Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.3% → 84.0%

Time: 11.3s

Alternatives: 18

Speedup: 1.7×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(M, \left|D\right|\right)\\ t_1 := \frac{t\_0}{d + d}\\ t_2 := \mathsf{min}\left(M, \left|D\right|\right)\\ t_3 := t\_2 \cdot t\_0\\ t_4 := \frac{t\_3}{d}\\ \mathbf{if}\;d \leq -2.05 \cdot 10^{-102}:\\ \;\;\;\;\frac{\frac{\left|d\right|}{\sqrt{-\ell}}}{\sqrt{-h}} \cdot \left(1 - \left(t\_1 \cdot t\_2\right) \cdot \left(\left(t\_1 \cdot \left(t\_2 \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq 4.9 \cdot 10^{-178}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{-0.25 \cdot \left(t\_4 \cdot h\right)}{\ell} \cdot t\_3, \frac{0.5}{d}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - \frac{t\_3 \cdot \left(\left(t\_4 \cdot 0.25\right) \cdot h\right)}{\left(d + d\right) \cdot \ell}\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (fmax M (fabs D)))
        (t_1 (/ t_0 (+ d d)))
        (t_2 (fmin M (fabs D)))
        (t_3 (* t_2 t_0))
        (t_4 (/ t_3 d)))
   (if (<= d -2.05e-102)
     (*
      (/ (/ (fabs d) (sqrt (- l))) (sqrt (- h)))
      (- 1.0 (* (* t_1 t_2) (* (* t_1 (* t_2 0.5)) (/ h l)))))
     (if (<= d 4.9e-178)
       (*
        (/ (fabs d) (sqrt (* l h)))
        (fma (* (/ (* -0.25 (* t_4 h)) l) t_3) (/ 0.5 d) 1.0))
       (*
        (/ d (* (sqrt l) (sqrt h)))
        (- 1.0 (/ (* t_3 (* (* t_4 0.25) h)) (* (+ d d) l))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fmax(M, fabs(D));
	double t_1 = t_0 / (d + d);
	double t_2 = fmin(M, fabs(D));
	double t_3 = t_2 * t_0;
	double t_4 = t_3 / d;
	double tmp;
	if (d <= -2.05e-102) {
		tmp = ((fabs(d) / sqrt(-l)) / sqrt(-h)) * (1.0 - ((t_1 * t_2) * ((t_1 * (t_2 * 0.5)) * (h / l))));
	} else if (d <= 4.9e-178) {
		tmp = (fabs(d) / sqrt((l * h))) * fma((((-0.25 * (t_4 * h)) / l) * t_3), (0.5 / d), 1.0);
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - ((t_3 * ((t_4 * 0.25) * h)) / ((d + d) * l)));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = fmax(M, abs(D))
	t_1 = Float64(t_0 / Float64(d + d))
	t_2 = fmin(M, abs(D))
	t_3 = Float64(t_2 * t_0)
	t_4 = Float64(t_3 / d)
	tmp = 0.0
	if (d <= -2.05e-102)
		tmp = Float64(Float64(Float64(abs(d) / sqrt(Float64(-l))) / sqrt(Float64(-h))) * Float64(1.0 - Float64(Float64(t_1 * t_2) * Float64(Float64(t_1 * Float64(t_2 * 0.5)) * Float64(h / l)))));
	elseif (d <= 4.9e-178)
		tmp = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * fma(Float64(Float64(Float64(-0.25 * Float64(t_4 * h)) / l) * t_3), Float64(0.5 / d), 1.0));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 - Float64(Float64(t_3 * Float64(Float64(t_4 * 0.25) * h)) / Float64(Float64(d + d) * l))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Max[M, N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(d + d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Min[M, N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 / d), $MachinePrecision]}, If[LessEqual[d, -2.05e-102], N[(N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(t$95$1 * t$95$2), $MachinePrecision] * N[(N[(t$95$1 * N[(t$95$2 * 0.5), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.9e-178], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(-0.25 * N[(t$95$4 * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(0.5 / d), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(t$95$3 * N[(N[(t$95$4 * 0.25), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / N[(N[(d + d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.0500000000000001e-102

   1. Initial program 66.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 67.3%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. frac-2neg N/A

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

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

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

      14. lift-neg.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

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

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

      19. lift-neg.f64 N/A

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

      20. associate-/l/ N/A

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

      21. lift-/.f64 N/A

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

      22. sqrt-div N/A

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

   5. Applied rewrites 41.1%

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

   if -2.0500000000000001e-102 < d < 4.9000000000000002e-178

   1. Initial program 66.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 67.3%

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

   5. Applied rewrites 75.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. count-2-rev N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. count-2-rev N/A

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

      18. lift-+.f64 N/A

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

      19. associate-/l* N/A

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

   7. Applied rewrites 76.1%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   9. Applied rewrites 77.6%

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

   if 4.9000000000000002e-178 < d

   1. Initial program 66.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 67.3%

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

   5. Applied rewrites 75.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

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

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

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

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

      6. lower-unsound-sqrt.f64 41.5%

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

   7. Applied rewrites 41.5%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-unprod N/A

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

      4. rem-square-sqrt 41.5%

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

   9. Applied rewrites 41.5%

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

Alternative 2: 83.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ t_1 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_2 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_3 := t\_1 \cdot t\_2\\ t_4 := t\_0 \cdot \mathsf{fma}\left(\frac{-0.25 \cdot \left(\frac{t\_3}{d} \cdot h\right)}{\ell} \cdot t\_3, \frac{0.5}{d}, 1\right)\\ t_5 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_3}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_5 \leq -2 \cdot 10^{+85}:\\ \;\;\;\;t\_0 \cdot \left(1 - \frac{t\_1 \cdot \left(\frac{t\_2}{d} \cdot \left(0.25 \cdot h\right)\right)}{\ell} \cdot \left(\frac{t\_2}{d + d} \cdot t\_1\right)\right)\\ \mathbf{elif}\;t\_5 \leq 0:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_5 \leq 2 \cdot 10^{+261}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ (fabs d) (sqrt (* l h))))
        (t_1 (fmin (fabs M) (fabs D)))
        (t_2 (fmax (fabs M) (fabs D)))
        (t_3 (* t_1 t_2))
        (t_4
         (* t_0 (fma (* (/ (* -0.25 (* (/ t_3 d) h)) l) t_3) (/ 0.5 d) 1.0)))
        (t_5
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ t_3 (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_5 -2e+85)
     (*
      t_0
      (-
       1.0
       (* (/ (* t_1 (* (/ t_2 d) (* 0.25 h))) l) (* (/ t_2 (+ d d)) t_1))))
     (if (<= t_5 0.0)
       t_4
       (if (<= t_5 2e+261) (* (sqrt (/ d l)) (sqrt (/ d h))) t_4)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fabs(d) / sqrt((l * h));
	double t_1 = fmin(fabs(M), fabs(D));
	double t_2 = fmax(fabs(M), fabs(D));
	double t_3 = t_1 * t_2;
	double t_4 = t_0 * fma((((-0.25 * ((t_3 / d) * h)) / l) * t_3), (0.5 / d), 1.0);
	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_3 / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_5 <= -2e+85) {
		tmp = t_0 * (1.0 - (((t_1 * ((t_2 / d) * (0.25 * h))) / l) * ((t_2 / (d + d)) * t_1)));
	} else if (t_5 <= 0.0) {
		tmp = t_4;
	} else if (t_5 <= 2e+261) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(abs(d) / sqrt(Float64(l * h)))
	t_1 = fmin(abs(M), abs(D))
	t_2 = fmax(abs(M), abs(D))
	t_3 = Float64(t_1 * t_2)
	t_4 = Float64(t_0 * fma(Float64(Float64(Float64(-0.25 * Float64(Float64(t_3 / d) * h)) / l) * t_3), Float64(0.5 / d), 1.0))
	t_5 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(t_3 / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_5 <= -2e+85)
		tmp = Float64(t_0 * Float64(1.0 - Float64(Float64(Float64(t_1 * Float64(Float64(t_2 / d) * Float64(0.25 * h))) / l) * Float64(Float64(t_2 / Float64(d + d)) * t_1))));
	elseif (t_5 <= 0.0)
		tmp = t_4;
	elseif (t_5 <= 2e+261)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 * N[(N[(N[(N[(-0.25 * N[(N[(t$95$3 / d), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(0.5 / d), $MachinePrecision] + 1.0), $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[(t$95$3 / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, -2e+85], N[(t$95$0 * N[(1.0 - N[(N[(N[(t$95$1 * N[(N[(t$95$2 / d), $MachinePrecision] * N[(0.25 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(t$95$2 / N[(d + d), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 0.0], t$95$4, If[LessEqual[t$95$5, 2e+261], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -2e85

