Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.7% → 83.0%

Time: 10.5s

Alternatives: 20

Speedup: 0.3×

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: 66.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 83.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_1 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_2 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_1 \cdot t\_0}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_3 := \frac{t\_1}{\left(d \cdot d\right) \cdot \ell}\\ t_4 := \left(\frac{-0.5}{d} \cdot t\_1\right) \cdot t\_0\\ \mathbf{if}\;t\_2 \leq 2 \cdot 10^{+204}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot t\_4\right) \cdot \left(t\_4 \cdot 0.5\right)\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot h, t\_0 \cdot \left(\left(0.25 \cdot t\_0\right) \cdot \left(t\_1 \cdot t\_3\right)\right), 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(\left(t\_0 \cdot t\_0\right) \cdot 0.25\right) \cdot t\_1\right) \cdot t\_3, -0.5 \cdot h, 1\right) \cdot \left|d\right|}{\sqrt{h \cdot \ell}}\\ \end{array} \]

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
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* t_1 t_0) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_3 (/ t_1 (* (* d d) l)))
        (t_4 (* (* (/ -0.5 d) t_1) t_0)))
   (if (<= t_2 2e+204)
     (*
      (* (sqrt (/ d l)) (sqrt (/ d h)))
      (- 1.0 (* (* (/ h l) t_4) (* t_4 0.5))))
     (if (<= t_2 INFINITY)
       (*
        (fma (* -0.5 h) (* t_0 (* (* 0.25 t_0) (* t_1 t_3))) 1.0)
        (/ (fabs d) (sqrt (* l h))))
       (/
        (* (fma (* (* (* (* t_0 t_0) 0.25) t_1) t_3) (* -0.5 h) 1.0) (fabs d))
        (sqrt (* 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 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((t_1 * t_0) / (2.0 * d)), 2.0)) * (h / l)));
	double t_3 = t_1 / ((d * d) * l);
	double t_4 = ((-0.5 / d) * t_1) * t_0;
	double tmp;
	if (t_2 <= 2e+204) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (((h / l) * t_4) * (t_4 * 0.5)));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = fma((-0.5 * h), (t_0 * ((0.25 * t_0) * (t_1 * t_3))), 1.0) * (fabs(d) / sqrt((l * h)));
	} else {
		tmp = (fma(((((t_0 * t_0) * 0.25) * t_1) * t_3), (-0.5 * h), 1.0) * fabs(d)) / sqrt((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((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(t_1 * t_0) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_3 = Float64(t_1 / Float64(Float64(d * d) * l))
	t_4 = Float64(Float64(Float64(-0.5 / d) * t_1) * t_0)
	tmp = 0.0
	if (t_2 <= 2e+204)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(Float64(h / l) * t_4) * Float64(t_4 * 0.5))));
	elseif (t_2 <= Inf)
		tmp = Float64(fma(Float64(-0.5 * h), Float64(t_0 * Float64(Float64(0.25 * t_0) * Float64(t_1 * t_3))), 1.0) * Float64(abs(d) / sqrt(Float64(l * h))));
	else
		tmp = Float64(Float64(fma(Float64(Float64(Float64(Float64(t_0 * t_0) * 0.25) * t_1) * t_3), Float64(-0.5 * h), 1.0) * abs(d)) / sqrt(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[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(t$95$1 * t$95$0), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(-0.5 / d), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$2, 2e+204], 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] * t$95$4), $MachinePrecision] * N[(t$95$4 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(N[(-0.5 * h), $MachinePrecision] * N[(t$95$0 * N[(N[(0.25 * t$95$0), $MachinePrecision] * N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 0.25), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(-0.5 * h), $MachinePrecision] + 1.0), $MachinePrecision] * N[Abs[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2e204

   1. Initial program 66.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 66.7%

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 66.7%

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 66.7%

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

   3. Applied rewrites 66.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. count-2-rev N/A

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

      6. lift-+.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. mult-flip N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 66.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 66.2%

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 66.2%

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 66.2%

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

      20. lift-/.f64 N/A

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

      21. metadata-eval 66.2%

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

   7. Applied rewrites 66.2%

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

      1. metadata-eval N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval 67.7%

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

   9. Applied rewrites 67.7%

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

   if 2e204 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 66.7%

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

      1. lift-*.f64 N/A

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

      2. 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

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

      9. frac-2neg N/A

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

      10. associate-*r/ N/A

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

      11. sqrt-div N/A

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

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

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

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

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lower-/.f64 N/A

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

      17. distribute-rgt-neg-out N/A

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

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

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

      19. lower-*.f64 N/A

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

      20. lower-neg.f64 N/A

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

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

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

      22. lower-neg.f64 24.2%

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

   3. Applied rewrites 24.2%

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

   5. Applied rewrites 49.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5 \cdot h, \frac{\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 61.7%

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

   7. Applied rewrites 61.7%

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

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

   1. Initial program 66.7%

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

      1. lift-*.f64 N/A

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

      2. 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

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

      9. frac-2neg N/A

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

      10. associate-*r/ N/A

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

      11. sqrt-div N/A

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

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

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

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

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lower-/.f64 N/A

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

      17. distribute-rgt-neg-out N/A

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

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

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

      19. lower-*.f64 N/A

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

      20. lower-neg.f64 N/A

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

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

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

      22. lower-neg.f64 24.2%

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

   3. Applied rewrites 24.2%

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

   5. Applied rewrites 49.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5 \cdot h, \frac{\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 56.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}, -0.5 \cdot h, 1\right) \cdot \left|d\right|}{\sqrt{h \cdot \ell}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 81.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_1 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_2 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_1 \cdot t\_0}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_3 := \frac{t\_1}{\left(d \cdot d\right) \cdot \ell}\\ t_4 := \left(t\_1 \cdot \frac{-0.5}{d}\right) \cdot t\_0\\ \mathbf{if}\;t\_2 \leq 2 \cdot 10^{+204}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(t\_4 \cdot \left(t\_4 \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot h, t\_0 \cdot \left(\left(0.25 \cdot t\_0\right) \cdot \left(t\_1 \cdot t\_3\right)\right), 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(\left(t\_0 \cdot t\_0\right) \cdot 0.25\right) \cdot t\_1\right) \cdot t\_3, -0.5 \cdot h, 1\right) \cdot \left|d\right|}{\sqrt{h \cdot \ell}}\\ \end{array} \]

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
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* t_1 t_0) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_3 (/ t_1 (* (* d d) l)))
        (t_4 (* (* t_1 (/ -0.5 d)) t_0)))
   (if (<= t_2 2e+204)
     (*
      (* (sqrt (/ d l)) (sqrt (/ d h)))
      (- 1.0 (* (* t_4 (* t_4 0.5)) (/ h l))))
     (if (<= t_2 INFINITY)
       (*
        (fma (* -0.5 h) (* t_0 (* (* 0.25 t_0) (* t_1 t_3))) 1.0)
        (/ (fabs d) (sqrt (* l h))))
       (/
        (* (fma (* (* (* (* t_0 t_0) 0.25) t_1) t_3) (* -0.5 h) 1.0) (fabs d))
        (sqrt (* 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 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((t_1 * t_0) / (2.0 * d)), 2.0)) * (h / l)));
	double t_3 = t_1 / ((d * d) * l);
	double t_4 = (t_1 * (-0.5 / d)) * t_0;
	double tmp;
	if (t_2 <= 2e+204) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((t_4 * (t_4 * 0.5)) * (h / l)));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = fma((-0.5 * h), (t_0 * ((0.25 * t_0) * (t_1 * t_3))), 1.0) * (fabs(d) / sqrt((l * h)));
	} else {
		tmp = (fma(((((t_0 * t_0) * 0.25) * t_1) * t_3), (-0.5 * h), 1.0) * fabs(d)) / sqrt((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((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(t_1 * t_0) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_3 = Float64(t_1 / Float64(Float64(d * d) * l))
	t_4 = Float64(Float64(t_1 * Float64(-0.5 / d)) * t_0)
	tmp = 0.0
	if (t_2 <= 2e+204)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(t_4 * Float64(t_4 * 0.5)) * Float64(h / l))));
	elseif (t_2 <= Inf)
		tmp = Float64(fma(Float64(-0.5 * h), Float64(t_0 * Float64(Float64(0.25 * t_0) * Float64(t_1 * t_3))), 1.0) * Float64(abs(d) / sqrt(Float64(l * h))));
	else
		tmp = Float64(Float64(fma(Float64(Float64(Float64(Float64(t_0 * t_0) * 0.25) * t_1) * t_3), Float64(-0.5 * h), 1.0) * abs(d)) / sqrt(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[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(t$95$1 * t$95$0), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$1 * N[(-0.5 / d), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$2, 2e+204], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(t$95$4 * N[(t$95$4 * 0.5), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(N[(-0.5 * h), $MachinePrecision] * N[(t$95$0 * N[(N[(0.25 * t$95$0), $MachinePrecision] * N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 0.25), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(-0.5 * h), $MachinePrecision] + 1.0), $MachinePrecision] * N[Abs[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2e204

   1. Initial program 66.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 66.7%

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 66.7%

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 66.7%

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

   3. Applied rewrites 66.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. count-2-rev N/A

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

      6. lift-+.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. mult-flip N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 66.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 66.2%

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 66.2%

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 66.2%

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

      20. lift-/.f64 N/A

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

      21. metadata-eval 66.2%

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

   7. Applied rewrites 66.2%

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

   if 2e204 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 66.7%

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

      1. lift-*.f64 N/A

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

      2. 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

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

      9. frac-2neg N/A

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

      10. associate-*r/ N/A

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

      11. sqrt-div N/A

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

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

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

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

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lower-/.f64 N/A

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

      17. distribute-rgt-neg-out N/A

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

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

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

      19. lower-*.f64 N/A

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

      20. lower-neg.f64 N/A

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

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

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

      22. lower-neg.f64 24.2%

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

   3. Applied rewrites 24.2%

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

   5. Applied rewrites 49.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5 \cdot h, \frac{\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 61.7%

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

   7. Applied rewrites 61.7%

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

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

   1. Initial program 66.7%

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

      1. lift-*.f64 N/A

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

      2. 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

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

      9. frac-2neg N/A

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

      10. associate-*r/ N/A

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

      11. sqrt-div N/A

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

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

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

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

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lower-/.f64 N/A

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

      17. distribute-rgt-neg-out N/A

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

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

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

      19. lower-*.f64 N/A

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

      20. lower-neg.f64 N/A

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

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

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

      22. lower-neg.f64 24.2%

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

   3. Applied rewrites 24.2%

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

   5. Applied rewrites 49.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5 \cdot h, \frac{\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 56.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}, -0.5 \cdot h, 1\right) \cdot \left|d\right|}{\sqrt{h \cdot \ell}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 81.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_1 := \frac{t\_0}{\left(d \cdot d\right) \cdot \ell}\\ t_2 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_3 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_0 \cdot t\_2}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_3 \leq 2 \cdot 10^{+204}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(t\_0 \cdot \frac{-0.5}{d}\right) \cdot t\_2\right) \cdot \left(-0.25 \cdot \frac{t\_2 \cdot t\_0}{d}\right)\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot h, t\_2 \cdot \left(\left(0.25 \cdot t\_2\right) \cdot \left(t\_0 \cdot t\_1\right)\right), 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(\left(t\_2 \cdot t\_2\right) \cdot 0.25\right) \cdot t\_0\right) \cdot t\_1, -0.5 \cdot h, 1\right) \cdot \left|d\right|}{\sqrt{h \cdot \ell}}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (fmin (fabs M) (fabs D)))
        (t_1 (/ t_0 (* (* d d) l)))
        (t_2 (fmax (fabs M) (fabs D)))
        (t_3
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* t_0 t_2) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_3 2e+204)
     (*
      (* (sqrt (/ d l)) (sqrt (/ d h)))
      (-
       1.0
       (* (* (* (* t_0 (/ -0.5 d)) t_2) (* -0.25 (/ (* t_2 t_0) d))) (/ h l))))
     (if (<= t_3 INFINITY)
       (*
        (fma (* -0.5 h) (* t_2 (* (* 0.25 t_2) (* t_0 t_1))) 1.0)
        (/ (fabs d) (sqrt (* l h))))
       (/
        (* (fma (* (* (* (* t_2 t_2) 0.25) t_0) t_1) (* -0.5 h) 1.0) (fabs d))
        (sqrt (* 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 = t_0 / ((d * d) * l);
	double t_2 = fmax(fabs(M), fabs(D));
	double t_3 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((t_0 * t_2) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_3 <= 2e+204) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((((t_0 * (-0.5 / d)) * t_2) * (-0.25 * ((t_2 * t_0) / d))) * (h / l)));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = fma((-0.5 * h), (t_2 * ((0.25 * t_2) * (t_0 * t_1))), 1.0) * (fabs(d) / sqrt((l * h)));
	} else {
		tmp = (fma(((((t_2 * t_2) * 0.25) * t_0) * t_1), (-0.5 * h), 1.0) * fabs(d)) / sqrt((h * l));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = fmin(abs(M), abs(D))
	t_1 = Float64(t_0 / Float64(Float64(d * d) * l))
	t_2 = fmax(abs(M), abs(D))
	t_3 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(t_0 * t_2) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_3 <= 2e+204)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(Float64(Float64(t_0 * Float64(-0.5 / d)) * t_2) * Float64(-0.25 * Float64(Float64(t_2 * t_0) / d))) * Float64(h / l))));
	elseif (t_3 <= Inf)
		tmp = Float64(fma(Float64(-0.5 * h), Float64(t_2 * Float64(Float64(0.25 * t_2) * Float64(t_0 * t_1))), 1.0) * Float64(abs(d) / sqrt(Float64(l * h))));
	else
		tmp = Float64(Float64(fma(Float64(Float64(Float64(Float64(t_2 * t_2) * 0.25) * t_0) * t_1), Float64(-0.5 * h), 1.0) * abs(d)) / sqrt(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[(t$95$0 / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(t$95$0 * t$95$2), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 2e+204], 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 * N[(-0.5 / d), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(-0.25 * N[(N[(t$95$2 * t$95$0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(N[(-0.5 * h), $MachinePrecision] * N[(t$95$2 * N[(N[(0.25 * t$95$2), $MachinePrecision] * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(t$95$2 * t$95$2), $MachinePrecision] * 0.25), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(-0.5 * h), $MachinePrecision] + 1.0), $MachinePrecision] * N[Abs[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2e204

