt-unprod N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      6. lower-*.f64 21.6%

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      8. *-commutative N/A

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

      9. lower-*.f64 21.6%

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   6. Applied rewrites 21.6%

      \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{\color{blue}{h}} \]

   7. Taylor expanded in d around 0

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      3. lower-/.f64 37.1%

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   9. Applied rewrites 37.1%

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   10. Step-by-step derivation

       1. lift-sqrt.f64 N/A

          \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

       3. sqrt-div N/A

          \[\leadsto \frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h} \]

       4. lower-unsound-/.f64 N/A

          \[\leadsto \frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h} \]

       5. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h} \]

       6. lower-unsound-sqrt.f64 22.3%

          \[\leadsto \frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h} \]

   11. Applied rewrites 22.3%

       \[\leadsto \frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 41.0% accurate, 5.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;d \leq 1.4 \cdot 10^{-302}:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d 1.4e-302)
   (/ (* d (sqrt (/ h l))) h)
   (/ (* d (/ (sqrt h) (sqrt l))) h)))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= 1.4e-302) {
		tmp = (d * sqrt((h / l))) / h;
	} else {
		tmp = (d * (sqrt(h) / sqrt(l))) / h;
	}
	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 <= 1.4d-302) then
        tmp = (d * sqrt((h / l))) / h
    else
        tmp = (d * (sqrt(h) / sqrt(l))) / h
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= 1.4e-302) {
		tmp = (d * Math.sqrt((h / l))) / h;
	} else {
		tmp = (d * (Math.sqrt(h) / Math.sqrt(l))) / h;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= 1.4e-302:
		tmp = (d * math.sqrt((h / l))) / h
	else:
		tmp = (d * (math.sqrt(h) / math.sqrt(l))) / h
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= 1.4e-302)
		tmp = Float64(Float64(d * sqrt(Float64(h / l))) / h);
	else
		tmp = Float64(Float64(d * Float64(sqrt(h) / sqrt(l))) / h);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= 1.4e-302)
		tmp = (d * sqrt((h / l))) / h;
	else
		tmp = (d * (sqrt(h) / sqrt(l))) / h;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, 1.4e-302], N[(N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(N[(d * N[(N[Sqrt[h], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.4e-302

   1. Initial program 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. 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. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. sqrt-unprod N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      6. lower-*.f64 21.6%

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      8. *-commutative N/A

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

      9. lower-*.f64 21.6%

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   6. Applied rewrites 21.6%

      \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{\color{blue}{h}} \]

   7. Taylor expanded in d around 0

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      3. lower-/.f64 37.1%

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   9. Applied rewrites 37.1%

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   if 1.4e-302 < d

   1. Initial program 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. 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. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. sqrt-unprod N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      6. lower-*.f64 21.6%

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      8. *-commutative N/A

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

      9. lower-*.f64 21.6%

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   6. Applied rewrites 21.6%

      \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{\color{blue}{h}} \]

   7. Taylor expanded in d around 0

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      3. lower-/.f64 37.1%

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   9. Applied rewrites 37.1%

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   10. Step-by-step derivation

       1. lift-sqrt.f64 N/A

          \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

       3. sqrt-div N/A

          \[\leadsto \frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h} \]

       4. lower-unsound-/.f64 N/A

          \[\leadsto \frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h} \]

       5. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h} \]

       6. lower-unsound-sqrt.f64 22.3%

          \[\leadsto \frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h} \]

   11. Applied rewrites 22.3%

       \[\leadsto \frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 38.1% accurate, 5.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\ell \leq 1.9 \cdot 10^{-150}:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{d} \cdot \sqrt{\frac{d}{h \cdot \ell}}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l 1.9e-150)
   (/ (* d (sqrt (/ h l))) h)
   (* (sqrt d) (sqrt (/ d (* h l))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 1.9e-150) {
		tmp = (d * sqrt((h / l))) / h;
	} else {
		tmp = sqrt(d) * sqrt((d / (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) :: tmp
    if (l <= 1.9d-150) then
        tmp = (d * sqrt((h / l))) / h
    else
        tmp = sqrt(d) * sqrt((d / (h * l)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 1.9e-150) {
		tmp = (d * Math.sqrt((h / l))) / h;
	} else {
		tmp = Math.sqrt(d) * Math.sqrt((d / (h * l)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= 1.9e-150:
		tmp = (d * math.sqrt((h / l))) / h
	else:
		tmp = math.sqrt(d) * math.sqrt((d / (h * l)))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 1.9e-150)
		tmp = Float64(Float64(d * sqrt(Float64(h / l))) / h);
	else
		tmp = Float64(sqrt(d) * sqrt(Float64(d / Float64(h * l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= 1.9e-150)
		tmp = (d * sqrt((h / l))) / h;
	else
		tmp = sqrt(d) * sqrt((d / (h * l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, 1.9e-150], N[(N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(N[Sqrt[d], $MachinePrecision] * N[Sqrt[N[(d / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.8999999999999999e-150