   1. Initial program 66.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 67.3%

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

   5. Applied rewrites 75.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. count-2-rev N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. count-2-rev N/A

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

      18. lift-+.f64 N/A

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

      19. associate-/l* N/A

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

   7. Applied rewrites 76.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 75.1%

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

   9. Applied rewrites 75.1%

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

   if -2e85 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 1.9999999999999999e261 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 67.3%

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

   5. Applied rewrites 75.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. count-2-rev N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. count-2-rev N/A

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

      18. lift-+.f64 N/A

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

      19. associate-/l* N/A

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

   7. Applied rewrites 76.1%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   9. Applied rewrites 77.6%

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

   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)))) < 1.9999999999999999e261

   1. Initial program 66.3%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 23.4%

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

   4. Applied rewrites 23.4%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-sqrt.f64 N/A

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

      4. pow1/2 N/A

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

      5. metadata-eval N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow1/2 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. lower-/.f64 23.4%

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

   6. Applied rewrites 23.4%

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

   7. Taylor expanded in h around inf

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 39.3%

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

   9. Applied rewrites 39.3%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 83.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_0 := 1 - \frac{\left(M \cdot D\right) \cdot \left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot h\right)}{\left(d + d\right) \cdot \ell}\\ \mathbf{if}\;d \leq -1.15 \cdot 10^{+66}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{M \cdot \left(\frac{D}{d} \cdot \left(0.25 \cdot h\right)\right)}{\ell} \cdot \left(\frac{D}{d + d} \cdot M\right)\right)\\ \mathbf{elif}\;d \leq -7 \cdot 10^{-309}:\\ \;\;\;\;\frac{\frac{\left|d\right|}{\sqrt{-\ell}} \cdot t\_0}{\sqrt{-h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot t\_0\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (- 1.0 (/ (* (* M D) (* (* (/ (* M D) d) 0.25) h)) (* (+ d d) l)))))
   (if (<= d -1.15e+66)
     (*
      (/ (fabs d) (sqrt (* l h)))
      (- 1.0 (* (/ (* M (* (/ D d) (* 0.25 h))) l) (* (/ D (+ d d)) M))))
     (if (<= d -7e-309)
       (/ (* (/ (fabs d) (sqrt (- l))) t_0) (sqrt (- h)))
       (* (/ d (* (sqrt l) (sqrt h))) t_0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 - (((M * D) * ((((M * D) / d) * 0.25) * h)) / ((d + d) * l));
	double tmp;
	if (d <= -1.15e+66) {
		tmp = (fabs(d) / sqrt((l * h))) * (1.0 - (((M * ((D / d) * (0.25 * h))) / l) * ((D / (d + d)) * M)));
	} else if (d <= -7e-309) {
		tmp = ((fabs(d) / sqrt(-l)) * t_0) / sqrt(-h);
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * 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 = 1.0d0 - (((m * d_1) * ((((m * d_1) / d) * 0.25d0) * h)) / ((d + d) * l))
    if (d <= (-1.15d+66)) then
        tmp = (abs(d) / sqrt((l * h))) * (1.0d0 - (((m * ((d_1 / d) * (0.25d0 * h))) / l) * ((d_1 / (d + d)) * m)))
    else if (d <= (-7d-309)) then
        tmp = ((abs(d) / sqrt(-l)) * t_0) / sqrt(-h)
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * 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 = 1.0 - (((M * D) * ((((M * D) / d) * 0.25) * h)) / ((d + d) * l));
	double tmp;
	if (d <= -1.15e+66) {
		tmp = (Math.abs(d) / Math.sqrt((l * h))) * (1.0 - (((M * ((D / d) * (0.25 * h))) / l) * ((D / (d + d)) * M)));
	} else if (d <= -7e-309) {
		tmp = ((Math.abs(d) / Math.sqrt(-l)) * t_0) / Math.sqrt(-h);
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = 1.0 - (((M * D) * ((((M * D) / d) * 0.25) * h)) / ((d + d) * l))
	tmp = 0
	if d <= -1.15e+66:
		tmp = (math.fabs(d) / math.sqrt((l * h))) * (1.0 - (((M * ((D / d) * (0.25 * h))) / l) * ((D / (d + d)) * M)))
	elif d <= -7e-309:
		tmp = ((math.fabs(d) / math.sqrt(-l)) * t_0) / math.sqrt(-h)
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(1.0 - Float64(Float64(Float64(M * D) * Float64(Float64(Float64(Float64(M * D) / d) * 0.25) * h)) / Float64(Float64(d + d) * l)))
	tmp = 0.0
	if (d <= -1.15e+66)
		tmp = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * Float64(1.0 - Float64(Float64(Float64(M * Float64(Float64(D / d) * Float64(0.25 * h))) / l) * Float64(Float64(D / Float64(d + d)) * M))));
	elseif (d <= -7e-309)
		tmp = Float64(Float64(Float64(abs(d) / sqrt(Float64(-l))) * t_0) / sqrt(Float64(-h)));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = 1.0 - (((M * D) * ((((M * D) / d) * 0.25) * h)) / ((d + d) * l));
	tmp = 0.0;
	if (d <= -1.15e+66)
		tmp = (abs(d) / sqrt((l * h))) * (1.0 - (((M * ((D / d) * (0.25 * h))) / l) * ((D / (d + d)) * M)));
	elseif (d <= -7e-309)
		tmp = ((abs(d) / sqrt(-l)) * t_0) / sqrt(-h);
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(1.0 - N[(N[(N[(M * D), $MachinePrecision] * N[(N[(N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision] * 0.25), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / N[(N[(d + d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.15e+66], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(M * N[(N[(D / d), $MachinePrecision] * N[(0.25 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(D / N[(d + d), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -7e-309], N[(N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.15e66

   1. Initial program 66.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 67.3%

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

   5. Applied rewrites 75.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. count-2-rev N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. count-2-rev N/A