   1. Initial program 66.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 66.7%

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 66.7%

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 66.7%

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

   3. Applied rewrites 66.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. count-2-rev N/A

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

      6. lift-+.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. mult-flip N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 66.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 66.2%

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 66.2%

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 66.2%

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

      20. lift-/.f64 N/A

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

      21. metadata-eval 66.2%

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

   7. Applied rewrites 66.2%

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

   8. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 65.9%

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

   10. Applied rewrites 65.9%

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

   if 2e204 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 66.7%

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

      1. lift-*.f64 N/A

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

      2. 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

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

      9. frac-2neg N/A

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

      10. associate-*r/ N/A

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

      11. sqrt-div N/A

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

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

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

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

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lower-/.f64 N/A

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

      17. distribute-rgt-neg-out N/A

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

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

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

      19. lower-*.f64 N/A

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

      20. lower-neg.f64 N/A

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

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

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

      22. lower-neg.f64 24.2%

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

   3. Applied rewrites 24.2%

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

   5. Applied rewrites 49.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5 \cdot h, \frac{\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 61.7%

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

   7. Applied rewrites 61.7%

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

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

   1. Initial program 66.7%

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

      1. lift-*.f64 N/A

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

      2. 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

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

      9. frac-2neg N/A

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

      10. associate-*r/ N/A

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

      11. sqrt-div N/A

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

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

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

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

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lower-/.f64 N/A

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

      17. distribute-rgt-neg-out N/A

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

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

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

      19. lower-*.f64 N/A

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

      20. lower-neg.f64 N/A

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

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

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

      22. lower-neg.f64 24.2%

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

   3. Applied rewrites 24.2%

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

   5. Applied rewrites 49.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5 \cdot h, \frac{\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 56.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}, -0.5 \cdot h, 1\right) \cdot \left|d\right|}{\sqrt{h \cdot \ell}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 80.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ t_1 := \mathsf{max}\left(M, \left|D\right|\right)\\ t_2 := \mathsf{min}\left(M, \left|D\right|\right)\\ t_3 := \frac{t\_2}{\left(d \cdot d\right) \cdot \ell}\\ t_4 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_2 \cdot t\_1}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_5 := \frac{t\_1}{d + d} \cdot t\_2\\ \mathbf{if}\;t\_4 \leq 0:\\ \;\;\;\;t\_0 \cdot \left(1 - \left(t\_5 \cdot \left(t\_5 \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+204}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot h, t\_1 \cdot \left(\left(0.25 \cdot t\_1\right) \cdot \left(t\_2 \cdot t\_3\right)\right), 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(\left(t\_1 \cdot t\_1\right) \cdot 0.25\right) \cdot t\_2\right) \cdot t\_3, -0.5 \cdot h, 1\right) \cdot \left|d\right|}{\sqrt{h \cdot \ell}}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ (fabs d) (sqrt (* l h))))
        (t_1 (fmax M (fabs D)))
        (t_2 (fmin M (fabs D)))
        (t_3 (/ t_2 (* (* d d) l)))
        (t_4
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* t_2 t_1) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_5 (* (/ t_1 (+ d d)) t_2)))
   (if (<= t_4 0.0)
     (* t_0 (- 1.0 (* (* t_5 (* t_5 0.5)) (/ h l))))
     (if (<= t_4 2e+204)
       (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0)
       (if (<= t_4 INFINITY)
         (* (fma (* -0.5 h) (* t_1 (* (* 0.25 t_1) (* t_2 t_3))) 1.0) t_0)
         (/
          (*
           (fma (* (* (* (* t_1 t_1) 0.25) t_2) t_3) (* -0.5 h) 1.0)
           (fabs d))
          (sqrt (* h l))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fabs(d) / sqrt((l * h));
	double t_1 = fmax(M, fabs(D));
	double t_2 = fmin(M, fabs(D));
	double t_3 = t_2 / ((d * d) * l);
	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_2 * t_1) / (2.0 * d)), 2.0)) * (h / l)));
	double t_5 = (t_1 / (d + d)) * t_2;
	double tmp;
	if (t_4 <= 0.0) {
		tmp = t_0 * (1.0 - ((t_5 * (t_5 * 0.5)) * (h / l)));
	} else if (t_4 <= 2e+204) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = fma((-0.5 * h), (t_1 * ((0.25 * t_1) * (t_2 * t_3))), 1.0) * t_0;
	} else {
		tmp = (fma(((((t_1 * t_1) * 0.25) * t_2) * t_3), (-0.5 * h), 1.0) * fabs(d)) / sqrt((h * l));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(abs(d) / sqrt(Float64(l * h)))
	t_1 = fmax(M, abs(D))
	t_2 = fmin(M, abs(D))
	t_3 = Float64(t_2 / Float64(Float64(d * d) * l))
	t_4 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(t_2 * t_1) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_5 = Float64(Float64(t_1 / Float64(d + d)) * t_2)
	tmp = 0.0
	if (t_4 <= 0.0)
		tmp = Float64(t_0 * Float64(1.0 - Float64(Float64(t_5 * Float64(t_5 * 0.5)) * Float64(h / l))));
	elseif (t_4 <= 2e+204)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	elseif (t_4 <= Inf)
		tmp = Float64(fma(Float64(-0.5 * h), Float64(t_1 * Float64(Float64(0.25 * t_1) * Float64(t_2 * t_3))), 1.0) * t_0);
	else
		tmp = Float64(Float64(fma(Float64(Float64(Float64(Float64(t_1 * t_1) * 0.25) * t_2) * t_3), Float64(-0.5 * h), 1.0) * abs(d)) / sqrt(Float64(h * l)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Max[M, N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Min[M, N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(t$95$2 * t$95$1), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$1 / N[(d + d), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[(t$95$0 * N[(1.0 - N[(N[(t$95$5 * N[(t$95$5 * 0.5), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2e+204], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[(N[(-0.5 * h), $MachinePrecision] * N[(t$95$1 * N[(N[(0.25 * t$95$1), $MachinePrecision] * N[(t$95$2 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] * 0.25), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(-0.5 * h), $MachinePrecision] + 1.0), $MachinePrecision] * N[Abs[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 66.7%

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

      1. lift-*.f64 N/A

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

      2. 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

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

      9. frac-2neg N/A

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

      10. associate-*r/ N/A

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

      11. sqrt-div N/A

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

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

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

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

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lower-/.f64 N/A

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

      17. distribute-rgt-neg-out N/A

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

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

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

      19. lower-*.f64 N/A

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

      20. lower-neg.f64 N/A

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

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

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

      22. lower-neg.f64 24.2%

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

   3. Applied rewrites 24.2%

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

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

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

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

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

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

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

      4. sqrt-div N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lift-/.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. frac-2neg N/A

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

      14. frac-times N/A

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

      15. lift-*.f64 N/A

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

      16. sqrt-div N/A

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

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

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

      18. lower-sqrt.f64 N/A

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

      19. lift-*.f64 N/A

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

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

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

   5. Applied rewrites 70.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. count-2-rev N/A

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

      16. lower-+.f64 N/A

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

      17. lower-*.f64 N/A

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

   7. Applied rewrites 69.5%

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

   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)))) < 2e204

   1. Initial program 66.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 66.7%

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 66.7%

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 66.7%

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

   3. Applied rewrites 66.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. count-2-rev N/A

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

      6. lift-+.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. mult-flip N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 66.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 66.2%

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 66.2%

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 66.2%

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

      20. lift-/.f64 N/A

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

      21. metadata-eval 66.2%

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

   7. Applied rewrites 66.2%

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

   8. Taylor expanded in d around inf

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

   10. Applied rewrites 40.0%

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

   if 2e204 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 66.7%

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

      1. lift-*.f64 N/A

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

      2. 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

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

      9. frac-2neg N/A

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

      10. associate-*r/ N/A

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

      11. sqrt-div N/A

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

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

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

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

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lower-/.f64 N/A

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

      17. distribute-rgt-neg-out N/A

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

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

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

      19. lower-*.f64 N/A

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

      20. lower-neg.f64 N/A

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

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

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

      22. lower-neg.f64 24.2%

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

   3. Applied rewrites 24.2%

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

   5. Applied rewrites 49.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5 \cdot h, \frac{\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 61.7%

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

   7. Applied rewrites 61.7%

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

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

   1. Initial program 66.7%

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

      1. lift-*.f64 N/A

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

      2. 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

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

      9. frac-2neg N/A

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

      10. associate-*r/ N/A

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

      11. sqrt-div N/A

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

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

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

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

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lower-/.f64 N/A

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

      17. distribute-rgt-neg-out N/A

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

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

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

      19. lower-*.f64 N/A

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

      20. lower-neg.f64 N/A

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

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

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

      22. lower-neg.f64 24.2%

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

   3. Applied rewrites 24.2%

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

   5. Applied rewrites 49.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5 \cdot h, \frac{\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 56.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}, -0.5 \cdot h, 1\right) \cdot \left|d\right|}{\sqrt{h \cdot \ell}}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 78.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_1 := \frac{t\_0}{\left(d \cdot d\right) \cdot \ell}\\ t_2 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_3 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_0 \cdot t\_2}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_4 := \left(\left(t\_2 \cdot t\_2\right) \cdot 0.25\right) \cdot t\_0\\ t_5 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot h, \frac{t\_4}{\ell \cdot d} \cdot \frac{t\_0}{d}, 1\right) \cdot t\_5\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+204}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot h, t\_2 \cdot \left(\left(0.25 \cdot t\_2\right) \cdot \left(t\_0 \cdot t\_1\right)\right), 1\right) \cdot t\_5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_4 \cdot t\_1, -0.5 \cdot h, 1\right) \cdot \left|d\right|}{\sqrt{h \cdot \ell}}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (fmin (fabs M) (fabs D)))
        (t_1 (/ t_0 (* (* d d) l)))
        (t_2 (fmax (fabs M) (fabs D)))
        (t_3
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* t_0 t_2) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_4 (* (* (* t_2 t_2) 0.25) t_0))
        (t_5 (/ (fabs d) (sqrt (* l h)))))
   (if (<= t_3 0.0)
     (* (fma (* -0.5 h) (* (/ t_4 (* l d)) (/ t_0 d)) 1.0) t_5)
     (if (<= t_3 2e+204)
       (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0)
       (if (<= t_3 INFINITY)
         (* (fma (* -0.5 h) (* t_2 (* (* 0.25 t_2) (* t_0 t_1))) 1.0) t_5)
         (/ (* (fma (* t_4 t_1) (* -0.5 h) 1.0) (fabs d)) (sqrt (* 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 = t_0 / ((d * d) * l);
	double t_2 = fmax(fabs(M), fabs(D));
	double t_3 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((t_0 * t_2) / (2.0 * d)), 2.0)) * (h / l)));
	double t_4 = ((t_2 * t_2) * 0.25) * t_0;
	double t_5 = fabs(d) / sqrt((l * h));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = fma((-0.5 * h), ((t_4 / (l * d)) * (t_0 / d)), 1.0) * t_5;
	} else if (t_3 <= 2e+204) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = fma((-0.5 * h), (t_2 * ((0.25 * t_2) * (t_0 * t_1))), 1.0) * t_5;
	} else {
		tmp = (fma((t_4 * t_1), (-0.5 * h), 1.0) * fabs(d)) / sqrt((h * l));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = fmin(abs(M), abs(D))
	t_1 = Float64(t_0 / Float64(Float64(d * d) * l))
	t_2 = fmax(abs(M), abs(D))
	t_3 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(t_0 * t_2) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_4 = Float64(Float64(Float64(t_2 * t_2) * 0.25) * t_0)
	t_5 = Float64(abs(d) / sqrt(Float64(l * h)))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(fma(Float64(-0.5 * h), Float64(Float64(t_4 / Float64(l * d)) * Float64(t_0 / d)), 1.0) * t_5);
	elseif (t_3 <= 2e+204)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	elseif (t_3 <= Inf)
		tmp = Float64(fma(Float64(-0.5 * h), Float64(t_2 * Float64(Float64(0.25 * t_2) * Float64(t_0 * t_1))), 1.0) * t_5);
	else
		tmp = Float64(Float64(fma(Float64(t_4 * t_1), Float64(-0.5 * h), 1.0) * abs(d)) / sqrt(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[(t$95$0 / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(t$95$0 * t$95$2), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(t$95$2 * t$95$2), $MachinePrecision] * 0.25), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$5 = N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[(N[(N[(-0.5 * h), $MachinePrecision] * N[(N[(t$95$4 / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$5), $MachinePrecision], If[LessEqual[t$95$3, 2e+204], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(N[(-0.5 * h), $MachinePrecision] * N[(t$95$2 * N[(N[(0.25 * t$95$2), $MachinePrecision] * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$5), $MachinePrecision], N[(N[(N[(N[(t$95$4 * t$95$1), $MachinePrecision] * N[(-0.5 * h), $MachinePrecision] + 1.0), $MachinePrecision] * N[Abs[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 66.7%

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

      1. lift-*.f64 N/A

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

      2. 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

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

      9. frac-2neg N/A

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

      10. associate-*r/ N/A

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

      11. sqrt-div N/A

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

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

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

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

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lower-/.f64 N/A

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

      17. distribute-rgt-neg-out N/A

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

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

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

      19. lower-*.f64 N/A

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

      20. lower-neg.f64 N/A

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

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

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

      22. lower-neg.f64 24.2%

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

   3. Applied rewrites 24.2%

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

   5. Applied rewrites 49.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5 \cdot h, \frac{\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 65.5%

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

   7. Applied rewrites 65.5%

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

   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)))) < 2e204

   1. Initial program 66.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 66.7%

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 66.7%

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 66.7%

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

   3. Applied rewrites 66.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. count-2-rev N/A

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

      6. lift-+.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. mult-flip N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 66.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 66.2%

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 66.2%

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 66.2%

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

      20. lift-/.f64 N/A

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

      21. metadata-eval 66.2%

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

   7. Applied rewrites 66.2%

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

   8. Taylor expanded in d around inf

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

   10. Applied rewrites 40.0%

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

   if 2e204 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 66.7%

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

      1. lift-*.f64 N/A

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

      2. 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

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

      9. frac-2neg N/A

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

      10. associate-*r/ N/A

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

      11. sqrt-div N/A

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

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

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

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

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lower-/.f64 N/A

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

      17. distribute-rgt-neg-out N/A

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

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

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

      19. lower-*.f64 N/A

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

      20. lower-neg.f64 N/A

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

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

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

      22. lower-neg.f64 24.2%

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

   3. Applied rewrites 24.2%

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

   5. Applied rewrites 49.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5 \cdot h, \frac{\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 61.7%