   1. Initial program 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. 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. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. sqrt-unprod N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      6. lower-*.f64 21.6%

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      8. *-commutative N/A

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

      9. lower-*.f64 21.6%

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   6. Applied rewrites 21.6%

      \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{\color{blue}{h}} \]

   7. Taylor expanded in d around 0

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      3. lower-/.f64 37.1%

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   9. Applied rewrites 37.1%

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   if 1.8999999999999999e-150 < l

   1. Initial program 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. mult-flip N/A

         \[\leadsto \sqrt{\color{blue}{\left(d \cdot \frac{1}{h}\right)} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. associate-*l* N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \left(\frac{1}{h} \cdot \frac{d}{\ell}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. sqrt-prod N/A

          \[\leadsto \color{blue}{\left(\sqrt{d} \cdot \sqrt{\frac{1}{h} \cdot \frac{d}{\ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-unsound-*.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{d} \cdot \sqrt{\frac{1}{h} \cdot \frac{d}{\ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-unsound-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{d}} \cdot \sqrt{\frac{1}{h} \cdot \frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-unsound-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{d} \cdot \color{blue}{\sqrt{\frac{1}{h} \cdot \frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{1}{h} \cdot \color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. frac-2neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{1}{h} \cdot \color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. frac-2neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{1}{h} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. frac-times N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\color{blue}{\frac{1 \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)\right)}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. remove-double-neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{1 \cdot \color{blue}{d}}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. *-lft-identity N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{\color{blue}{d}}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. remove-double-neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{d}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(h\right)\right)\right)\right)} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      22. remove-double-neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{d}{\color{blue}{h} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      23. remove-double-neg N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\frac{d}{h \cdot \color{blue}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      24. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{d} \cdot \sqrt{\color{blue}{\frac{d}{h \cdot \ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 32.2%

      \[\leadsto \color{blue}{\left(\sqrt{d} \cdot \sqrt{\frac{d}{h \cdot \ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \color{blue}{{d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {d}^{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{d}}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \color{blue}{\sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\color{blue}{\frac{1}{d \cdot \left(h \cdot \ell\right)}}}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{\color{blue}{1}}{d \cdot \left(h \cdot \ell\right)}}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right) \]

      9. lower-*.f64 13.9%

         \[\leadsto {d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right) \]

   6. Applied rewrites 13.9%

      \[\leadsto \color{blue}{{d}^{2} \cdot \left(\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d \cdot \left(h \cdot \ell\right)}}\right)} \]

   7. Taylor expanded in d around 0

      \[\leadsto \sqrt{d} \cdot \color{blue}{\sqrt{\frac{d}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{d} \cdot \sqrt{\frac{d}{h \cdot \ell}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{d} \cdot \sqrt{\frac{d}{h \cdot \ell}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{d} \cdot \sqrt{\frac{d}{h \cdot \ell}} \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{d} \cdot \sqrt{\frac{d}{h \cdot \ell}} \]

      5. lower-*.f64 19.4%

         \[\leadsto \sqrt{d} \cdot \sqrt{\frac{d}{h \cdot \ell}} \]

   9. Applied rewrites 19.4%

      \[\leadsto \sqrt{d} \cdot \color{blue}{\sqrt{\frac{d}{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 37.1% accurate, 7.4× speedup

Math

\[\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

FPCore

(FPCore (d h l M D) :precision binary64 (/ (* d (sqrt (/ h l))) h))

C

double code(double d, double h, double l, double M, double D) {
	return (d * sqrt((h / l))) / h;
}

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((h / l))) / h
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return (d * Math.sqrt((h / l))) / h;
}

Python

def code(d, h, l, M, D):
	return (d * math.sqrt((h / l))) / h

Julia

function code(d, h, l, M, D)
	return Float64(Float64(d * sqrt(Float64(h / l))) / h)
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = (d * sqrt((h / l))) / h;
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]

Derivation

1. Initial program 66.4%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

2. Taylor expanded in h around 0

   \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

   2. 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. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   2. lift-sqrt.f64 N/A

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   3. lift-sqrt.f64 N/A

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. sqrt-unprod N/A

      \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

   6. lower-*.f64 21.6%

      \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

   8. *-commutative N/A

      \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   9. lower-*.f64 21.6%

      \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

6. Applied rewrites 21.6%

   \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{\color{blue}{h}} \]

7. Taylor expanded in d around 0

   \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   2. lower-sqrt.f64 N/A

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   3. lower-/.f64 37.1%

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

9. Applied rewrites 37.1%

   \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025191 
(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)))))