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

      18. lift-+.f64 N/A

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

      19. associate-/l* N/A

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

   7. Applied rewrites 76.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 75.1%

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

   9. Applied rewrites 75.1%

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

   if -1.15e66 < d < -6.9999999999999984e-309

   1. Initial program 66.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 67.3%

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

   5. Applied rewrites 41.9%

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

   if -6.9999999999999984e-309 < d

   1. Initial program 66.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 67.3%

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

   5. Applied rewrites 75.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

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

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

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

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

      6. lower-unsound-sqrt.f64 41.5%

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

   7. Applied rewrites 41.5%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-unprod N/A

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

      4. rem-square-sqrt 41.5%

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

   9. Applied rewrites 41.5%

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

Alternative 4: 83.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_1 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_2 := \left(d + d\right) \cdot \ell\\ t_3 := t\_0 \cdot t\_1\\ \mathbf{if}\;\ell \leq -2 \cdot 10^{+90}:\\ \;\;\;\;\frac{\left(1 - \left(\left(t\_0 \cdot \left(0.25 \cdot \frac{t\_1 \cdot t\_0}{d}\right)\right) \cdot t\_1\right) \cdot \frac{h}{t\_2}\right) \cdot \frac{\left|d\right|}{\sqrt{-h}}}{\sqrt{-\ell}}\\ \mathbf{elif}\;\ell \leq 6.8 \cdot 10^{-293}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{t\_0 \cdot \left(\frac{t\_1}{d} \cdot \left(0.25 \cdot h\right)\right)}{\ell} \cdot \left(\frac{t\_1}{d + d} \cdot t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - \frac{t\_3 \cdot \left(\left(\frac{t\_3}{d} \cdot 0.25\right) \cdot h\right)}{t\_2}\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (fmin (fabs M) (fabs D)))
        (t_1 (fmax (fabs M) (fabs D)))
        (t_2 (* (+ d d) l))
        (t_3 (* t_0 t_1)))
   (if (<= l -2e+90)
     (/
      (*
       (- 1.0 (* (* (* t_0 (* 0.25 (/ (* t_1 t_0) d))) t_1) (/ h t_2)))
       (/ (fabs d) (sqrt (- h))))
      (sqrt (- l)))
     (if (<= l 6.8e-293)
       (*
        (/ (fabs d) (sqrt (* l h)))
        (-
         1.0
         (* (/ (* t_0 (* (/ t_1 d) (* 0.25 h))) l) (* (/ t_1 (+ d d)) t_0))))
       (*
        (/ d (* (sqrt l) (sqrt h)))
        (- 1.0 (/ (* t_3 (* (* (/ t_3 d) 0.25) h)) t_2)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fmin(fabs(M), fabs(D));
	double t_1 = fmax(fabs(M), fabs(D));
	double t_2 = (d + d) * l;
	double t_3 = t_0 * t_1;
	double tmp;
	if (l <= -2e+90) {
		tmp = ((1.0 - (((t_0 * (0.25 * ((t_1 * t_0) / d))) * t_1) * (h / t_2))) * (fabs(d) / sqrt(-h))) / sqrt(-l);
	} else if (l <= 6.8e-293) {
		tmp = (fabs(d) / sqrt((l * h))) * (1.0 - (((t_0 * ((t_1 / d) * (0.25 * h))) / l) * ((t_1 / (d + d)) * t_0)));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - ((t_3 * (((t_3 / d) * 0.25) * h)) / t_2));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = fmin(abs(m), abs(d_1))
    t_1 = fmax(abs(m), abs(d_1))
    t_2 = (d + d) * l
    t_3 = t_0 * t_1
    if (l <= (-2d+90)) then
        tmp = ((1.0d0 - (((t_0 * (0.25d0 * ((t_1 * t_0) / d))) * t_1) * (h / t_2))) * (abs(d) / sqrt(-h))) / sqrt(-l)
    else if (l <= 6.8d-293) then
        tmp = (abs(d) / sqrt((l * h))) * (1.0d0 - (((t_0 * ((t_1 / d) * (0.25d0 * h))) / l) * ((t_1 / (d + d)) * t_0)))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 - ((t_3 * (((t_3 / d) * 0.25d0) * h)) / t_2))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = fmin(Math.abs(M), Math.abs(D));
	double t_1 = fmax(Math.abs(M), Math.abs(D));
	double t_2 = (d + d) * l;
	double t_3 = t_0 * t_1;
	double tmp;
	if (l <= -2e+90) {
		tmp = ((1.0 - (((t_0 * (0.25 * ((t_1 * t_0) / d))) * t_1) * (h / t_2))) * (Math.abs(d) / Math.sqrt(-h))) / Math.sqrt(-l);
	} else if (l <= 6.8e-293) {
		tmp = (Math.abs(d) / Math.sqrt((l * h))) * (1.0 - (((t_0 * ((t_1 / d) * (0.25 * h))) / l) * ((t_1 / (d + d)) * t_0)));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 - ((t_3 * (((t_3 / d) * 0.25) * h)) / t_2));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = fmin(math.fabs(M), math.fabs(D))
	t_1 = fmax(math.fabs(M), math.fabs(D))
	t_2 = (d + d) * l
	t_3 = t_0 * t_1
	tmp = 0
	if l <= -2e+90:
		tmp = ((1.0 - (((t_0 * (0.25 * ((t_1 * t_0) / d))) * t_1) * (h / t_2))) * (math.fabs(d) / math.sqrt(-h))) / math.sqrt(-l)
	elif l <= 6.8e-293:
		tmp = (math.fabs(d) / math.sqrt((l * h))) * (1.0 - (((t_0 * ((t_1 / d) * (0.25 * h))) / l) * ((t_1 / (d + d)) * t_0)))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 - ((t_3 * (((t_3 / d) * 0.25) * h)) / t_2))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = fmin(abs(M), abs(D))
	t_1 = fmax(abs(M), abs(D))
	t_2 = Float64(Float64(d + d) * l)
	t_3 = Float64(t_0 * t_1)
	tmp = 0.0
	if (l <= -2e+90)
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(Float64(t_0 * Float64(0.25 * Float64(Float64(t_1 * t_0) / d))) * t_1) * Float64(h / t_2))) * Float64(abs(d) / sqrt(Float64(-h)))) / sqrt(Float64(-l)));
	elseif (l <= 6.8e-293)
		tmp = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * Float64(1.0 - Float64(Float64(Float64(t_0 * Float64(Float64(t_1 / d) * Float64(0.25 * h))) / l) * Float64(Float64(t_1 / Float64(d + d)) * t_0))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 - Float64(Float64(t_3 * Float64(Float64(Float64(t_3 / d) * 0.25) * h)) / t_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = min(abs(M), abs(D));
	t_1 = max(abs(M), abs(D));
	t_2 = (d + d) * l;
	t_3 = t_0 * t_1;
	tmp = 0.0;
	if (l <= -2e+90)
		tmp = ((1.0 - (((t_0 * (0.25 * ((t_1 * t_0) / d))) * t_1) * (h / t_2))) * (abs(d) / sqrt(-h))) / sqrt(-l);
	elseif (l <= 6.8e-293)
		tmp = (abs(d) / sqrt((l * h))) * (1.0 - (((t_0 * ((t_1 / d) * (0.25 * h))) / l) * ((t_1 / (d + d)) * t_0)));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - ((t_3 * (((t_3 / d) * 0.25) * h)) / t_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Min[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(d + d), $MachinePrecision] * l), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 * t$95$1), $MachinePrecision]}, If[LessEqual[l, -2e+90], N[(N[(N[(1.0 - N[(N[(N[(t$95$0 * N[(0.25 * N[(N[(t$95$1 * t$95$0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(h / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.8e-293], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(t$95$0 * N[(N[(t$95$1 / d), $MachinePrecision] * N[(0.25 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(t$95$1 / N[(d + d), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(t$95$3 * N[(N[(N[(t$95$3 / d), $MachinePrecision] * 0.25), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.9999999999999999e90

   1. Initial program 66.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 67.3%

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

   5. Applied rewrites 75.4%

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

   7. Applied rewrites 39.2%

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

   if -1.9999999999999999e90 < l < 6.7999999999999999e-293

   1. Initial program 66.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 67.3%

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

   5. Applied rewrites 75.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. count-2-rev N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. count-2-rev N/A