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

   7. Applied rewrites 61.7%

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

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

   1. Initial program 66.7%

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

      1. lift-*.f64 N/A

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

      2. 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

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

      9. frac-2neg N/A

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

      10. associate-*r/ N/A

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

      11. sqrt-div N/A

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

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

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

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

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lower-/.f64 N/A

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

      17. distribute-rgt-neg-out N/A

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

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

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

      19. lower-*.f64 N/A

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

      20. lower-neg.f64 N/A

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

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

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

      22. lower-neg.f64 24.2%

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

   3. Applied rewrites 24.2%

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

   5. Applied rewrites 49.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5 \cdot h, \frac{\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 56.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}, -0.5 \cdot h, 1\right) \cdot \left|d\right|}{\sqrt{h \cdot \ell}}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 6: 73.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_1 := \frac{t\_0}{\left(d \cdot d\right) \cdot \ell}\\ t_2 := \sqrt{\frac{d}{h}}\\ t_3 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_4 := \mathsf{fma}\left(-0.5 \cdot h, t\_3 \cdot \left(\left(0.25 \cdot t\_3\right) \cdot \left(t\_0 \cdot t\_1\right)\right), 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ t_5 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_0 \cdot t\_3}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_5 \leq -2 \cdot 10^{-162}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_5 \leq 0:\\ \;\;\;\;\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{d \cdot \ell}}\right)\right) \cdot t\_2\\ \mathbf{elif}\;t\_5 \leq 2 \cdot 10^{+204}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot t\_2\right) \cdot 1\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(\left(t\_3 \cdot t\_3\right) \cdot 0.25\right) \cdot t\_0\right) \cdot t\_1, -0.5 \cdot h, 1\right) \cdot \left|d\right|}{\sqrt{h \cdot \ell}}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (fmin (fabs M) (fabs D)))
        (t_1 (/ t_0 (* (* d d) l)))
        (t_2 (sqrt (/ d h)))
        (t_3 (fmax (fabs M) (fabs D)))
        (t_4
         (*
          (fma (* -0.5 h) (* t_3 (* (* 0.25 t_3) (* t_0 t_1))) 1.0)
          (/ (fabs d) (sqrt (* l h)))))
        (t_5
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* t_0 t_3) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_5 -2e-162)
     t_4
     (if (<= t_5 0.0)
       (* (* -1.0 (* d (sqrt (/ 1.0 (* d l))))) t_2)
       (if (<= t_5 2e+204)
         (* (* (sqrt (/ d l)) t_2) 1.0)
         (if (<= t_5 INFINITY)
           t_4
           (/
            (*
             (fma (* (* (* (* t_3 t_3) 0.25) t_0) t_1) (* -0.5 h) 1.0)
             (fabs d))
            (sqrt (* 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 = t_0 / ((d * d) * l);
	double t_2 = sqrt((d / h));
	double t_3 = fmax(fabs(M), fabs(D));
	double t_4 = fma((-0.5 * h), (t_3 * ((0.25 * t_3) * (t_0 * t_1))), 1.0) * (fabs(d) / sqrt((l * h)));
	double t_5 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((t_0 * t_3) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_5 <= -2e-162) {
		tmp = t_4;
	} else if (t_5 <= 0.0) {
		tmp = (-1.0 * (d * sqrt((1.0 / (d * l))))) * t_2;
	} else if (t_5 <= 2e+204) {
		tmp = (sqrt((d / l)) * t_2) * 1.0;
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = (fma(((((t_3 * t_3) * 0.25) * t_0) * t_1), (-0.5 * h), 1.0) * fabs(d)) / sqrt((h * l));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = fmin(abs(M), abs(D))
	t_1 = Float64(t_0 / Float64(Float64(d * d) * l))
	t_2 = sqrt(Float64(d / h))
	t_3 = fmax(abs(M), abs(D))
	t_4 = Float64(fma(Float64(-0.5 * h), Float64(t_3 * Float64(Float64(0.25 * t_3) * Float64(t_0 * t_1))), 1.0) * Float64(abs(d) / sqrt(Float64(l * h))))
	t_5 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(t_0 * t_3) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_5 <= -2e-162)
		tmp = t_4;
	elseif (t_5 <= 0.0)
		tmp = Float64(Float64(-1.0 * Float64(d * sqrt(Float64(1.0 / Float64(d * l))))) * t_2);
	elseif (t_5 <= 2e+204)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * t_2) * 1.0);
	elseif (t_5 <= Inf)
		tmp = t_4;
	else
		tmp = Float64(Float64(fma(Float64(Float64(Float64(Float64(t_3 * t_3) * 0.25) * t_0) * t_1), Float64(-0.5 * h), 1.0) * abs(d)) / sqrt(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[(t$95$0 / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Max[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(-0.5 * h), $MachinePrecision] * N[(t$95$3 * N[(N[(0.25 * t$95$3), $MachinePrecision] * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(t$95$0 * t$95$3), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, -2e-162], t$95$4, If[LessEqual[t$95$5, 0.0], N[(N[(-1.0 * N[(d * N[Sqrt[N[(1.0 / N[(d * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$5, 2e+204], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$5, Infinity], t$95$4, N[(N[(N[(N[(N[(N[(N[(t$95$3 * t$95$3), $MachinePrecision] * 0.25), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(-0.5 * h), $MachinePrecision] + 1.0), $MachinePrecision] * N[Abs[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.9999999999999999e-162 or 2e204 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 66.7%

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

      1. lift-*.f64 N/A

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

      2. 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

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

      9. frac-2neg N/A

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

      10. associate-*r/ N/A

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

      11. sqrt-div N/A

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

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

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

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

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lower-/.f64 N/A

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

      17. distribute-rgt-neg-out N/A

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

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

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

      19. lower-*.f64 N/A

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

      20. lower-neg.f64 N/A

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

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

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

      22. lower-neg.f64 24.2%

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

   3. Applied rewrites 24.2%

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

   5. Applied rewrites 49.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5 \cdot h, \frac{\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 61.7%

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

   7. Applied rewrites 61.7%

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

   if -1.9999999999999999e-162 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -0.0

   1. Initial program 66.7%

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

   3. Applied rewrites 43.9%

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

   4. Taylor expanded in d around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 24.3%

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

   6. Applied rewrites 24.3%

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

   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)))) < 2e204

   1. Initial program 66.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 66.7%

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 66.7%

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 66.7%

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

   3. Applied rewrites 66.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. count-2-rev N/A

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

      6. lift-+.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. mult-flip N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 66.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left({\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{{\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)}^{2}} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{\left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)\right)} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)} \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{\left(\left(\frac{\frac{-1}{2}}{d} \cdot M\right) \cdot D\right)} \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{\left(\left(\frac{\frac{-1}{2}}{d} \cdot M\right) \cdot D\right)} \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\color{blue}{\left(\frac{\frac{-1}{2}}{d} \cdot M\right)} \cdot D\right) \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      11. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\color{blue}{\left(M \cdot \frac{\frac{-1}{2}}{d}\right)} \cdot D\right) \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\color{blue}{\left(M \cdot \frac{\frac{-1}{2}}{d}\right)} \cdot D\right) \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-*.f64 66.2%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \color{blue}{\left(\left(D \cdot \left(\frac{-0.5}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \left(\color{blue}{\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)} \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      15. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \left(\color{blue}{\left(\left(\frac{\frac{-1}{2}}{d} \cdot M\right) \cdot D\right)} \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-*.f64 66.2%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \left(\color{blue}{\left(\left(\frac{-0.5}{d} \cdot M\right) \cdot D\right)} \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \left(\left(\color{blue}{\left(\frac{\frac{-1}{2}}{d} \cdot M\right)} \cdot D\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      18. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \left(\left(\color{blue}{\left(M \cdot \frac{\frac{-1}{2}}{d}\right)} \cdot D\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      19. lower-*.f64 66.2%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \left(\left(\color{blue}{\left(M \cdot \frac{-0.5}{d}\right)} \cdot D\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      20. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \color{blue}{\frac{1}{2}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      21. metadata-eval 66.2%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \color{blue}{0.5}\right)\right) \cdot \frac{h}{\ell}\right) \]

   7. Applied rewrites 66.2%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot 0.5\right)\right)} \cdot \frac{h}{\ell}\right) \]

   8. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   9. Step-by-step derivation

   10. Applied rewrites 40.0%

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. 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

         \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. frac-2neg N/A

         \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. associate-*r/ N/A

          \[\leadsto \sqrt{\color{blue}{\frac{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}{\mathsf{neg}\left(\ell\right)}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-unsound-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{h}} \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. associate-*l/ N/A

          \[\leadsto \frac{\sqrt{\color{blue}{\frac{d \cdot \left(\mathsf{neg}\left(d\right)\right)}{h}}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{\frac{d \cdot \left(\mathsf{neg}\left(d\right)\right)}{h}}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. distribute-rgt-neg-out N/A

          \[\leadsto \frac{\sqrt{\frac{\color{blue}{\mathsf{neg}\left(d \cdot d\right)}}{h}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. distribute-lft-neg-out N/A

          \[\leadsto \frac{\sqrt{\frac{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot d}}{h}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot d}}{h}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. lower-neg.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{\color{blue}{\left(-d\right)} \cdot d}{h}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{\left(-d\right) \cdot d}{h}}}{\color{blue}{\sqrt{\mathsf{neg}\left(\ell\right)}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      22. lower-neg.f64 24.2%

          \[\leadsto \frac{\sqrt{\frac{\left(-d\right) \cdot d}{h}}}{\sqrt{\color{blue}{-\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 24.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{\left(-d\right) \cdot d}{h}}}{\sqrt{-\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 49.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5 \cdot h, \frac{\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{2} \cdot h, \frac{\left(\left(D \cdot D\right) \cdot \frac{1}{4}\right) \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot h, \frac{\left(\left(D \cdot D\right) \cdot \frac{1}{4}\right) \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{-1}{2} \cdot h, \frac{\left(\left(D \cdot D\right) \cdot \frac{1}{4}\right) \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \left|d\right|}{\sqrt{\ell \cdot h}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{-1}{2} \cdot h, \frac{\left(\left(D \cdot D\right) \cdot \frac{1}{4}\right) \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \left|d\right|}{\sqrt{\ell \cdot h}}} \]

   7. Applied rewrites 56.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}, -0.5 \cdot h, 1\right) \cdot \left|d\right|}{\sqrt{h \cdot \ell}}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 72.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \mathsf{fma}\left(-0.5 \cdot h, D \cdot \left(\left(0.25 \cdot D\right) \cdot \left(M \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right)\right), 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-162}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{d \cdot \ell}}\right)\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+204}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot t\_1\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (sqrt (/ d h)))
        (t_2
         (*
          (fma (* -0.5 h) (* D (* (* 0.25 D) (* M (/ M (* (* d d) l))))) 1.0)
          (/ (fabs d) (sqrt (* l h))))))
   (if (<= t_0 -2e-162)
     t_2
     (if (<= t_0 0.0)
       (* (* -1.0 (* d (sqrt (/ 1.0 (* d l))))) t_1)
       (if (<= t_0 2e+204) (* (* (sqrt (/ d l)) t_1) 1.0) t_2)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = sqrt((d / h));
	double t_2 = fma((-0.5 * h), (D * ((0.25 * D) * (M * (M / ((d * d) * l))))), 1.0) * (fabs(d) / sqrt((l * h)));
	double tmp;
	if (t_0 <= -2e-162) {
		tmp = t_2;
	} else if (t_0 <= 0.0) {
		tmp = (-1.0 * (d * sqrt((1.0 / (d * l))))) * t_1;
	} else if (t_0 <= 2e+204) {
		tmp = (sqrt((d / l)) * t_1) * 1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = sqrt(Float64(d / h))
	t_2 = Float64(fma(Float64(-0.5 * h), Float64(D * Float64(Float64(0.25 * D) * Float64(M * Float64(M / Float64(Float64(d * d) * l))))), 1.0) * Float64(abs(d) / sqrt(Float64(l * h))))
	tmp = 0.0
	if (t_0 <= -2e-162)
		tmp = t_2;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(-1.0 * Float64(d * sqrt(Float64(1.0 / Float64(d * l))))) * t_1);
	elseif (t_0 <= 2e+204)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * t_1) * 1.0);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-0.5 * h), $MachinePrecision] * N[(D * N[(N[(0.25 * D), $MachinePrecision] * N[(M * N[(M / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-162], t$95$2, If[LessEqual[t$95$0, 0.0], N[(N[(-1.0 * N[(d * N[Sqrt[N[(1.0 / N[(d * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 2e+204], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] * 1.0), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.9999999999999999e-162 or 2e204 < (*.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 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. 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