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

      18. lift-+.f64 N/A

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

      19. associate-/l* N/A

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

   7. Applied rewrites 76.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 75.1%

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

   9. Applied rewrites 75.1%

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

   if 6.7999999999999999e-293 < l

   1. Initial program 66.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 67.3%

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

   5. Applied rewrites 75.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

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

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

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

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

      6. lower-unsound-sqrt.f64 41.5%

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

   7. Applied rewrites 41.5%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-unprod N/A

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

      4. rem-square-sqrt 41.5%

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

   9. Applied rewrites 41.5%

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

Alternative 5: 82.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{-0.25 \cdot \left(\frac{M \cdot D}{d} \cdot h\right)}{\ell} \cdot \left(M \cdot D\right), \frac{0.5}{d}, 1\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+261}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1
         (*
          (/ (fabs d) (sqrt (* l h)))
          (fma
           (* (/ (* -0.25 (* (/ (* M D) d) h)) l) (* M D))
           (/ 0.5 d)
           1.0))))
   (if (<= t_0 0.0)
     t_1
     (if (<= t_0 2e+261) (* (sqrt (/ d l)) (sqrt (/ d h))) t_1))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = (fabs(d) / sqrt((l * h))) * fma((((-0.25 * (((M * D) / d) * h)) / l) * (M * D)), (0.5 / d), 1.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2e+261) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * fma(Float64(Float64(Float64(-0.25 * Float64(Float64(Float64(M * D) / d) * h)) / l) * Float64(M * D)), Float64(0.5 / d), 1.0))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 2e+261)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(-0.25 * N[(N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * N[(0.5 / d), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 2e+261], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 1.9999999999999999e261 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 67.3%

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

   5. Applied rewrites 75.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. count-2-rev N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. count-2-rev N/A

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

      18. lift-+.f64 N/A

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

      19. associate-/l* N/A

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

   7. Applied rewrites 76.1%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   9. Applied rewrites 77.6%

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

   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)))) < 1.9999999999999999e261

   1. Initial program 66.3%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 23.4%

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

   4. Applied rewrites 23.4%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-sqrt.f64 N/A

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

      4. pow1/2 N/A

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

      5. metadata-eval N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow1/2 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. lower-/.f64 23.4%

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

   6. Applied rewrites 23.4%

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

   7. Taylor expanded in h around inf

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 39.3%

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

   9. Applied rewrites 39.3%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 82.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(M, D\right) \cdot \mathsf{max}\left(M, D\right)\\ t_1 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{-0.25 \cdot \left(\frac{t\_0}{d} \cdot h\right)}{\ell}, \frac{\mathsf{max}\left(M, D\right)}{d + d} \cdot \mathsf{min}\left(M, D\right), 1\right)\\ t_2 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_0}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+261}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (fmin M D) (fmax M D)))
        (t_1
         (*
          (/ (fabs d) (sqrt (* l h)))
          (fma
           (/ (* -0.25 (* (/ t_0 d) h)) l)
           (* (/ (fmax M D) (+ d d)) (fmin M D))
           1.0)))
        (t_2
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ t_0 (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_2 0.0)
     t_1
     (if (<= t_2 2e+261) (* (sqrt (/ d l)) (sqrt (/ d h))) t_1))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fmin(M, D) * fmax(M, D);
	double t_1 = (fabs(d) / sqrt((l * h))) * fma(((-0.25 * ((t_0 / d) * h)) / l), ((fmax(M, D) / (d + d)) * fmin(M, D)), 1.0);
	double t_2 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow((t_0 / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = t_1;
	} else if (t_2 <= 2e+261) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(fmin(M, D) * fmax(M, D))
	t_1 = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * fma(Float64(Float64(-0.25 * Float64(Float64(t_0 / d) * h)) / l), Float64(Float64(fmax(M, D) / Float64(d + d)) * fmin(M, D)), 1.0))
	t_2 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(t_0 / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = t_1;
	elseif (t_2 <= 2e+261)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Min[M, D], $MachinePrecision] * N[Max[M, D], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-0.25 * N[(N[(t$95$0 / d), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Max[M, D], $MachinePrecision] / N[(d + d), $MachinePrecision]), $MachinePrecision] * N[Min[M, D], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(t$95$0 / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], t$95$1, If[LessEqual[t$95$2, 2e+261], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 1.9999999999999999e261 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 67.3%

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

   5. Applied rewrites 75.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. count-2-rev N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. count-2-rev N/A

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

      18. lift-+.f64 N/A

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

      19. associate-/l* N/A

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

   7. Applied rewrites 76.1%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

   9. Applied rewrites 76.1%

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

   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)))) < 1.9999999999999999e261

   1. Initial program 66.3%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 23.4%

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

   4. Applied rewrites 23.4%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-sqrt.f64 N/A

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

      4. pow1/2 N/A

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

      5. metadata-eval N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow1/2 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. lower-/.f64 23.4%

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

   6. Applied rewrites 23.4%

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

   7. Taylor expanded in h around inf

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 39.3%

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

   9. Applied rewrites 39.3%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 81.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left(\left(0.25 \cdot \frac{D \cdot M}{d}\right) \cdot h\right) \cdot \frac{D \cdot M}{\left(d + d\right) \cdot \ell}\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+261}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1
         (*
          (/ (fabs d) (sqrt (* l h)))
          (- 1.0 (* (* (* 0.25 (/ (* D M) d)) h) (/ (* D M) (* (+ d d) l)))))))
   (if (<= t_0 0.0)
     t_1
     (if (<= t_0 2e+261) (* (sqrt (/ d l)) (sqrt (/ d h))) t_1))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = (fabs(d) / sqrt((l * h))) * (1.0 - (((0.25 * ((D * M) / d)) * h) * ((D * M) / ((d + d) * l))));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2e+261) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    t_1 = (abs(d) / sqrt((l * h))) * (1.0d0 - (((0.25d0 * ((d_1 * m) / d)) * h) * ((d_1 * m) / ((d + d) * l))))
    if (t_0 <= 0.0d0) then
        tmp = t_1
    else if (t_0 <= 2d+261) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = (Math.abs(d) / Math.sqrt((l * h))) * (1.0 - (((0.25 * ((D * M) / d)) * h) * ((D * M) / ((d + d) * l))));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2e+261) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_1 = (math.fabs(d) / math.sqrt((l * h))) * (1.0 - (((0.25 * ((D * M) / d)) * h) * ((D * M) / ((d + d) * l))))
	tmp = 0
	if t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 2e+261:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	else:
		tmp = t_1
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * Float64(1.0 - Float64(Float64(Float64(0.25 * Float64(Float64(D * M) / d)) * h) * Float64(Float64(D * M) / Float64(Float64(d + d) * l)))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 2e+261)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_1 = (abs(d) / sqrt((l * h))) * (1.0 - (((0.25 * ((D * M) / d)) * h) * ((D * M) / ((d + d) * l))));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 2e+261)
		tmp = sqrt((d / l)) * sqrt((d / h));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(0.25 * N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] * N[(N[(D * M), $MachinePrecision] / N[(N[(d + d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 2e+261], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 1.9999999999999999e261 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 67.3%

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

   5. Applied rewrites 75.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 76.3%

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 76.3%

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

   7. Applied rewrites 76.3%

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

   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)))) < 1.9999999999999999e261

   1. Initial program 66.3%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 23.4%

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

   4. Applied rewrites 23.4%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-sqrt.f64 N/A

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

      4. pow1/2 N/A

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

      5. metadata-eval N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow1/2 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. lower-/.f64 23.4%