         \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. frac-2neg N/A

         \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. associate-*r/ N/A

          \[\leadsto \sqrt{\color{blue}{\frac{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}{\mathsf{neg}\left(\ell\right)}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-unsound-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{h}} \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. associate-*l/ N/A

          \[\leadsto \frac{\sqrt{\color{blue}{\frac{d \cdot \left(\mathsf{neg}\left(d\right)\right)}{h}}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{\frac{d \cdot \left(\mathsf{neg}\left(d\right)\right)}{h}}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. distribute-rgt-neg-out N/A

          \[\leadsto \frac{\sqrt{\frac{\color{blue}{\mathsf{neg}\left(d \cdot d\right)}}{h}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. distribute-lft-neg-out N/A

          \[\leadsto \frac{\sqrt{\frac{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot d}}{h}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot d}}{h}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. lower-neg.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{\color{blue}{\left(-d\right)} \cdot d}{h}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{\left(-d\right) \cdot d}{h}}}{\color{blue}{\sqrt{\mathsf{neg}\left(\ell\right)}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      22. lower-neg.f64 24.2%

          \[\leadsto \frac{\sqrt{\frac{\left(-d\right) \cdot d}{h}}}{\sqrt{\color{blue}{-\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 24.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{\left(-d\right) \cdot d}{h}}}{\sqrt{-\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 49.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5 \cdot h, \frac{\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot h, \color{blue}{\frac{\left(\left(D \cdot D\right) \cdot \frac{1}{4}\right) \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot h, \frac{\color{blue}{\left(\left(D \cdot D\right) \cdot \frac{1}{4}\right) \cdot \left(M \cdot M\right)}}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot h, \color{blue}{\left(\left(D \cdot D\right) \cdot \frac{1}{4}\right) \cdot \frac{M \cdot M}{\left(d \cdot d\right) \cdot \ell}}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot h, \color{blue}{\left(\left(D \cdot D\right) \cdot \frac{1}{4}\right)} \cdot \frac{M \cdot M}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot h, \left(\color{blue}{\left(D \cdot D\right)} \cdot \frac{1}{4}\right) \cdot \frac{M \cdot M}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot h, \color{blue}{\left(D \cdot \left(D \cdot \frac{1}{4}\right)\right)} \cdot \frac{M \cdot M}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot h, \color{blue}{D \cdot \left(\left(D \cdot \frac{1}{4}\right) \cdot \frac{M \cdot M}{\left(d \cdot d\right) \cdot \ell}\right)}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot h, \color{blue}{D \cdot \left(\left(D \cdot \frac{1}{4}\right) \cdot \frac{M \cdot M}{\left(d \cdot d\right) \cdot \ell}\right)}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot h, D \cdot \color{blue}{\left(\left(D \cdot \frac{1}{4}\right) \cdot \frac{M \cdot M}{\left(d \cdot d\right) \cdot \ell}\right)}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot h, D \cdot \left(\color{blue}{\left(\frac{1}{4} \cdot D\right)} \cdot \frac{M \cdot M}{\left(d \cdot d\right) \cdot \ell}\right), 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot h, D \cdot \left(\color{blue}{\left(\frac{1}{4} \cdot D\right)} \cdot \frac{M \cdot M}{\left(d \cdot d\right) \cdot \ell}\right), 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot h, D \cdot \left(\left(\frac{1}{4} \cdot D\right) \cdot \frac{\color{blue}{M \cdot M}}{\left(d \cdot d\right) \cdot \ell}\right), 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot h, D \cdot \left(\left(\frac{1}{4} \cdot D\right) \cdot \color{blue}{\left(M \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right)}\right), 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot h, D \cdot \left(\left(\frac{1}{4} \cdot D\right) \cdot \color{blue}{\left(M \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right)}\right), 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      15. lower-/.f64 61.7%

          \[\leadsto \mathsf{fma}\left(-0.5 \cdot h, D \cdot \left(\left(0.25 \cdot D\right) \cdot \left(M \cdot \color{blue}{\frac{M}{\left(d \cdot d\right) \cdot \ell}}\right)\right), 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

   7. Applied rewrites 61.7%

      \[\leadsto \mathsf{fma}\left(-0.5 \cdot h, \color{blue}{D \cdot \left(\left(0.25 \cdot D\right) \cdot \left(M \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right)\right)}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

   if -1.9999999999999999e-162 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -0.0

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 43.9%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.5, \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right)}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{d \cdot \ell}}\right)\right)} \cdot \sqrt{\frac{d}{h}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{d \cdot \ell}}\right)}\right) \cdot \sqrt{\frac{d}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{d \cdot \ell}}}\right)\right) \cdot \sqrt{\frac{d}{h}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{d \cdot \ell}}\right)\right) \cdot \sqrt{\frac{d}{h}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{d \cdot \ell}}\right)\right) \cdot \sqrt{\frac{d}{h}} \]

      5. lower-*.f64 24.3%

         \[\leadsto \left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{d \cdot \ell}}\right)\right) \cdot \sqrt{\frac{d}{h}} \]

   6. Applied rewrites 24.3%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{d \cdot \ell}}\right)\right)} \cdot \sqrt{\frac{d}{h}} \]

   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)))) < 2e204

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 66.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 66.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 66.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 66.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. count-2-rev N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{D \cdot M}{\color{blue}{d + d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-+.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{D \cdot M}{\color{blue}{d + d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. associate-*l/ N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{D}{d + d} \cdot M\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\frac{D}{d + d}} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\frac{D}{d + d}} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. mult-flip N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\left(D \cdot \frac{1}{d + d}\right)} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. associate-*l* N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{d + d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{d + d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \color{blue}{\left(\frac{1}{d + d} \cdot M\right)}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 66.2%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{-0.5}{d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left({\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{{\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)}^{2}} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{\left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)\right)} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)} \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{\left(\left(\frac{\frac{-1}{2}}{d} \cdot M\right) \cdot D\right)} \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{\left(\left(\frac{\frac{-1}{2}}{d} \cdot M\right) \cdot D\right)} \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\color{blue}{\left(\frac{\frac{-1}{2}}{d} \cdot M\right)} \cdot D\right) \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      11. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\color{blue}{\left(M \cdot \frac{\frac{-1}{2}}{d}\right)} \cdot D\right) \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\color{blue}{\left(M \cdot \frac{\frac{-1}{2}}{d}\right)} \cdot D\right) \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-*.f64 66.2%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \color{blue}{\left(\left(D \cdot \left(\frac{-0.5}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \left(\color{blue}{\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)} \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      15. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \left(\color{blue}{\left(\left(\frac{\frac{-1}{2}}{d} \cdot M\right) \cdot D\right)} \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-*.f64 66.2%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \left(\color{blue}{\left(\left(\frac{-0.5}{d} \cdot M\right) \cdot D\right)} \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \left(\left(\color{blue}{\left(\frac{\frac{-1}{2}}{d} \cdot M\right)} \cdot D\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      18. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \left(\left(\color{blue}{\left(M \cdot \frac{\frac{-1}{2}}{d}\right)} \cdot D\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      19. lower-*.f64 66.2%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \left(\left(\color{blue}{\left(M \cdot \frac{-0.5}{d}\right)} \cdot D\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      20. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \color{blue}{\frac{1}{2}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      21. metadata-eval 66.2%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \color{blue}{0.5}\right)\right) \cdot \frac{h}{\ell}\right) \]

   7. Applied rewrites 66.2%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot 0.5\right)\right)} \cdot \frac{h}{\ell}\right) \]

   8. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   9. Step-by-step derivation

   10. Applied rewrites 40.0%

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 54.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \sqrt{\frac{d}{h}}\\ t_3 := d \cdot \sqrt{\frac{1}{d \cdot \ell}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-130}:\\ \;\;\;\;t\_3 \cdot \left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{d \cdot h}}\right)\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\left(-1 \cdot t\_3\right) \cdot t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+212}:\\ \;\;\;\;\left(t\_0 \cdot t\_2\right) \cdot 1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot t\_0}{h}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l)))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_2 (sqrt (/ d h)))
        (t_3 (* d (sqrt (/ 1.0 (* d l))))))
   (if (<= t_1 -1e-130)
     (* t_3 (* -1.0 (* d (sqrt (/ 1.0 (* d h))))))
     (if (<= t_1 0.0)
       (* (* -1.0 t_3) t_2)
       (if (<= t_1 1e+212)
         (* (* t_0 t_2) 1.0)
         (if (<= t_1 INFINITY)
           (/ (/ (* (sqrt (* d h)) (sqrt (* d l))) l) h)
           (/ (* (* -1.0 (* d (sqrt (/ h d)))) t_0) h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / l));
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = sqrt((d / h));
	double t_3 = d * sqrt((1.0 / (d * l)));
	double tmp;
	if (t_1 <= -1e-130) {
		tmp = t_3 * (-1.0 * (d * sqrt((1.0 / (d * h)))));
	} else if (t_1 <= 0.0) {
		tmp = (-1.0 * t_3) * t_2;
	} else if (t_1 <= 1e+212) {
		tmp = (t_0 * t_2) * 1.0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((sqrt((d * h)) * sqrt((d * l))) / l) / h;
	} else {
		tmp = ((-1.0 * (d * sqrt((h / d)))) * t_0) / h;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((d / l));
	double t_1 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = Math.sqrt((d / h));
	double t_3 = d * Math.sqrt((1.0 / (d * l)));
	double tmp;
	if (t_1 <= -1e-130) {
		tmp = t_3 * (-1.0 * (d * Math.sqrt((1.0 / (d * h)))));
	} else if (t_1 <= 0.0) {
		tmp = (-1.0 * t_3) * t_2;
	} else if (t_1 <= 1e+212) {
		tmp = (t_0 * t_2) * 1.0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = ((Math.sqrt((d * h)) * Math.sqrt((d * l))) / l) / h;
	} else {
		tmp = ((-1.0 * (d * Math.sqrt((h / d)))) * t_0) / h;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / l))
	t_1 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_2 = math.sqrt((d / h))
	t_3 = d * math.sqrt((1.0 / (d * l)))
	tmp = 0
	if t_1 <= -1e-130:
		tmp = t_3 * (-1.0 * (d * math.sqrt((1.0 / (d * h)))))
	elif t_1 <= 0.0:
		tmp = (-1.0 * t_3) * t_2
	elif t_1 <= 1e+212:
		tmp = (t_0 * t_2) * 1.0
	elif t_1 <= math.inf:
		tmp = ((math.sqrt((d * h)) * math.sqrt((d * l))) / l) / h
	else:
		tmp = ((-1.0 * (d * math.sqrt((h / d)))) * t_0) / h
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / l))
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_2 = sqrt(Float64(d / h))
	t_3 = Float64(d * sqrt(Float64(1.0 / Float64(d * l))))
	tmp = 0.0
	if (t_1 <= -1e-130)
		tmp = Float64(t_3 * Float64(-1.0 * Float64(d * sqrt(Float64(1.0 / Float64(d * h))))));
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(-1.0 * t_3) * t_2);
	elseif (t_1 <= 1e+212)
		tmp = Float64(Float64(t_0 * t_2) * 1.0);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(sqrt(Float64(d * h)) * sqrt(Float64(d * l))) / l) / h);
	else
		tmp = Float64(Float64(Float64(-1.0 * Float64(d * sqrt(Float64(h / d)))) * t_0) / h);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / l));
	t_1 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_2 = sqrt((d / h));
	t_3 = d * sqrt((1.0 / (d * l)));
	tmp = 0.0;
	if (t_1 <= -1e-130)
		tmp = t_3 * (-1.0 * (d * sqrt((1.0 / (d * h)))));
	elseif (t_1 <= 0.0)
		tmp = (-1.0 * t_3) * t_2;
	elseif (t_1 <= 1e+212)
		tmp = (t_0 * t_2) * 1.0;
	elseif (t_1 <= Inf)
		tmp = ((sqrt((d * h)) * sqrt((d * l))) / l) / h;
	else
		tmp = ((-1.0 * (d * sqrt((h / d)))) * t_0) / h;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(d * N[Sqrt[N[(1.0 / N[(d * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-130], N[(t$95$3 * N[(-1.0 * N[(d * N[Sqrt[N[(1.0 / N[(d * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(-1.0 * t$95$3), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$1, 1e+212], N[(N[(t$95$0 * t$95$2), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(N[Sqrt[N[(d * h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(-1.0 * N[(d * N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] / h), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 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)))) < -1.0000000000000001e-130

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 43.9%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.5, \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right)}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]

   4. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{d \cdot \ell}}\right)} \cdot \sqrt{\frac{d}{h}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(d \cdot \color{blue}{\sqrt{\frac{1}{d \cdot \ell}}}\right) \cdot \sqrt{\frac{d}{h}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{d \cdot \ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      3. lower-/.f64 N/A

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{d \cdot \ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      4. lower-*.f64 24.1%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{d \cdot \ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

   6. Applied rewrites 24.1%

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{d \cdot \ell}}\right)} \cdot \sqrt{\frac{d}{h}} \]

   7. Taylor expanded in d around -inf

      \[\leadsto \left(d \cdot \sqrt{\frac{1}{d \cdot \ell}}\right) \cdot \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{d \cdot h}}\right)\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{d \cdot \ell}}\right) \cdot \left(-1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{d \cdot h}}\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{d \cdot \ell}}\right) \cdot \left(-1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{d \cdot h}}}\right)\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{d \cdot \ell}}\right) \cdot \left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{d \cdot h}}\right)\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{d \cdot \ell}}\right) \cdot \left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{d \cdot h}}\right)\right) \]

      5. lower-*.f64 16.2%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{d \cdot \ell}}\right) \cdot \left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{d \cdot h}}\right)\right) \]

   9. Applied rewrites 16.2%

      \[\leadsto \left(d \cdot \sqrt{\frac{1}{d \cdot \ell}}\right) \cdot \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{d \cdot h}}\right)\right)} \]

   if -1.0000000000000001e-130 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -0.0

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 43.9%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.5, \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right)}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{d \cdot \ell}}\right)\right)} \cdot \sqrt{\frac{d}{h}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{d \cdot \ell}}\right)}\right) \cdot \sqrt{\frac{d}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{d \cdot \ell}}}\right)\right) \cdot \sqrt{\frac{d}{h}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{d \cdot \ell}}\right)\right) \cdot \sqrt{\frac{d}{h}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{d \cdot \ell}}\right)\right) \cdot \sqrt{\frac{d}{h}} \]

      5. lower-*.f64 24.3%

         \[\leadsto \left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{d \cdot \ell}}\right)\right) \cdot \sqrt{\frac{d}{h}} \]

   6. Applied rewrites 24.3%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{d \cdot \ell}}\right)\right)} \cdot \sqrt{\frac{d}{h}} \]

   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)))) < 9.9999999999999991e211

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 66.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 66.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 66.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 66.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. count-2-rev N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{D \cdot M}{\color{blue}{d + d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-+.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{D \cdot M}{\color{blue}{d + d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. associate-*l/ N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{D}{d + d} \cdot M\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\frac{D}{d + d}} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\frac{D}{d + d}} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. mult-flip N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\left(D \cdot \frac{1}{d + d}\right)} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. associate-*l* N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{d + d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{d + d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \color{blue}{\left(\frac{1}{d + d} \cdot M\right)}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 66.2%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{-0.5}{d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left({\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{{\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)}^{2}} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{\left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)\right)} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)} \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{\left(\left(\frac{\frac{-1}{2}}{d} \cdot M\right) \cdot D\right)} \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{\left(\left(\frac{\frac{-1}{2}}{d} \cdot M\right) \cdot D\right)} \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\color{blue}{\left(\frac{\frac{-1}{2}}{d} \cdot M\right)} \cdot D\right) \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      11. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\color{blue}{\left(M \cdot \frac{\frac{-1}{2}}{d}\right)} \cdot D\right) \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\color{blue}{\left(M \cdot \frac{\frac{-1}{2}}{d}\right)} \cdot D\right) \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-*.f64 66.2%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \color{blue}{\left(\left(D \cdot \left(\frac{-0.5}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \left(\color{blue}{\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)} \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      15. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \left(\color{blue}{\left(\left(\frac{\frac{-1}{2}}{d} \cdot M\right) \cdot D\right)} \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-*.f64 66.2%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \left(\color{blue}{\left(\left(\frac{-0.5}{d} \cdot M\right) \cdot D\right)} \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \left(\left(\color{blue}{\left(\frac{\frac{-1}{2}}{d} \cdot M\right)} \cdot D\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      18. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \left(\left(\color{blue}{\left(M \cdot \frac{\frac{-1}{2}}{d}\right)} \cdot D\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      19. lower-*.f64 66.2%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \left(\left(\color{blue}{\left(M \cdot \frac{-0.5}{d}\right)} \cdot D\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      20. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \color{blue}{\frac{1}{2}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      21. metadata-eval 66.2%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \color{blue}{0.5}\right)\right) \cdot \frac{h}{\ell}\right) \]