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

   6. Applied rewrites 23.4%

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

   7. Taylor expanded in h around inf

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 39.3%

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

   9. Applied rewrites 39.3%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 78.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := 1 - \left(\left(\mathsf{min}\left(M, D\right) \cdot \left(0.25 \cdot \frac{\mathsf{max}\left(M, D\right) \cdot \mathsf{min}\left(M, D\right)}{d}\right)\right) \cdot \mathsf{max}\left(M, D\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\mathsf{min}\left(M, D\right) \cdot \mathsf{max}\left(M, D\right)}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \sqrt{h \cdot \ell}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_0 \cdot \frac{\left|d\right|}{t\_2}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+261}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|d\right| \cdot \frac{t\_0}{t\_2}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (-
          1.0
          (*
           (*
            (* (fmin M D) (* 0.25 (/ (* (fmax M D) (fmin M D)) d)))
            (fmax M D))
           (/ h (* (+ d 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 (/ (* (fmin M D) (fmax M D)) (* 2.0 d)) 2.0))
            (/ h l)))))
        (t_2 (sqrt (* h l))))
   (if (<= t_1 0.0)
     (* t_0 (/ (fabs d) t_2))
     (if (<= t_1 2e+261)
       (* (sqrt (/ d l)) (sqrt (/ d h)))
       (* (fabs d) (/ t_0 t_2))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 - (((fmin(M, D) * (0.25 * ((fmax(M, D) * fmin(M, D)) / d))) * fmax(M, D)) * (h / ((d + 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(((fmin(M, D) * fmax(M, D)) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = sqrt((h * l));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0 * (fabs(d) / t_2);
	} else if (t_1 <= 2e+261) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = fabs(d) * (t_0 / t_2);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 - (((fmin(m, d_1) * (0.25d0 * ((fmax(m, d_1) * fmin(m, d_1)) / d))) * fmax(m, d_1)) * (h / ((d + d) * l)))
    t_1 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((fmin(m, d_1) * fmax(m, d_1)) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    t_2 = sqrt((h * l))
    if (t_1 <= 0.0d0) then
        tmp = t_0 * (abs(d) / t_2)
    else if (t_1 <= 2d+261) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else
        tmp = abs(d) * (t_0 / t_2)
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 - (((fmin(M, D) * (0.25 * ((fmax(M, D) * fmin(M, D)) / d))) * fmax(M, D)) * (h / ((d + 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(((fmin(M, D) * fmax(M, D)) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = Math.sqrt((h * l));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0 * (Math.abs(d) / t_2);
	} else if (t_1 <= 2e+261) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else {
		tmp = Math.abs(d) * (t_0 / t_2);
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = 1.0 - (((fmin(M, D) * (0.25 * ((fmax(M, D) * fmin(M, D)) / d))) * fmax(M, D)) * (h / ((d + 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(((fmin(M, D) * fmax(M, D)) / (2.0 * d)), 2.0)) * (h / l)))
	t_2 = math.sqrt((h * l))
	tmp = 0
	if t_1 <= 0.0:
		tmp = t_0 * (math.fabs(d) / t_2)
	elif t_1 <= 2e+261:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	else:
		tmp = math.fabs(d) * (t_0 / t_2)
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(1.0 - Float64(Float64(Float64(fmin(M, D) * Float64(0.25 * Float64(Float64(fmax(M, D) * fmin(M, D)) / d))) * fmax(M, D)) * Float64(h / Float64(Float64(d + 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(fmin(M, D) * fmax(M, D)) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_2 = sqrt(Float64(h * l))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(t_0 * Float64(abs(d) / t_2));
	elseif (t_1 <= 2e+261)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = Float64(abs(d) * Float64(t_0 / t_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = 1.0 - (((min(M, D) * (0.25 * ((max(M, D) * min(M, D)) / d))) * max(M, D)) * (h / ((d + d) * l)));
	t_1 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((min(M, D) * max(M, D)) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_2 = sqrt((h * l));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = t_0 * (abs(d) / t_2);
	elseif (t_1 <= 2e+261)
		tmp = sqrt((d / l)) * sqrt((d / h));
	else
		tmp = abs(d) * (t_0 / t_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(1.0 - N[(N[(N[(N[Min[M, D], $MachinePrecision] * N[(0.25 * N[(N[(N[Max[M, D], $MachinePrecision] * N[Min[M, D], $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Max[M, D], $MachinePrecision]), $MachinePrecision] * N[(h / N[(N[(d + d), $MachinePrecision] * l), $MachinePrecision]), $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[(N[Min[M, D], $MachinePrecision] * N[Max[M, D], $MachinePrecision]), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(t$95$0 * N[(N[Abs[d], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+261], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Abs[d], $MachinePrecision] * N[(t$95$0 / t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 66.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 67.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]

   5. Applied rewrites 75.4%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot \left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot h\right)}{\left(d + d\right) \cdot \ell}\right)} \]

   7. Applied rewrites 69.8%

      \[\leadsto \color{blue}{\left(1 - \left(\left(M \cdot \left(0.25 \cdot \frac{D \cdot M}{d}\right)\right) \cdot D\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \]

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1.9999999999999999e261

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.4%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.4%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. mult-flip N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{\frac{1}{h}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \frac{1}{h} \]

      4. pow1/2 N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot {\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right) \cdot \frac{1}{h} \]

      5. metadata-eval N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{h} \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{h} \]

      7. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{h} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{\color{blue}{1}}{h} \]

      9. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{d \cdot h}\right) \cdot \frac{\color{blue}{1}}{h} \]

      10. associate-*l* N/A

          \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{d \cdot h} \cdot \frac{1}{h}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{d \cdot h} \cdot \frac{1}{h}\right)} \]

      12. lift-pow.f64 N/A

          \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\color{blue}{\sqrt{d \cdot h}} \cdot \frac{1}{h}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{d \cdot h} \cdot \frac{1}{h}\right) \]

      14. metadata-eval N/A

          \[\leadsto {\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left(\sqrt{d \cdot h} \cdot \frac{1}{h}\right) \]

      15. pow1/2 N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\sqrt{d \cdot h}} \cdot \frac{1}{h}\right) \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\sqrt{d \cdot h}} \cdot \frac{1}{h}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{d \cdot h} \cdot \color{blue}{\frac{1}{h}}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{d \cdot h} \cdot \frac{1}{h}\right) \]

      19. *-commutative N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{h \cdot d} \cdot \frac{1}{h}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{h \cdot d} \cdot \frac{1}{h}\right) \]

      21. lower-/.f64 23.4%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{h \cdot d} \cdot \frac{1}{\color{blue}{h}}\right) \]

   6. Applied rewrites 23.4%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\sqrt{h \cdot d} \cdot \frac{1}{h}\right)} \]

   7. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]
   8. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]

      2. lower-/.f64 39.3%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]

   9. Applied rewrites 39.3%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]

   if 1.9999999999999999e261 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

   3. Applied rewrites 67.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]

   5. Applied rewrites 75.4%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot \left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot h\right)}{\left(d + d\right) \cdot \ell}\right)} \]