   7. Applied rewrites 66.2%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot 0.5\right)\right)} \cdot \frac{h}{\ell}\right) \]

   8. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   9. Step-by-step derivation

   10. Applied rewrites 40.0%

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]

   if 9.9999999999999991e211 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 24.3%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 24.3%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      6. lower-*.f64 32.1%

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   7. Applied rewrites 32.1%

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 24.3%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 24.3%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in d around -inf

      \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-/.f64 11.8%

         \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   7. Applied rewrites 11.8%

      \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 9: 53.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-130}:\\ \;\;\;\;-1 \cdot \sqrt{\frac{{d}^{2}}{h \cdot \ell}}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{d \cdot \ell}}\right)\right) \cdot t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+212}:\\ \;\;\;\;\left(t\_0 \cdot t\_2\right) \cdot 1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot t\_0}{h}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l)))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_2 (sqrt (/ d h))))
   (if (<= t_1 -1e-130)
     (* -1.0 (sqrt (/ (pow d 2.0) (* h l))))
     (if (<= t_1 0.0)
       (* (* -1.0 (* d (sqrt (/ 1.0 (* d l))))) t_2)
       (if (<= t_1 1e+212)
         (* (* t_0 t_2) 1.0)
         (if (<= t_1 INFINITY)
           (/ (/ (* (sqrt (* d h)) (sqrt (* d l))) l) h)
           (/ (* (* -1.0 (* d (sqrt (/ h d)))) t_0) h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / l));
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = sqrt((d / h));
	double tmp;
	if (t_1 <= -1e-130) {
		tmp = -1.0 * sqrt((pow(d, 2.0) / (h * l)));
	} else if (t_1 <= 0.0) {
		tmp = (-1.0 * (d * sqrt((1.0 / (d * l))))) * t_2;
	} else if (t_1 <= 1e+212) {
		tmp = (t_0 * t_2) * 1.0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((sqrt((d * h)) * sqrt((d * l))) / l) / h;
	} else {
		tmp = ((-1.0 * (d * sqrt((h / d)))) * t_0) / h;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((d / l));
	double t_1 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = Math.sqrt((d / h));
	double tmp;
	if (t_1 <= -1e-130) {
		tmp = -1.0 * Math.sqrt((Math.pow(d, 2.0) / (h * l)));
	} else if (t_1 <= 0.0) {
		tmp = (-1.0 * (d * Math.sqrt((1.0 / (d * l))))) * t_2;
	} else if (t_1 <= 1e+212) {
		tmp = (t_0 * t_2) * 1.0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = ((Math.sqrt((d * h)) * Math.sqrt((d * l))) / l) / h;
	} else {
		tmp = ((-1.0 * (d * Math.sqrt((h / d)))) * t_0) / h;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / l))
	t_1 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_2 = math.sqrt((d / h))
	tmp = 0
	if t_1 <= -1e-130:
		tmp = -1.0 * math.sqrt((math.pow(d, 2.0) / (h * l)))
	elif t_1 <= 0.0:
		tmp = (-1.0 * (d * math.sqrt((1.0 / (d * l))))) * t_2
	elif t_1 <= 1e+212:
		tmp = (t_0 * t_2) * 1.0
	elif t_1 <= math.inf:
		tmp = ((math.sqrt((d * h)) * math.sqrt((d * l))) / l) / h
	else:
		tmp = ((-1.0 * (d * math.sqrt((h / d)))) * t_0) / h
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / l))
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_2 = sqrt(Float64(d / h))
	tmp = 0.0
	if (t_1 <= -1e-130)
		tmp = Float64(-1.0 * sqrt(Float64((d ^ 2.0) / Float64(h * l))));
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(-1.0 * Float64(d * sqrt(Float64(1.0 / Float64(d * l))))) * t_2);
	elseif (t_1 <= 1e+212)
		tmp = Float64(Float64(t_0 * t_2) * 1.0);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(sqrt(Float64(d * h)) * sqrt(Float64(d * l))) / l) / h);
	else
		tmp = Float64(Float64(Float64(-1.0 * Float64(d * sqrt(Float64(h / d)))) * t_0) / h);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / l));
	t_1 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_2 = sqrt((d / h));
	tmp = 0.0;
	if (t_1 <= -1e-130)
		tmp = -1.0 * sqrt(((d ^ 2.0) / (h * l)));
	elseif (t_1 <= 0.0)
		tmp = (-1.0 * (d * sqrt((1.0 / (d * l))))) * t_2;
	elseif (t_1 <= 1e+212)
		tmp = (t_0 * t_2) * 1.0;
	elseif (t_1 <= Inf)
		tmp = ((sqrt((d * h)) * sqrt((d * l))) / l) / h;
	else
		tmp = ((-1.0 * (d * sqrt((h / d)))) * t_0) / h;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -1e-130], N[(-1.0 * N[Sqrt[N[(N[Power[d, 2.0], $MachinePrecision] / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(-1.0 * N[(d * N[Sqrt[N[(1.0 / N[(d * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$1, 1e+212], N[(N[(t$95$0 * t$95$2), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(N[Sqrt[N[(d * h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(-1.0 * N[(d * N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] / h), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 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)))) < -1.0000000000000001e-130

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 24.3%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 24.3%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   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 22.2%

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      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 22.2%

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   6. Applied rewrites 22.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h}} \]

   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 12.0%

         \[\leadsto -1 \cdot \sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]

   9. Applied rewrites 12.0%

      \[\leadsto -1 \cdot \color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell}}} \]

   if -1.0000000000000001e-130 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -0.0

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 43.9%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.5, \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right)}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{d \cdot \ell}}\right)\right)} \cdot \sqrt{\frac{d}{h}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{d \cdot \ell}}\right)}\right) \cdot \sqrt{\frac{d}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{d \cdot \ell}}}\right)\right) \cdot \sqrt{\frac{d}{h}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{d \cdot \ell}}\right)\right) \cdot \sqrt{\frac{d}{h}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{d \cdot \ell}}\right)\right) \cdot \sqrt{\frac{d}{h}} \]

      5. lower-*.f64 24.3%

         \[\leadsto \left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{d \cdot \ell}}\right)\right) \cdot \sqrt{\frac{d}{h}} \]

   6. Applied rewrites 24.3%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{d \cdot \ell}}\right)\right)} \cdot \sqrt{\frac{d}{h}} \]

   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)))) < 9.9999999999999991e211

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 66.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 66.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 66.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 66.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. count-2-rev N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{D \cdot M}{\color{blue}{d + d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-+.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{D \cdot M}{\color{blue}{d + d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. associate-*l/ N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{D}{d + d} \cdot M\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\frac{D}{d + d}} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\frac{D}{d + d}} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. mult-flip N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\left(D \cdot \frac{1}{d + d}\right)} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. associate-*l* N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{d + d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{d + d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \color{blue}{\left(\frac{1}{d + d} \cdot M\right)}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 66.2%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{-0.5}{d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left({\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{{\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)}^{2}} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{\left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)\right)} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)} \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{\left(\left(\frac{\frac{-1}{2}}{d} \cdot M\right) \cdot D\right)} \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{\left(\left(\frac{\frac{-1}{2}}{d} \cdot M\right) \cdot D\right)} \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\color{blue}{\left(\frac{\frac{-1}{2}}{d} \cdot M\right)} \cdot D\right) \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      11. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\color{blue}{\left(M \cdot \frac{\frac{-1}{2}}{d}\right)} \cdot D\right) \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\color{blue}{\left(M \cdot \frac{\frac{-1}{2}}{d}\right)} \cdot D\right) \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-*.f64 66.2%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \color{blue}{\left(\left(D \cdot \left(\frac{-0.5}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \left(\color{blue}{\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)} \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      15. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \left(\color{blue}{\left(\left(\frac{\frac{-1}{2}}{d} \cdot M\right) \cdot D\right)} \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-*.f64 66.2%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \left(\color{blue}{\left(\left(\frac{-0.5}{d} \cdot M\right) \cdot D\right)} \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \left(\left(\color{blue}{\left(\frac{\frac{-1}{2}}{d} \cdot M\right)} \cdot D\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      18. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \left(\left(\color{blue}{\left(M \cdot \frac{\frac{-1}{2}}{d}\right)} \cdot D\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      19. lower-*.f64 66.2%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \left(\left(\color{blue}{\left(M \cdot \frac{-0.5}{d}\right)} \cdot D\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      20. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \color{blue}{\frac{1}{2}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      21. metadata-eval 66.2%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \color{blue}{0.5}\right)\right) \cdot \frac{h}{\ell}\right) \]

   7. Applied rewrites 66.2%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot 0.5\right)\right)} \cdot \frac{h}{\ell}\right) \]

   8. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   9. Step-by-step derivation

   10. Applied rewrites 40.0%

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]

   if 9.9999999999999991e211 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 24.3%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 24.3%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      6. lower-*.f64 32.1%

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   7. Applied rewrites 32.1%

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 24.3%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 24.3%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in d around -inf

      \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-/.f64 11.8%

         \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   7. Applied rewrites 11.8%

      \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 10: 50.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{d \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}{h}\\ \mathbf{elif}\;t\_0 \leq 10^{+212}:\\ \;\;\;\;\left(t\_1 \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot t\_1}{h}\\ \end{array} \]

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 (sqrt (/ d l))))
   (if (<= t_0 0.0)
     (/ (* d (* h (sqrt (/ 1.0 (* h l))))) h)
     (if (<= t_0 1e+212)
       (* (* t_1 (sqrt (/ d h))) 1.0)
       (if (<= t_0 INFINITY)
         (/ (/ (* (sqrt (* d h)) (sqrt (* d l))) l) h)
         (/ (* (* -1.0 (* d (sqrt (/ h d)))) t_1) h))))))

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 = sqrt((d / l));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (d * (h * sqrt((1.0 / (h * l))))) / h;
	} else if (t_0 <= 1e+212) {
		tmp = (t_1 * sqrt((d / h))) * 1.0;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = ((sqrt((d * h)) * sqrt((d * l))) / l) / h;
	} else {
		tmp = ((-1.0 * (d * sqrt((h / d)))) * t_1) / h;
	}
	return tmp;
}

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.sqrt((d / l));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (d * (h * Math.sqrt((1.0 / (h * l))))) / h;
	} else if (t_0 <= 1e+212) {
		tmp = (t_1 * Math.sqrt((d / h))) * 1.0;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = ((Math.sqrt((d * h)) * Math.sqrt((d * l))) / l) / h;
	} else {
		tmp = ((-1.0 * (d * Math.sqrt((h / d)))) * t_1) / h;
	}
	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.sqrt((d / l))
	tmp = 0
	if t_0 <= 0.0:
		tmp = (d * (h * math.sqrt((1.0 / (h * l))))) / h
	elif t_0 <= 1e+212:
		tmp = (t_1 * math.sqrt((d / h))) * 1.0
	elif t_0 <= math.inf:
		tmp = ((math.sqrt((d * h)) * math.sqrt((d * l))) / l) / h
	else:
		tmp = ((-1.0 * (d * math.sqrt((h / d)))) * t_1) / h
	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 = sqrt(Float64(d / l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(d * Float64(h * sqrt(Float64(1.0 / Float64(h * l))))) / h);
	elseif (t_0 <= 1e+212)
		tmp = Float64(Float64(t_1 * sqrt(Float64(d / h))) * 1.0);
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(Float64(sqrt(Float64(d * h)) * sqrt(Float64(d * l))) / l) / h);
	else
		tmp = Float64(Float64(Float64(-1.0 * Float64(d * sqrt(Float64(h / d)))) * t_1) / h);
	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 = sqrt((d / l));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = (d * (h * sqrt((1.0 / (h * l))))) / h;
	elseif (t_0 <= 1e+212)
		tmp = (t_1 * sqrt((d / h))) * 1.0;
	elseif (t_0 <= Inf)
		tmp = ((sqrt((d * h)) * sqrt((d * l))) / l) / h;
	else
		tmp = ((-1.0 * (d * sqrt((h / d)))) * t_1) / h;
	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[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(d * N[(h * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[t$95$0, 1e+212], N[(N[(t$95$1 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(N[Sqrt[N[(d * h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(-1.0 * N[(d * N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] / h), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -0.0

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 24.3%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 24.3%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   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 22.2%

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      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 22.2%

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   6. Applied rewrites 22.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h}} \]

   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 37.5%

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   9. Applied rewrites 37.5%

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   10. Taylor expanded in h around inf

       \[\leadsto \frac{d \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}{h} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{d \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}{h} \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \frac{d \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}{h} \]

       3. lower-/.f64 N/A

          \[\leadsto \frac{d \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}{h} \]

       4. lower-*.f64 25.7%

          \[\leadsto \frac{d \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}{h} \]

   12. Applied rewrites 25.7%

       \[\leadsto \frac{d \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}{h} \]

   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)))) < 9.9999999999999991e211