   7. Applied rewrites 71.0%

      \[\leadsto \color{blue}{\left|d\right| \cdot \frac{1 - \left(\left(M \cdot \left(0.25 \cdot \frac{D \cdot M}{d}\right)\right) \cdot D\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}}{\sqrt{h \cdot \ell}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 78.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := 1 - \left(\left(\mathsf{min}\left(M, D\right) \cdot \left(0.25 \cdot \frac{\mathsf{max}\left(M, D\right) \cdot \mathsf{min}\left(M, D\right)}{d}\right)\right) \cdot \mathsf{max}\left(M, D\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\mathsf{min}\left(M, D\right) \cdot \mathsf{max}\left(M, D\right)}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \sqrt{h \cdot \ell}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_0 \cdot \frac{\left|d\right|}{t\_2}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+261}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot \left|d\right|}{t\_2}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (-
          1.0
          (*
           (*
            (* (fmin M D) (* 0.25 (/ (* (fmax M D) (fmin M D)) d)))
            (fmax M D))
           (/ h (* (+ d 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 (/ (* (fmin M D) (fmax M D)) (* 2.0 d)) 2.0))
            (/ h l)))))
        (t_2 (sqrt (* h l))))
   (if (<= t_1 0.0)
     (* t_0 (/ (fabs d) t_2))
     (if (<= t_1 2e+261)
       (* (sqrt (/ d l)) (sqrt (/ d h)))
       (/ (* t_0 (fabs d)) t_2)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 - (((fmin(M, D) * (0.25 * ((fmax(M, D) * fmin(M, D)) / d))) * fmax(M, D)) * (h / ((d + 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(((fmin(M, D) * fmax(M, D)) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = sqrt((h * l));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0 * (fabs(d) / t_2);
	} else if (t_1 <= 2e+261) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = (t_0 * fabs(d)) / t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 - (((fmin(m, d_1) * (0.25d0 * ((fmax(m, d_1) * fmin(m, d_1)) / d))) * fmax(m, d_1)) * (h / ((d + d) * l)))
    t_1 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((fmin(m, d_1) * fmax(m, d_1)) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    t_2 = sqrt((h * l))
    if (t_1 <= 0.0d0) then
        tmp = t_0 * (abs(d) / t_2)
    else if (t_1 <= 2d+261) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else
        tmp = (t_0 * abs(d)) / t_2
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 - (((fmin(M, D) * (0.25 * ((fmax(M, D) * fmin(M, D)) / d))) * fmax(M, D)) * (h / ((d + 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(((fmin(M, D) * fmax(M, D)) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = Math.sqrt((h * l));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0 * (Math.abs(d) / t_2);
	} else if (t_1 <= 2e+261) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else {
		tmp = (t_0 * Math.abs(d)) / t_2;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = 1.0 - (((fmin(M, D) * (0.25 * ((fmax(M, D) * fmin(M, D)) / d))) * fmax(M, D)) * (h / ((d + 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(((fmin(M, D) * fmax(M, D)) / (2.0 * d)), 2.0)) * (h / l)))
	t_2 = math.sqrt((h * l))
	tmp = 0
	if t_1 <= 0.0:
		tmp = t_0 * (math.fabs(d) / t_2)
	elif t_1 <= 2e+261:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	else:
		tmp = (t_0 * math.fabs(d)) / t_2
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(1.0 - Float64(Float64(Float64(fmin(M, D) * Float64(0.25 * Float64(Float64(fmax(M, D) * fmin(M, D)) / d))) * fmax(M, D)) * Float64(h / Float64(Float64(d + 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(fmin(M, D) * fmax(M, D)) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_2 = sqrt(Float64(h * l))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(t_0 * Float64(abs(d) / t_2));
	elseif (t_1 <= 2e+261)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = Float64(Float64(t_0 * abs(d)) / t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = 1.0 - (((min(M, D) * (0.25 * ((max(M, D) * min(M, D)) / d))) * max(M, D)) * (h / ((d + d) * l)));
	t_1 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((min(M, D) * max(M, D)) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_2 = sqrt((h * l));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = t_0 * (abs(d) / t_2);
	elseif (t_1 <= 2e+261)
		tmp = sqrt((d / l)) * sqrt((d / h));
	else
		tmp = (t_0 * abs(d)) / t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(1.0 - N[(N[(N[(N[Min[M, D], $MachinePrecision] * N[(0.25 * N[(N[(N[Max[M, D], $MachinePrecision] * N[Min[M, D], $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Max[M, D], $MachinePrecision]), $MachinePrecision] * N[(h / N[(N[(d + d), $MachinePrecision] * l), $MachinePrecision]), $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[(N[Min[M, D], $MachinePrecision] * N[Max[M, D], $MachinePrecision]), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(t$95$0 * N[(N[Abs[d], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+261], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[Abs[d], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

   3. Applied rewrites 67.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]

   5. Applied rewrites 75.4%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot \left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot h\right)}{\left(d + d\right) \cdot \ell}\right)} \]

   7. Applied rewrites 69.8%

      \[\leadsto \color{blue}{\left(1 - \left(\left(M \cdot \left(0.25 \cdot \frac{D \cdot M}{d}\right)\right) \cdot D\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \]

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1.9999999999999999e261

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.4%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.4%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. mult-flip N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{\frac{1}{h}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \frac{1}{h} \]

      4. pow1/2 N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot {\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right) \cdot \frac{1}{h} \]

      5. metadata-eval N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{h} \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{h} \]

      7. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{h} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{\color{blue}{1}}{h} \]

      9. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{d \cdot h}\right) \cdot \frac{\color{blue}{1}}{h} \]

      10. associate-*l* N/A

          \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{d \cdot h} \cdot \frac{1}{h}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{d \cdot h} \cdot \frac{1}{h}\right)} \]

      12. lift-pow.f64 N/A

          \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\color{blue}{\sqrt{d \cdot h}} \cdot \frac{1}{h}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{d \cdot h} \cdot \frac{1}{h}\right) \]

      14. metadata-eval N/A

          \[\leadsto {\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left(\sqrt{d \cdot h} \cdot \frac{1}{h}\right) \]

      15. pow1/2 N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\sqrt{d \cdot h}} \cdot \frac{1}{h}\right) \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\sqrt{d \cdot h}} \cdot \frac{1}{h}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{d \cdot h} \cdot \color{blue}{\frac{1}{h}}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{d \cdot h} \cdot \frac{1}{h}\right) \]

      19. *-commutative N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{h \cdot d} \cdot \frac{1}{h}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{h \cdot d} \cdot \frac{1}{h}\right) \]

      21. lower-/.f64 23.4%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{h \cdot d} \cdot \frac{1}{\color{blue}{h}}\right) \]

   6. Applied rewrites 23.4%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\sqrt{h \cdot d} \cdot \frac{1}{h}\right)} \]

   7. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]
   8. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]

      2. lower-/.f64 39.3%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]

   9. Applied rewrites 39.3%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]

   if 1.9999999999999999e261 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

   3. Applied rewrites 67.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]

   5. Applied rewrites 75.4%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot \left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot h\right)}{\left(d + d\right) \cdot \ell}\right)} \]