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 66.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 66.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 66.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 66.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. count-2-rev N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{D \cdot M}{\color{blue}{d + d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-+.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{D \cdot M}{\color{blue}{d + d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. associate-*l/ N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{D}{d + d} \cdot M\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\frac{D}{d + d}} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\frac{D}{d + d}} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. mult-flip N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\left(D \cdot \frac{1}{d + d}\right)} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. associate-*l* N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{d + d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{d + d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \color{blue}{\left(\frac{1}{d + d} \cdot M\right)}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 66.2%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{-0.5}{d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left({\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{{\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)}^{2}} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{\left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)\right)} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)} \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{\left(\left(\frac{\frac{-1}{2}}{d} \cdot M\right) \cdot D\right)} \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{\left(\left(\frac{\frac{-1}{2}}{d} \cdot M\right) \cdot D\right)} \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\color{blue}{\left(\frac{\frac{-1}{2}}{d} \cdot M\right)} \cdot D\right) \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      11. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\color{blue}{\left(M \cdot \frac{\frac{-1}{2}}{d}\right)} \cdot D\right) \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\color{blue}{\left(M \cdot \frac{\frac{-1}{2}}{d}\right)} \cdot D\right) \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-*.f64 66.2%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \color{blue}{\left(\left(D \cdot \left(\frac{-0.5}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \left(\color{blue}{\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)} \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      15. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \left(\color{blue}{\left(\left(\frac{\frac{-1}{2}}{d} \cdot M\right) \cdot D\right)} \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-*.f64 66.2%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \left(\color{blue}{\left(\left(\frac{-0.5}{d} \cdot M\right) \cdot D\right)} \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \left(\left(\color{blue}{\left(\frac{\frac{-1}{2}}{d} \cdot M\right)} \cdot D\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      18. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \left(\left(\color{blue}{\left(M \cdot \frac{\frac{-1}{2}}{d}\right)} \cdot D\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      19. lower-*.f64 66.2%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \left(\left(\color{blue}{\left(M \cdot \frac{-0.5}{d}\right)} \cdot D\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      20. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \color{blue}{\frac{1}{2}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      21. metadata-eval 66.2%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \color{blue}{0.5}\right)\right) \cdot \frac{h}{\ell}\right) \]

   7. Applied rewrites 66.2%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot 0.5\right)\right)} \cdot \frac{h}{\ell}\right) \]

   8. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   9. Step-by-step derivation

   10. Applied rewrites 40.0%

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]

   if 9.9999999999999991e211 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 24.3%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 24.3%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      6. lower-*.f64 32.1%

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   7. Applied rewrites 32.1%

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 24.3%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 24.3%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in d around -inf

      \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-/.f64 11.8%

         \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   7. Applied rewrites 11.8%

      \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 11: 48.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{d \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}{h}\\ \mathbf{elif}\;t\_0 \leq 10^{+212}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h}\\ \end{array} \]

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))))))
   (if (<= t_0 0.0)
     (/ (* d (* h (sqrt (/ 1.0 (* h l))))) h)
     (if (<= t_0 1e+212)
       (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0)
       (if (<= t_0 INFINITY)
         (/ (/ (* (sqrt (* d h)) (sqrt (* d l))) l) h)
         (/ (* -1.0 (* d (sqrt (/ h l)))) h))))))

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 tmp;
	if (t_0 <= 0.0) {
		tmp = (d * (h * sqrt((1.0 / (h * l))))) / h;
	} else if (t_0 <= 1e+212) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = ((sqrt((d * h)) * sqrt((d * l))) / l) / h;
	} else {
		tmp = (-1.0 * (d * sqrt((h / l)))) / h;
	}
	return tmp;
}

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 tmp;
	if (t_0 <= 0.0) {
		tmp = (d * (h * Math.sqrt((1.0 / (h * l))))) / h;
	} else if (t_0 <= 1e+212) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * 1.0;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = ((Math.sqrt((d * h)) * Math.sqrt((d * l))) / l) / h;
	} else {
		tmp = (-1.0 * (d * Math.sqrt((h / l)))) / h;
	}
	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)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = (d * (h * math.sqrt((1.0 / (h * l))))) / h
	elif t_0 <= 1e+212:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * 1.0
	elif t_0 <= math.inf:
		tmp = ((math.sqrt((d * h)) * math.sqrt((d * l))) / l) / h
	else:
		tmp = (-1.0 * (d * math.sqrt((h / l)))) / h
	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))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(d * Float64(h * sqrt(Float64(1.0 / Float64(h * l))))) / h);
	elseif (t_0 <= 1e+212)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(Float64(sqrt(Float64(d * h)) * sqrt(Float64(d * l))) / l) / h);
	else
		tmp = Float64(Float64(-1.0 * Float64(d * sqrt(Float64(h / l)))) / h);
	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)));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = (d * (h * sqrt((1.0 / (h * l))))) / h;
	elseif (t_0 <= 1e+212)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	elseif (t_0 <= Inf)
		tmp = ((sqrt((d * h)) * sqrt((d * l))) / l) / h;
	else
		tmp = (-1.0 * (d * sqrt((h / l)))) / h;
	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]}, If[LessEqual[t$95$0, 0.0], N[(N[(d * N[(h * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[t$95$0, 1e+212], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(N[Sqrt[N[(d * h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / h), $MachinePrecision], N[(N[(-1.0 * N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -0.0

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 24.3%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 24.3%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   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 22.2%

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      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 22.2%

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   6. Applied rewrites 22.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h}} \]

   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 37.5%

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   9. Applied rewrites 37.5%

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   10. Taylor expanded in h around inf

       \[\leadsto \frac{d \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}{h} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{d \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}{h} \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \frac{d \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}{h} \]

       3. lower-/.f64 N/A

          \[\leadsto \frac{d \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}{h} \]

       4. lower-*.f64 25.7%

          \[\leadsto \frac{d \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}{h} \]

   12. Applied rewrites 25.7%

       \[\leadsto \frac{d \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}{h} \]

   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)))) < 9.9999999999999991e211

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 66.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 66.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 66.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 66.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. count-2-rev N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{D \cdot M}{\color{blue}{d + d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-+.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{D \cdot M}{\color{blue}{d + d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. associate-*l/ N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{D}{d + d} \cdot M\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\frac{D}{d + d}} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\frac{D}{d + d}} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. mult-flip N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\left(D \cdot \frac{1}{d + d}\right)} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. associate-*l* N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{d + d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{d + d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \color{blue}{\left(\frac{1}{d + d} \cdot M\right)}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 66.2%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{-0.5}{d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left({\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{{\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)}^{2}} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{\left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)\right)} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)} \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{\left(\left(\frac{\frac{-1}{2}}{d} \cdot M\right) \cdot D\right)} \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{\left(\left(\frac{\frac{-1}{2}}{d} \cdot M\right) \cdot D\right)} \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\color{blue}{\left(\frac{\frac{-1}{2}}{d} \cdot M\right)} \cdot D\right) \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      11. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\color{blue}{\left(M \cdot \frac{\frac{-1}{2}}{d}\right)} \cdot D\right) \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\color{blue}{\left(M \cdot \frac{\frac{-1}{2}}{d}\right)} \cdot D\right) \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-*.f64 66.2%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \color{blue}{\left(\left(D \cdot \left(\frac{-0.5}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \left(\color{blue}{\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)} \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      15. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \left(\color{blue}{\left(\left(\frac{\frac{-1}{2}}{d} \cdot M\right) \cdot D\right)} \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-*.f64 66.2%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \left(\color{blue}{\left(\left(\frac{-0.5}{d} \cdot M\right) \cdot D\right)} \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \left(\left(\color{blue}{\left(\frac{\frac{-1}{2}}{d} \cdot M\right)} \cdot D\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      18. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \left(\left(\color{blue}{\left(M \cdot \frac{\frac{-1}{2}}{d}\right)} \cdot D\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      19. lower-*.f64 66.2%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \left(\left(\color{blue}{\left(M \cdot \frac{-0.5}{d}\right)} \cdot D\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      20. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \color{blue}{\frac{1}{2}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      21. metadata-eval 66.2%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \color{blue}{0.5}\right)\right) \cdot \frac{h}{\ell}\right) \]

   7. Applied rewrites 66.2%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot 0.5\right)\right)} \cdot \frac{h}{\ell}\right) \]

   8. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   9. Step-by-step derivation

   10. Applied rewrites 40.0%

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]

   if 9.9999999999999991e211 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 24.3%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 24.3%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      6. lower-*.f64 32.1%

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   7. Applied rewrites 32.1%

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 24.3%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 24.3%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   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 22.2%

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      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 22.2%

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   6. Applied rewrites 22.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h}} \]

   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 13.6%

         \[\leadsto \frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]

   9. Applied rewrites 13.6%

      \[\leadsto \frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 12: 48.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{d \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}{h}\\ \mathbf{elif}\;t\_0 \leq 10^{+212}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h}\\ \end{array} \]

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))))))
   (if (<= t_0 0.0)
     (/ (* d (* h (sqrt (/ 1.0 (* h l))))) h)
     (if (<= t_0 1e+212)
       (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0)
       (if (<= t_0 INFINITY)
         (/ (* (sqrt (* d h)) (sqrt (* d l))) (* h l))
         (/ (* -1.0 (* d (sqrt (/ h l)))) h))))))

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 tmp;
	if (t_0 <= 0.0) {
		tmp = (d * (h * sqrt((1.0 / (h * l))))) / h;
	} else if (t_0 <= 1e+212) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (sqrt((d * h)) * sqrt((d * l))) / (h * l);
	} else {
		tmp = (-1.0 * (d * sqrt((h / l)))) / h;
	}
	return tmp;
}

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 tmp;
	if (t_0 <= 0.0) {
		tmp = (d * (h * Math.sqrt((1.0 / (h * l))))) / h;
	} else if (t_0 <= 1e+212) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * 1.0;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (Math.sqrt((d * h)) * Math.sqrt((d * l))) / (h * l);
	} else {
		tmp = (-1.0 * (d * Math.sqrt((h / l)))) / h;
	}
	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)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = (d * (h * math.sqrt((1.0 / (h * l))))) / h
	elif t_0 <= 1e+212:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * 1.0
	elif t_0 <= math.inf:
		tmp = (math.sqrt((d * h)) * math.sqrt((d * l))) / (h * l)
	else:
		tmp = (-1.0 * (d * math.sqrt((h / l)))) / h
	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))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(d * Float64(h * sqrt(Float64(1.0 / Float64(h * l))))) / h);
	elseif (t_0 <= 1e+212)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(d * h)) * sqrt(Float64(d * l))) / Float64(h * l));
	else
		tmp = Float64(Float64(-1.0 * Float64(d * sqrt(Float64(h / l)))) / h);
	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)));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = (d * (h * sqrt((1.0 / (h * l))))) / h;
	elseif (t_0 <= 1e+212)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	elseif (t_0 <= Inf)
		tmp = (sqrt((d * h)) * sqrt((d * l))) / (h * l);
	else
		tmp = (-1.0 * (d * sqrt((h / l)))) / h;
	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]}, If[LessEqual[t$95$0, 0.0], N[(N[(d * N[(h * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[t$95$0, 1e+212], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[Sqrt[N[(d * h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(h * l), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -0.0

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 24.3%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 24.3%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   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 22.2%

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      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 22.2%

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   6. Applied rewrites 22.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h}} \]

   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 37.5%

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   9. Applied rewrites 37.5%

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   10. Taylor expanded in h around inf

       \[\leadsto \frac{d \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}{h} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{d \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}{h} \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \frac{d \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}{h} \]

       3. lower-/.f64 N/A

          \[\leadsto \frac{d \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}{h} \]

       4. lower-*.f64 25.7%

          \[\leadsto \frac{d \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}{h} \]

   12. Applied rewrites 25.7%

       \[\leadsto \frac{d \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}{h} \]

   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)))) < 9.9999999999999991e211

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 66.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 66.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 66.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 66.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. count-2-rev N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{D \cdot M}{\color{blue}{d + d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-+.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{D \cdot M}{\color{blue}{d + d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. associate-*l/ N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{D}{d + d} \cdot M\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\frac{D}{d + d}} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\frac{D}{d + d}} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. mult-flip N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\left(D \cdot \frac{1}{d + d}\right)} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. associate-*l* N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{d + d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{d + d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \color{blue}{\left(\frac{1}{d + d} \cdot M\right)}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 66.2%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{-0.5}{d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left({\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{{\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)}^{2}} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{\left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)\right)} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)} \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{\left(\left(\frac{\frac{-1}{2}}{d} \cdot M\right) \cdot D\right)} \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{\left(\left(\frac{\frac{-1}{2}}{d} \cdot M\right) \cdot D\right)} \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\color{blue}{\left(\frac{\frac{-1}{2}}{d} \cdot M\right)} \cdot D\right) \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      11. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\color{blue}{\left(M \cdot \frac{\frac{-1}{2}}{d}\right)} \cdot D\right) \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\color{blue}{\left(M \cdot \frac{\frac{-1}{2}}{d}\right)} \cdot D\right) \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-*.f64 66.2%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \color{blue}{\left(\left(D \cdot \left(\frac{-0.5}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \left(\color{blue}{\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)} \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      15. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \left(\color{blue}{\left(\left(\frac{\frac{-1}{2}}{d} \cdot M\right) \cdot D\right)} \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-*.f64 66.2%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \left(\color{blue}{\left(\left(\frac{-0.5}{d} \cdot M\right) \cdot D\right)} \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \left(\left(\color{blue}{\left(\frac{\frac{-1}{2}}{d} \cdot M\right)} \cdot D\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      18. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \left(\left(\color{blue}{\left(M \cdot \frac{\frac{-1}{2}}{d}\right)} \cdot D\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      19. lower-*.f64 66.2%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \left(\left(\color{blue}{\left(M \cdot \frac{-0.5}{d}\right)} \cdot D\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      20. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \color{blue}{\frac{1}{2}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      21. metadata-eval 66.2%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \color{blue}{0.5}\right)\right) \cdot \frac{h}{\ell}\right) \]

   7. Applied rewrites 66.2%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot 0.5\right)\right)} \cdot \frac{h}{\ell}\right) \]

   8. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   9. Step-by-step derivation

   10. Applied rewrites 40.0%

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]

   if 9.9999999999999991e211 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 24.3%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 24.3%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\color{blue}{h \cdot \ell}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \color{blue}{\ell}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      7. lower-*.f64 29.8%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

   7. Applied rewrites 29.8%

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\color{blue}{h \cdot \ell}} \]

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 24.3%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 24.3%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   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 22.2%

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      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 22.2%

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   6. Applied rewrites 22.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h}} \]

   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 13.6%

         \[\leadsto \frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]

   9. Applied rewrites 13.6%

      \[\leadsto \frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 13: 48.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := d \cdot \sqrt{\frac{h}{\ell}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{d \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}{h}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+181}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{t\_1}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot t\_1}{h}\\ \end{array} \]

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 (* d (sqrt (/ h l)))))
   (if (<= t_0 0.0)
     (/ (* d (* h (sqrt (/ 1.0 (* h l))))) h)
     (if (<= t_0 2e+181)
       (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0)
       (if (<= t_0 INFINITY) (/ t_1 h) (/ (* -1.0 t_1) h))))))

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 = d * sqrt((h / l));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (d * (h * sqrt((1.0 / (h * l))))) / h;
	} else if (t_0 <= 2e+181) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_1 / h;
	} else {
		tmp = (-1.0 * t_1) / h;
	}
	return tmp;
}

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 = d * Math.sqrt((h / l));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (d * (h * Math.sqrt((1.0 / (h * l))))) / h;
	} else if (t_0 <= 2e+181) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * 1.0;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_1 / h;
	} else {
		tmp = (-1.0 * t_1) / h;
	}
	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 = d * math.sqrt((h / l))
	tmp = 0
	if t_0 <= 0.0:
		tmp = (d * (h * math.sqrt((1.0 / (h * l))))) / h
	elif t_0 <= 2e+181:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * 1.0
	elif t_0 <= math.inf:
		tmp = t_1 / h
	else:
		tmp = (-1.0 * t_1) / h
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(d * sqrt(Float64(h / l)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(d * Float64(h * sqrt(Float64(1.0 / Float64(h * l))))) / h);
	elseif (t_0 <= 2e+181)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	elseif (t_0 <= Inf)
		tmp = Float64(t_1 / h);
	else
		tmp = Float64(Float64(-1.0 * t_1) / h);
	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 = d * sqrt((h / l));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = (d * (h * sqrt((1.0 / (h * l))))) / h;
	elseif (t_0 <= 2e+181)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	elseif (t_0 <= Inf)
		tmp = t_1 / h;
	else
		tmp = (-1.0 * t_1) / h;
	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[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(d * N[(h * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[t$95$0, 2e+181], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(t$95$1 / h), $MachinePrecision], N[(N[(-1.0 * t$95$1), $MachinePrecision] / h), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -0.0

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 24.3%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 24.3%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   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 22.2%

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      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 22.2%

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   6. Applied rewrites 22.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h}} \]

   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 37.5%

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   9. Applied rewrites 37.5%

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   10. Taylor expanded in h around inf

       \[\leadsto \frac{d \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}{h} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{d \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}{h} \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \frac{d \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}{h} \]

       3. lower-/.f64 N/A

          \[\leadsto \frac{d \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}{h} \]

       4. lower-*.f64 25.7%

          \[\leadsto \frac{d \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}{h} \]

   12. Applied rewrites 25.7%

       \[\leadsto \frac{d \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}{h} \]

   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)))) < 1.9999999999999998e181

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 66.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 66.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 66.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 66.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. count-2-rev N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{D \cdot M}{\color{blue}{d + d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-+.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{D \cdot M}{\color{blue}{d + d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. associate-*l/ N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{D}{d + d} \cdot M\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\frac{D}{d + d}} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\frac{D}{d + d}} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. mult-flip N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\left(D \cdot \frac{1}{d + d}\right)} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. associate-*l* N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{d + d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{d + d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \color{blue}{\left(\frac{1}{d + d} \cdot M\right)}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 66.2%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{-0.5}{d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left({\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{{\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)}^{2}} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{\left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)\right)} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)} \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{\left(\left(\frac{\frac{-1}{2}}{d} \cdot M\right) \cdot D\right)} \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{\left(\left(\frac{\frac{-1}{2}}{d} \cdot M\right) \cdot D\right)} \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\color{blue}{\left(\frac{\frac{-1}{2}}{d} \cdot M\right)} \cdot D\right) \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      11. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\color{blue}{\left(M \cdot \frac{\frac{-1}{2}}{d}\right)} \cdot D\right) \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\color{blue}{\left(M \cdot \frac{\frac{-1}{2}}{d}\right)} \cdot D\right) \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-*.f64 66.2%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \color{blue}{\left(\left(D \cdot \left(\frac{-0.5}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \left(\color{blue}{\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)} \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      15. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \left(\color{blue}{\left(\left(\frac{\frac{-1}{2}}{d} \cdot M\right) \cdot D\right)} \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-*.f64 66.2%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \left(\color{blue}{\left(\left(\frac{-0.5}{d} \cdot M\right) \cdot D\right)} \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \left(\left(\color{blue}{\left(\frac{\frac{-1}{2}}{d} \cdot M\right)} \cdot D\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      18. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \left(\left(\color{blue}{\left(M \cdot \frac{\frac{-1}{2}}{d}\right)} \cdot D\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      19. lower-*.f64 66.2%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \left(\left(\color{blue}{\left(M \cdot \frac{-0.5}{d}\right)} \cdot D\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      20. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \color{blue}{\frac{1}{2}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      21. metadata-eval 66.2%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \color{blue}{0.5}\right)\right) \cdot \frac{h}{\ell}\right) \]

   7. Applied rewrites 66.2%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot 0.5\right)\right)} \cdot \frac{h}{\ell}\right) \]

   8. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   9. Step-by-step derivation

   10. Applied rewrites 40.0%

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]

   if 1.9999999999999998e181 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 24.3%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 24.3%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   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 22.2%

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      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 22.2%

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   6. Applied rewrites 22.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h}} \]

   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 37.5%

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   9. Applied rewrites 37.5%

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 24.3%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 24.3%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   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 22.2%

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      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 22.2%

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   6. Applied rewrites 22.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h}} \]

   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 13.6%

         \[\leadsto \frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]

   9. Applied rewrites 13.6%

      \[\leadsto \frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 14: 47.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_3 := d \cdot \sqrt{\frac{h}{\ell}}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-162}:\\ \;\;\;\;-1 \cdot \left(t\_0 \cdot t\_1\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+181}:\\ \;\;\;\;\left(t\_1 \cdot t\_0\right) \cdot 1\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{t\_3}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot t\_3}{h}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h)))
        (t_1 (sqrt (/ d l)))
        (t_2
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_3 (* d (sqrt (/ h l)))))
   (if (<= t_2 -2e-162)
     (* -1.0 (* t_0 t_1))
     (if (<= t_2 2e+181)
       (* (* t_1 t_0) 1.0)
       (if (<= t_2 INFINITY) (/ t_3 h) (/ (* -1.0 t_3) h))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h));
	double t_1 = sqrt((d / l));
	double t_2 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_3 = d * sqrt((h / l));
	double tmp;
	if (t_2 <= -2e-162) {
		tmp = -1.0 * (t_0 * t_1);
	} else if (t_2 <= 2e+181) {
		tmp = (t_1 * t_0) * 1.0;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_3 / h;
	} else {
		tmp = (-1.0 * t_3) / h;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((d / h));
	double t_1 = Math.sqrt((d / l));
	double t_2 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_3 = d * Math.sqrt((h / l));
	double tmp;
	if (t_2 <= -2e-162) {
		tmp = -1.0 * (t_0 * t_1);
	} else if (t_2 <= 2e+181) {
		tmp = (t_1 * t_0) * 1.0;
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = t_3 / h;
	} else {
		tmp = (-1.0 * t_3) / h;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / h))
	t_1 = math.sqrt((d / l))
	t_2 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_3 = d * math.sqrt((h / l))
	tmp = 0
	if t_2 <= -2e-162:
		tmp = -1.0 * (t_0 * t_1)
	elif t_2 <= 2e+181:
		tmp = (t_1 * t_0) * 1.0
	elif t_2 <= math.inf:
		tmp = t_3 / h
	else:
		tmp = (-1.0 * t_3) / h
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / h))
	t_1 = sqrt(Float64(d / l))
	t_2 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_3 = Float64(d * sqrt(Float64(h / l)))
	tmp = 0.0
	if (t_2 <= -2e-162)
		tmp = Float64(-1.0 * Float64(t_0 * t_1));
	elseif (t_2 <= 2e+181)
		tmp = Float64(Float64(t_1 * t_0) * 1.0);
	elseif (t_2 <= Inf)
		tmp = Float64(t_3 / h);
	else
		tmp = Float64(Float64(-1.0 * t_3) / h);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / h));
	t_1 = sqrt((d / l));
	t_2 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_3 = d * sqrt((h / l));
	tmp = 0.0;
	if (t_2 <= -2e-162)
		tmp = -1.0 * (t_0 * t_1);
	elseif (t_2 <= 2e+181)
		tmp = (t_1 * t_0) * 1.0;
	elseif (t_2 <= Inf)
		tmp = t_3 / h;
	else
		tmp = (-1.0 * t_3) / h;
	end
	tmp_2 = 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[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-162], N[(-1.0 * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+181], N[(N[(t$95$1 * t$95$0), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(t$95$3 / h), $MachinePrecision], N[(N[(-1.0 * t$95$3), $MachinePrecision] / h), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.9999999999999999e-162

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 24.3%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 24.3%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in h around -inf

      \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \]

      6. lower-/.f64 9.4%

         \[\leadsto -1 \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \]

   7. Applied rewrites 9.4%

      \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)} \]

   if -1.9999999999999999e-162 < (*.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)))) < 1.9999999999999998e181

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 66.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 66.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 66.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 66.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. count-2-rev N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{D \cdot M}{\color{blue}{d + d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-+.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{D \cdot M}{\color{blue}{d + d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. associate-*l/ N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{D}{d + d} \cdot M\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\frac{D}{d + d}} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\frac{D}{d + d}} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. mult-flip N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\color{blue}{\left(D \cdot \frac{1}{d + d}\right)} \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. associate-*l* N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{d + d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{1}{d + d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(D \cdot \color{blue}{\left(\frac{1}{d + d} \cdot M\right)}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 66.2%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(D \cdot \left(\frac{-0.5}{d} \cdot M\right)\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left({\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{{\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)}^{2}} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{\left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)\right)} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)} \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{\left(\left(\frac{\frac{-1}{2}}{d} \cdot M\right) \cdot D\right)} \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{\left(\left(\frac{\frac{-1}{2}}{d} \cdot M\right) \cdot D\right)} \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\color{blue}{\left(\frac{\frac{-1}{2}}{d} \cdot M\right)} \cdot D\right) \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      11. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\color{blue}{\left(M \cdot \frac{\frac{-1}{2}}{d}\right)} \cdot D\right) \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\color{blue}{\left(M \cdot \frac{\frac{-1}{2}}{d}\right)} \cdot D\right) \cdot \left(\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-*.f64 66.2%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \color{blue}{\left(\left(D \cdot \left(\frac{-0.5}{d} \cdot M\right)\right) \cdot \frac{1}{2}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \left(\color{blue}{\left(D \cdot \left(\frac{\frac{-1}{2}}{d} \cdot M\right)\right)} \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      15. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \left(\color{blue}{\left(\left(\frac{\frac{-1}{2}}{d} \cdot M\right) \cdot D\right)} \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-*.f64 66.2%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \left(\color{blue}{\left(\left(\frac{-0.5}{d} \cdot M\right) \cdot D\right)} \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \left(\left(\color{blue}{\left(\frac{\frac{-1}{2}}{d} \cdot M\right)} \cdot D\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      18. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \left(\left(\color{blue}{\left(M \cdot \frac{\frac{-1}{2}}{d}\right)} \cdot D\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      19. lower-*.f64 66.2%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \left(\left(\color{blue}{\left(M \cdot \frac{-0.5}{d}\right)} \cdot D\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      20. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \left(\left(\left(M \cdot \frac{\frac{-1}{2}}{d}\right) \cdot D\right) \cdot \color{blue}{\frac{1}{2}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      21. metadata-eval 66.2%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \color{blue}{0.5}\right)\right) \cdot \frac{h}{\ell}\right) \]

   7. Applied rewrites 66.2%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot 0.5\right)\right)} \cdot \frac{h}{\ell}\right) \]

   8. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   9. Step-by-step derivation

   10. Applied rewrites 40.0%

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]

   if 1.9999999999999998e181 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 24.3%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 24.3%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   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 22.2%

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      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 22.2%

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   6. Applied rewrites 22.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h}} \]

   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 37.5%

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   9. Applied rewrites 37.5%

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 24.3%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 24.3%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   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 22.2%

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      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 22.2%

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   6. Applied rewrites 22.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h}} \]

   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 13.6%

         \[\leadsto \frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]

   9. Applied rewrites 13.6%

      \[\leadsto \frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 15: 47.2% accurate, 4.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;d \leq -4.4 \cdot 10^{-159}:\\ \;\;\;\;-1 \cdot \frac{d \cdot \sqrt{\frac{-1}{h}}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq 8.5 \cdot 10^{-189}:\\ \;\;\;\;\frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -4.4e-159)
   (* -1.0 (/ (* d (sqrt (/ -1.0 h))) (sqrt (- l))))
   (if (<= d 8.5e-189)
     (/ (* -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 (d <= -4.4e-159) {
		tmp = -1.0 * ((d * sqrt((-1.0 / h))) / sqrt(-l));
	} else if (d <= 8.5e-189) {
		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 (d <= (-4.4d-159)) then
        tmp = (-1.0d0) * ((d * sqrt(((-1.0d0) / h))) / sqrt(-l))
    else if (d <= 8.5d-189) 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 (d <= -4.4e-159) {
		tmp = -1.0 * ((d * Math.sqrt((-1.0 / h))) / Math.sqrt(-l));
	} else if (d <= 8.5e-189) {
		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 d <= -4.4e-159:
		tmp = -1.0 * ((d * math.sqrt((-1.0 / h))) / math.sqrt(-l))
	elif d <= 8.5e-189:
		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 (d <= -4.4e-159)
		tmp = Float64(-1.0 * Float64(Float64(d * sqrt(Float64(-1.0 / h))) / sqrt(Float64(-l))));
	elseif (d <= 8.5e-189)
		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 (d <= -4.4e-159)
		tmp = -1.0 * ((d * sqrt((-1.0 / h))) / sqrt(-l));
	elseif (d <= 8.5e-189)
		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[d, -4.4e-159], N[(-1.0 * N[(N[(d * N[Sqrt[N[(-1.0 / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 8.5e-189], 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 d < -4.4000000000000002e-159

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. 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