   7. Applied rewrites 71.1%

      \[\leadsto \color{blue}{\frac{\left(1 - \left(\left(M \cdot \left(0.25 \cdot \frac{D \cdot M}{d}\right)\right) \cdot D\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right) \cdot \left|d\right|}{\sqrt{h \cdot \ell}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 77.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left(1 - \left(\left(\mathsf{min}\left(M, D\right) \cdot \left(0.25 \cdot \frac{\mathsf{max}\left(M, D\right) \cdot \mathsf{min}\left(M, D\right)}{d}\right)\right) \cdot \mathsf{max}\left(M, D\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\mathsf{min}\left(M, D\right) \cdot \mathsf{max}\left(M, D\right)}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+261}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (-
           1.0
           (*
            (*
             (* (fmin M D) (* 0.25 (/ (* (fmax M D) (fmin M D)) d)))
             (fmax M D))
            (/ h (* (+ d d) l))))
          (/ (fabs d) (sqrt (* h l)))))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (*
            (* (/ 1.0 2.0) (pow (/ (* (fmin M D) (fmax M D)) (* 2.0 d)) 2.0))
            (/ h l))))))
   (if (<= t_1 0.0)
     t_0
     (if (<= t_1 2e+261) (* (sqrt (/ d l)) (sqrt (/ d h))) t_0))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (1.0 - (((fmin(M, D) * (0.25 * ((fmax(M, D) * fmin(M, D)) / d))) * fmax(M, D)) * (h / ((d + d) * l)))) * (fabs(d) / sqrt((h * l)));
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((fmin(M, D) * fmax(M, D)) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 2e+261) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (1.0d0 - (((fmin(m, d_1) * (0.25d0 * ((fmax(m, d_1) * fmin(m, d_1)) / d))) * fmax(m, d_1)) * (h / ((d + d) * l)))) * (abs(d) / sqrt((h * l)))
    t_1 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((fmin(m, d_1) * fmax(m, d_1)) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_1 <= 0.0d0) then
        tmp = t_0
    else if (t_1 <= 2d+261) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (1.0 - (((fmin(M, D) * (0.25 * ((fmax(M, D) * fmin(M, D)) / d))) * fmax(M, D)) * (h / ((d + d) * l)))) * (Math.abs(d) / Math.sqrt((h * l)));
	double t_1 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((fmin(M, D) * fmax(M, D)) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 2e+261) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (1.0 - (((fmin(M, D) * (0.25 * ((fmax(M, D) * fmin(M, D)) / d))) * fmax(M, D)) * (h / ((d + d) * l)))) * (math.fabs(d) / math.sqrt((h * l)))
	t_1 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((fmin(M, D) * fmax(M, D)) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_1 <= 0.0:
		tmp = t_0
	elif t_1 <= 2e+261:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	else:
		tmp = t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(1.0 - Float64(Float64(Float64(fmin(M, D) * Float64(0.25 * Float64(Float64(fmax(M, D) * fmin(M, D)) / d))) * fmax(M, D)) * Float64(h / Float64(Float64(d + d) * l)))) * Float64(abs(d) / sqrt(Float64(h * l))))
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(fmin(M, D) * fmax(M, D)) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 2e+261)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (1.0 - (((min(M, D) * (0.25 * ((max(M, D) * min(M, D)) / d))) * max(M, D)) * (h / ((d + d) * l)))) * (abs(d) / sqrt((h * l)));
	t_1 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((min(M, D) * max(M, D)) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 2e+261)
		tmp = sqrt((d / l)) * sqrt((d / h));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(1.0 - N[(N[(N[(N[Min[M, D], $MachinePrecision] * N[(0.25 * N[(N[(N[Max[M, D], $MachinePrecision] * N[Min[M, D], $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Max[M, D], $MachinePrecision]), $MachinePrecision] * N[(h / N[(N[(d + d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $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[(N[Min[M, D], $MachinePrecision] * N[Max[M, D], $MachinePrecision]), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], t$95$0, If[LessEqual[t$95$1, 2e+261], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 1.9999999999999999e261 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

   3. Applied rewrites 67.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]

   5. Applied rewrites 75.4%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot \left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot h\right)}{\left(d + d\right) \cdot \ell}\right)} \]

   7. Applied rewrites 69.8%

      \[\leadsto \color{blue}{\left(1 - \left(\left(M \cdot \left(0.25 \cdot \frac{D \cdot M}{d}\right)\right) \cdot D\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \]

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1.9999999999999999e261

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.4%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.4%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. mult-flip N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{\frac{1}{h}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \frac{1}{h} \]

      4. pow1/2 N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot {\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right) \cdot \frac{1}{h} \]

      5. metadata-eval N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{h} \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{h} \]

      7. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{h} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{\color{blue}{1}}{h} \]

      9. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{d \cdot h}\right) \cdot \frac{\color{blue}{1}}{h} \]

      10. associate-*l* N/A

          \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{d \cdot h} \cdot \frac{1}{h}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{d \cdot h} \cdot \frac{1}{h}\right)} \]

      12. lift-pow.f64 N/A

          \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\color{blue}{\sqrt{d \cdot h}} \cdot \frac{1}{h}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{d \cdot h} \cdot \frac{1}{h}\right) \]

      14. metadata-eval N/A

          \[\leadsto {\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left(\sqrt{d \cdot h} \cdot \frac{1}{h}\right) \]

      15. pow1/2 N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\sqrt{d \cdot h}} \cdot \frac{1}{h}\right) \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\sqrt{d \cdot h}} \cdot \frac{1}{h}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{d \cdot h} \cdot \color{blue}{\frac{1}{h}}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{d \cdot h} \cdot \frac{1}{h}\right) \]

      19. *-commutative N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{h \cdot d} \cdot \frac{1}{h}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{h \cdot d} \cdot \frac{1}{h}\right) \]

      21. lower-/.f64 23.4%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{h \cdot d} \cdot \frac{1}{\color{blue}{h}}\right) \]

   6. Applied rewrites 23.4%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\sqrt{h \cdot d} \cdot \frac{1}{h}\right)} \]

   7. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]
   8. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]

      2. lower-/.f64 39.3%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]

   9. Applied rewrites 39.3%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 74.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\frac{\mathsf{max}\left(M, D\right) \cdot \left(\left(\frac{\mathsf{max}\left(M, D\right)}{d} \cdot \mathsf{min}\left(M, D\right)\right) \cdot \mathsf{min}\left(M, D\right)\right)}{d} \cdot -0.125, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\mathsf{min}\left(M, D\right) \cdot \mathsf{max}\left(M, D\right)}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+261}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (fma
           (*
            (/ (* (fmax M D) (* (* (/ (fmax M D) d) (fmin M D)) (fmin M D))) d)
            -0.125)
           (/ h l)
           1.0)
          (/ (fabs d) (sqrt (* l h)))))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (*
            (* (/ 1.0 2.0) (pow (/ (* (fmin M D) (fmax M D)) (* 2.0 d)) 2.0))
            (/ h l))))))
   (if (<= t_1 0.0)
     t_0
     (if (<= t_1 2e+261) (* (sqrt (/ d l)) (sqrt (/ d h))) t_0))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fma((((fmax(M, D) * (((fmax(M, D) / d) * fmin(M, D)) * fmin(M, D))) / d) * -0.125), (h / l), 1.0) * (fabs(d) / sqrt((l * h)));
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((fmin(M, D) * fmax(M, D)) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 2e+261) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(fma(Float64(Float64(Float64(fmax(M, D) * Float64(Float64(Float64(fmax(M, D) / d) * fmin(M, D)) * fmin(M, D))) / d) * -0.125), Float64(h / l), 1.0) * Float64(abs(d) / sqrt(Float64(l * h))))
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(fmin(M, D) * fmax(M, D)) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 2e+261)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[(N[(N[(N[Max[M, D], $MachinePrecision] * N[(N[(N[(N[Max[M, D], $MachinePrecision] / d), $MachinePrecision] * N[Min[M, D], $MachinePrecision]), $MachinePrecision] * N[Min[M, D], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $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[(N[Min[M, D], $MachinePrecision] * N[Max[M, D], $MachinePrecision]), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], t$95$0, If[LessEqual[t$95$1, 2e+261], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 1.9999999999999999e261 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      8. associate-/l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      9. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      10. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot \left(M \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right)\right)} \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{2 \cdot d} \cdot \left(M \cdot \left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \frac{1}{2}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{2 \cdot d} \cdot \left(M \cdot \left(\frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \frac{1}{2}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      13. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{2 \cdot d} \cdot \left(M \cdot \left(\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \frac{1}{2}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      14. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{2 \cdot d} \cdot \left(M \cdot \left(\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \frac{1}{2}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      15. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{2 \cdot d} \cdot \left(M \cdot \color{blue}{\left(\frac{D}{2 \cdot d} \cdot \left(M \cdot \frac{1}{2}\right)\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{2 \cdot d} \cdot \left(M \cdot \left(\frac{D}{2 \cdot d} \cdot \left(M \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      17. mult-flip N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{2 \cdot d} \cdot \left(M \cdot \left(\frac{D}{2 \cdot d} \cdot \color{blue}{\frac{M}{2}}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{2 \cdot d} \cdot \color{blue}{\left(\left(M \cdot \frac{D}{2 \cdot d}\right) \cdot \frac{M}{2}\right)}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 59.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right)\right)} \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 67.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{D \cdot \left(\left(\frac{D}{d} \cdot M\right) \cdot M\right)}{d} \cdot -0.125, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \]