         \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. frac-2neg N/A

         \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. associate-*r/ N/A

          \[\leadsto \sqrt{\color{blue}{\frac{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}{\mathsf{neg}\left(\ell\right)}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-unsound-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{h}} \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. associate-*l/ N/A

          \[\leadsto \frac{\sqrt{\color{blue}{\frac{d \cdot \left(\mathsf{neg}\left(d\right)\right)}{h}}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{\frac{d \cdot \left(\mathsf{neg}\left(d\right)\right)}{h}}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. distribute-rgt-neg-out N/A

          \[\leadsto \frac{\sqrt{\frac{\color{blue}{\mathsf{neg}\left(d \cdot d\right)}}{h}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. distribute-lft-neg-out N/A

          \[\leadsto \frac{\sqrt{\frac{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot d}}{h}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot d}}{h}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. lower-neg.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{\color{blue}{\left(-d\right)} \cdot d}{h}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{\left(-d\right) \cdot d}{h}}}{\color{blue}{\sqrt{\mathsf{neg}\left(\ell\right)}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      22. lower-neg.f64 24.2%

          \[\leadsto \frac{\sqrt{\frac{\left(-d\right) \cdot d}{h}}}{\sqrt{\color{blue}{-\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 24.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{\left(-d\right) \cdot d}{h}}}{\sqrt{-\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{d \cdot \sqrt{\frac{-1}{h}}}{\sqrt{\mathsf{neg}\left(\ell\right)}}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{d \cdot \sqrt{\frac{-1}{h}}}{\sqrt{\mathsf{neg}\left(\ell\right)}}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{-1}{h}}}{\color{blue}{\sqrt{\mathsf{neg}\left(\ell\right)}}} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{-1}{h}}}{\sqrt{\color{blue}{\mathsf{neg}\left(\ell\right)}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{-1}{h}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{-1}{h}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{-1}{h}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \]

      7. lower-neg.f64 24.6%

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{-1}{h}}}{\sqrt{-\ell}} \]

   6. Applied rewrites 24.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{d \cdot \sqrt{\frac{-1}{h}}}{\sqrt{-\ell}}} \]

   if -4.4000000000000002e-159 < d < 8.5000000000000007e-189

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 24.3%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 24.3%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   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 22.2%

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      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 22.2%

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   6. Applied rewrites 22.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h}} \]

   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 13.6%

         \[\leadsto \frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]

   9. Applied rewrites 13.6%

      \[\leadsto \frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]

   if 8.5000000000000007e-189 < d

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 24.3%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 24.3%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   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 22.2%

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      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 22.2%

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   6. Applied rewrites 22.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h}} \]

   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 37.5%

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   9. Applied rewrites 37.5%

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   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 23.1%

          \[\leadsto \frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h} \]

   11. Applied rewrites 23.1%

       \[\leadsto \frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 46.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := d \cdot \sqrt{\frac{h}{\ell}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-162}:\\ \;\;\;\;-1 \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{t\_1}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot t\_1}{h}\\ \end{array} \]

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 (* d (sqrt (/ h l)))))
   (if (<= t_0 -2e-162)
     (* -1.0 (* (sqrt (/ d h)) (sqrt (/ d l))))
     (if (<= t_0 INFINITY) (/ t_1 h) (/ (* -1.0 t_1) h)))))

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 = d * sqrt((h / l));
	double tmp;
	if (t_0 <= -2e-162) {
		tmp = -1.0 * (sqrt((d / h)) * sqrt((d / l)));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_1 / h;
	} else {
		tmp = (-1.0 * t_1) / h;
	}
	return tmp;
}

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 = d * Math.sqrt((h / l));
	double tmp;
	if (t_0 <= -2e-162) {
		tmp = -1.0 * (Math.sqrt((d / h)) * Math.sqrt((d / l)));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_1 / h;
	} else {
		tmp = (-1.0 * t_1) / h;
	}
	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 = d * math.sqrt((h / l))
	tmp = 0
	if t_0 <= -2e-162:
		tmp = -1.0 * (math.sqrt((d / h)) * math.sqrt((d / l)))
	elif t_0 <= math.inf:
		tmp = t_1 / h
	else:
		tmp = (-1.0 * t_1) / h
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(d * sqrt(Float64(h / l)))
	tmp = 0.0
	if (t_0 <= -2e-162)
		tmp = Float64(-1.0 * Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))));
	elseif (t_0 <= Inf)
		tmp = Float64(t_1 / h);
	else
		tmp = Float64(Float64(-1.0 * t_1) / h);
	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 = d * sqrt((h / l));
	tmp = 0.0;
	if (t_0 <= -2e-162)
		tmp = -1.0 * (sqrt((d / h)) * sqrt((d / l)));
	elseif (t_0 <= Inf)
		tmp = t_1 / h;
	else
		tmp = (-1.0 * t_1) / h;
	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[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-162], N[(-1.0 * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(t$95$1 / h), $MachinePrecision], N[(N[(-1.0 * t$95$1), $MachinePrecision] / h), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.9999999999999999e-162

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 24.3%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 24.3%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in h around -inf

      \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \]

      6. lower-/.f64 9.4%

         \[\leadsto -1 \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \]

   7. Applied rewrites 9.4%

      \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)} \]

   if -1.9999999999999999e-162 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 24.3%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 24.3%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   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 22.2%

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      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 22.2%

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   6. Applied rewrites 22.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h}} \]

   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 37.5%

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   9. Applied rewrites 37.5%

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 24.3%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 24.3%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   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 22.2%

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      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 22.2%

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   6. Applied rewrites 22.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h}} \]

   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 13.6%

         \[\leadsto \frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]

   9. Applied rewrites 13.6%

      \[\leadsto \frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 45.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := d \cdot \sqrt{\frac{h}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \frac{-1 \cdot t\_0}{h}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-162}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{t\_0}{h}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* d (sqrt (/ h l))))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_2 (/ (* -1.0 t_0) h)))
   (if (<= t_1 -2e-162) t_2 (if (<= t_1 INFINITY) (/ t_0 h) t_2))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = d * sqrt((h / l));
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = (-1.0 * t_0) / h;
	double tmp;
	if (t_1 <= -2e-162) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_0 / h;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = d * Math.sqrt((h / l));
	double t_1 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = (-1.0 * t_0) / h;
	double tmp;
	if (t_1 <= -2e-162) {
		tmp = t_2;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 / h;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = d * math.sqrt((h / l))
	t_1 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_2 = (-1.0 * t_0) / h
	tmp = 0
	if t_1 <= -2e-162:
		tmp = t_2
	elif t_1 <= math.inf:
		tmp = t_0 / h
	else:
		tmp = t_2
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(d * sqrt(Float64(h / l)))
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_2 = Float64(Float64(-1.0 * t_0) / h)
	tmp = 0.0
	if (t_1 <= -2e-162)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = Float64(t_0 / h);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = d * sqrt((h / l));
	t_1 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_2 = (-1.0 * t_0) / h;
	tmp = 0.0;
	if (t_1 <= -2e-162)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = t_0 / h;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-1.0 * t$95$0), $MachinePrecision] / h), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-162], t$95$2, If[LessEqual[t$95$1, Infinity], N[(t$95$0 / h), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.9999999999999999e-162 or +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 24.3%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 24.3%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   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 22.2%

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      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 22.2%

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   6. Applied rewrites 22.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h}} \]

   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 13.6%

         \[\leadsto \frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]

   9. Applied rewrites 13.6%

      \[\leadsto \frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]

   if -1.9999999999999999e-162 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 24.3%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 24.3%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   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 22.2%

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      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 22.2%

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   6. Applied rewrites 22.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h}} \]

   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 37.5%

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   9. Applied rewrites 37.5%

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 39.3% accurate, 3.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|M\right| \cdot \left|D\right| \leq 1.9 \cdot 10^{+194}:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d \cdot \sqrt{h \cdot \ell}}{\ell}}{h}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= (* (fabs M) (fabs D)) 1.9e+194)
   (/ (* d (sqrt (/ h l))) h)
   (/ (/ (* d (sqrt (* h l))) l) h)))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if ((fabs(M) * fabs(D)) <= 1.9e+194) {
		tmp = (d * sqrt((h / l))) / h;
	} else {
		tmp = ((d * sqrt((h * l))) / 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 ((abs(m) * abs(d_1)) <= 1.9d+194) then
        tmp = (d * sqrt((h / l))) / h
    else
        tmp = ((d * sqrt((h * l))) / 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 ((Math.abs(M) * Math.abs(D)) <= 1.9e+194) {
		tmp = (d * Math.sqrt((h / l))) / h;
	} else {
		tmp = ((d * Math.sqrt((h * l))) / l) / h;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if (math.fabs(M) * math.fabs(D)) <= 1.9e+194:
		tmp = (d * math.sqrt((h / l))) / h
	else:
		tmp = ((d * math.sqrt((h * l))) / l) / h
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (Float64(abs(M) * abs(D)) <= 1.9e+194)
		tmp = Float64(Float64(d * sqrt(Float64(h / l))) / h);
	else
		tmp = Float64(Float64(Float64(d * sqrt(Float64(h * l))) / l) / h);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if ((abs(M) * abs(D)) <= 1.9e+194)
		tmp = (d * sqrt((h / l))) / h;
	else
		tmp = ((d * sqrt((h * l))) / l) / h;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[N[(N[Abs[M], $MachinePrecision] * N[Abs[D], $MachinePrecision]), $MachinePrecision], 1.9e+194], N[(N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(d * N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / h), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 1.8999999999999999e194

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 24.3%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 24.3%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   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 22.2%

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      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 22.2%

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   6. Applied rewrites 22.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h}} \]

   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 37.5%

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   9. Applied rewrites 37.5%

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   if 1.8999999999999999e194 < (*.f64 M D)

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 24.3%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 24.3%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   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 22.2%

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      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 22.2%

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   6. Applied rewrites 22.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h}} \]

   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 37.5%

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   9. Applied rewrites 37.5%

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   10. Taylor expanded in l around 0

       \[\leadsto \frac{\frac{d \cdot \sqrt{h \cdot \ell}}{\ell}}{h} \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{\frac{d \cdot \sqrt{h \cdot \ell}}{\ell}}{h} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\frac{d \cdot \sqrt{h \cdot \ell}}{\ell}}{h} \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \frac{\frac{d \cdot \sqrt{h \cdot \ell}}{\ell}}{h} \]

       4. lower-*.f64 19.7%

          \[\leadsto \frac{\frac{d \cdot \sqrt{h \cdot \ell}}{\ell}}{h} \]

   12. Applied rewrites 19.7%

       \[\leadsto \frac{\frac{d \cdot \sqrt{h \cdot \ell}}{\ell}}{h} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 38.7% accurate, 4.2× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ \mathbf{if}\;\left|M\right| \cdot \left|D\right| \leq 5.8 \cdot 10^{+194}:\\ \;\;\;\;\frac{d \cdot t\_0}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\ell \cdot t\_0}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ h l))))
   (if (<= (* (fabs M) (fabs D)) 5.8e+194)
     (/ (* d t_0) h)
     (/ (fabs d) (* l t_0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h / l));
	double tmp;
	if ((fabs(M) * fabs(D)) <= 5.8e+194) {
		tmp = (d * t_0) / h;
	} else {
		tmp = fabs(d) / (l * t_0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((h / l))
    if ((abs(m) * abs(d_1)) <= 5.8d+194) then
        tmp = (d * t_0) / h
    else
        tmp = abs(d) / (l * t_0)
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((h / l));
	double tmp;
	if ((Math.abs(M) * Math.abs(D)) <= 5.8e+194) {
		tmp = (d * t_0) / h;
	} else {
		tmp = Math.abs(d) / (l * t_0);
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((h / l))
	tmp = 0
	if (math.fabs(M) * math.fabs(D)) <= 5.8e+194:
		tmp = (d * t_0) / h
	else:
		tmp = math.fabs(d) / (l * t_0)
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h / l))
	tmp = 0.0
	if (Float64(abs(M) * abs(D)) <= 5.8e+194)
		tmp = Float64(Float64(d * t_0) / h);
	else
		tmp = Float64(abs(d) / Float64(l * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((h / l));
	tmp = 0.0;
	if ((abs(M) * abs(D)) <= 5.8e+194)
		tmp = (d * t_0) / h;
	else
		tmp = abs(d) / (l * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Abs[M], $MachinePrecision] * N[Abs[D], $MachinePrecision]), $MachinePrecision], 5.8e+194], N[(N[(d * t$95$0), $MachinePrecision] / h), $MachinePrecision], N[(N[Abs[d], $MachinePrecision] / N[(l * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 5.8000000000000001e194

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 24.3%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 24.3%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   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 22.2%

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      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 22.2%

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   6. Applied rewrites 22.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h}} \]

   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 37.5%

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   9. Applied rewrites 37.5%

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   if 5.8000000000000001e194 < (*.f64 M D)

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. 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

         \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. frac-2neg N/A

         \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. associate-*r/ N/A

          \[\leadsto \sqrt{\color{blue}{\frac{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}{\mathsf{neg}\left(\ell\right)}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-unsound-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{h}} \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. associate-*l/ N/A

          \[\leadsto \frac{\sqrt{\color{blue}{\frac{d \cdot \left(\mathsf{neg}\left(d\right)\right)}{h}}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{\frac{d \cdot \left(\mathsf{neg}\left(d\right)\right)}{h}}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. distribute-rgt-neg-out N/A

          \[\leadsto \frac{\sqrt{\frac{\color{blue}{\mathsf{neg}\left(d \cdot d\right)}}{h}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. distribute-lft-neg-out N/A

          \[\leadsto \frac{\sqrt{\frac{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot d}}{h}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot d}}{h}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. lower-neg.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{\color{blue}{\left(-d\right)} \cdot d}{h}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{\left(-d\right) \cdot d}{h}}}{\color{blue}{\sqrt{\mathsf{neg}\left(\ell\right)}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      22. lower-neg.f64 24.2%

          \[\leadsto \frac{\sqrt{\frac{\left(-d\right) \cdot d}{h}}}{\sqrt{\color{blue}{-\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 24.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{\left(-d\right) \cdot d}{h}}}{\sqrt{-\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 49.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5 \cdot h, \frac{\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \]

   6. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\ell \cdot \sqrt{\frac{h}{\ell}}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\color{blue}{\ell \cdot \sqrt{\frac{h}{\ell}}}} \]

      2. lower-fabs.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\color{blue}{\ell} \cdot \sqrt{\frac{h}{\ell}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\ell \cdot \color{blue}{\sqrt{\frac{h}{\ell}}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\ell \cdot \sqrt{\frac{h}{\ell}}} \]

      5. lower-/.f64 24.1%

         \[\leadsto \frac{\left|d\right|}{\ell \cdot \sqrt{\frac{h}{\ell}}} \]

   8. Applied rewrites 24.1%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\ell \cdot \sqrt{\frac{h}{\ell}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 37.5% accurate, 7.3× 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 66.7%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

2. 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 24.3%

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

4. Applied rewrites 24.3%

   \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
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 22.2%

      \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

   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 22.2%

      \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

6. Applied rewrites 22.2%

   \[\leadsto \color{blue}{\frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h}} \]

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 37.5%

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

9. Applied rewrites 37.5%

   \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025202 
(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)))))