   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)))) < 1.9999999999999999e261

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.4%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.4%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. mult-flip N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{\frac{1}{h}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \frac{1}{h} \]

      4. pow1/2 N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot {\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right) \cdot \frac{1}{h} \]

      5. metadata-eval N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{h} \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{h} \]

      7. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{h} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{\color{blue}{1}}{h} \]

      9. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{d \cdot h}\right) \cdot \frac{\color{blue}{1}}{h} \]

      10. associate-*l* N/A

          \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{d \cdot h} \cdot \frac{1}{h}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{d \cdot h} \cdot \frac{1}{h}\right)} \]

      12. lift-pow.f64 N/A

          \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\color{blue}{\sqrt{d \cdot h}} \cdot \frac{1}{h}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{d \cdot h} \cdot \frac{1}{h}\right) \]

      14. metadata-eval N/A

          \[\leadsto {\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left(\sqrt{d \cdot h} \cdot \frac{1}{h}\right) \]

      15. pow1/2 N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\sqrt{d \cdot h}} \cdot \frac{1}{h}\right) \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\sqrt{d \cdot h}} \cdot \frac{1}{h}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{d \cdot h} \cdot \color{blue}{\frac{1}{h}}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{d \cdot h} \cdot \frac{1}{h}\right) \]

      19. *-commutative N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{h \cdot d} \cdot \frac{1}{h}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{h \cdot d} \cdot \frac{1}{h}\right) \]

      21. lower-/.f64 23.4%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{h \cdot d} \cdot \frac{1}{\color{blue}{h}}\right) \]

   6. Applied rewrites 23.4%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\sqrt{h \cdot d} \cdot \frac{1}{h}\right)} \]

   7. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]
   8. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]

      2. lower-/.f64 39.3%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]

   9. Applied rewrites 39.3%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 57.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-162}:\\ \;\;\;\;-1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+261}:\\ \;\;\;\;t\_2 \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot t\_2}{h}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ (fabs d) (sqrt (* l h))) 1.0))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_2 (sqrt (/ d l))))
   (if (<= t_1 -5e-162)
     (* -1.0 (/ (* d (sqrt (/ h l))) h))
     (if (<= t_1 0.0)
       t_0
       (if (<= t_1 2e+261)
         (* t_2 (sqrt (/ d h)))
         (if (<= t_1 INFINITY)
           t_0
           (/ (* (* -1.0 (* d (sqrt (/ h d)))) t_2) h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (fabs(d) / sqrt((l * h))) * 1.0;
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = sqrt((d / l));
	double tmp;
	if (t_1 <= -5e-162) {
		tmp = -1.0 * ((d * sqrt((h / l))) / h);
	} else if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 2e+261) {
		tmp = t_2 * sqrt((d / h));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = ((-1.0 * (d * sqrt((h / d)))) * t_2) / h;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.abs(d) / Math.sqrt((l * h))) * 1.0;
	double t_1 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = Math.sqrt((d / l));
	double tmp;
	if (t_1 <= -5e-162) {
		tmp = -1.0 * ((d * Math.sqrt((h / l))) / h);
	} else if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 2e+261) {
		tmp = t_2 * Math.sqrt((d / h));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else {
		tmp = ((-1.0 * (d * Math.sqrt((h / d)))) * t_2) / h;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.fabs(d) / math.sqrt((l * h))) * 1.0
	t_1 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_2 = math.sqrt((d / l))
	tmp = 0
	if t_1 <= -5e-162:
		tmp = -1.0 * ((d * math.sqrt((h / l))) / h)
	elif t_1 <= 0.0:
		tmp = t_0
	elif t_1 <= 2e+261:
		tmp = t_2 * math.sqrt((d / h))
	elif t_1 <= math.inf:
		tmp = t_0
	else:
		tmp = ((-1.0 * (d * math.sqrt((h / d)))) * t_2) / h
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * 1.0)
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_2 = sqrt(Float64(d / l))
	tmp = 0.0
	if (t_1 <= -5e-162)
		tmp = Float64(-1.0 * Float64(Float64(d * sqrt(Float64(h / l))) / h));
	elseif (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 2e+261)
		tmp = Float64(t_2 * sqrt(Float64(d / h)));
	elseif (t_1 <= Inf)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(-1.0 * Float64(d * sqrt(Float64(h / d)))) * t_2) / h);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (abs(d) / sqrt((l * h))) * 1.0;
	t_1 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_2 = sqrt((d / l));
	tmp = 0.0;
	if (t_1 <= -5e-162)
		tmp = -1.0 * ((d * sqrt((h / l))) / h);
	elseif (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 2e+261)
		tmp = t_2 * sqrt((d / h));
	elseif (t_1 <= Inf)
		tmp = t_0;
	else
		tmp = ((-1.0 * (d * sqrt((h / d)))) * t_2) / h;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -5e-162], N[(-1.0 * N[(N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], t$95$0, If[LessEqual[t$95$1, 2e+261], N[(t$95$2 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$0, N[(N[(N[(-1.0 * N[(d * N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $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)))) < -5.0000000000000001e-162

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. 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. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. mult-flip N/A

          \[\leadsto \sqrt{\color{blue}{\left(d \cdot \frac{1}{h}\right)} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. associate-*l* N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \left(\frac{1}{h} \cdot \frac{d}{\ell}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \left(\frac{1}{h} \cdot \frac{d}{\ell}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{d \cdot \left(\frac{1}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. frac-2neg N/A

          \[\leadsto \sqrt{d \cdot \left(\frac{1}{h} \cdot \color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. frac-2neg N/A

          \[\leadsto \sqrt{d \cdot \left(\frac{1}{h} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. frac-times N/A

          \[\leadsto \sqrt{d \cdot \color{blue}{\frac{1 \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)\right)}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. remove-double-neg N/A

          \[\leadsto \sqrt{d \cdot \frac{1 \cdot \color{blue}{d}}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. *-lft-identity N/A

          \[\leadsto \sqrt{d \cdot \frac{\color{blue}{d}}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. remove-double-neg N/A

          \[\leadsto \sqrt{d \cdot \frac{d}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(h\right)\right)\right)\right)} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. remove-double-neg N/A

          \[\leadsto \sqrt{d \cdot \frac{d}{\color{blue}{h} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. remove-double-neg N/A

          \[\leadsto \sqrt{d \cdot \frac{d}{h \cdot \color{blue}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      22. lower-/.f64 N/A

          \[\leadsto \sqrt{d \cdot \color{blue}{\frac{d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      23. lower-*.f64 50.7%

          \[\leadsto \sqrt{d \cdot \frac{d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 50.7%

      \[\leadsto \color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. lower-*.f64 26.3%

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   6. Applied rewrites 26.3%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   7. Taylor expanded in h around 0

      \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      4. lower-/.f64 14.1%

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   9. Applied rewrites 14.1%

      \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

   if -5.0000000000000001e-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 or 1.9999999999999999e261 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

   3. Applied rewrites 67.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]

   5. Applied rewrites 75.4%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot \left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot h\right)}{\left(d + d\right) \cdot \ell}\right)} \]

   6. Taylor expanded in d around inf

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 42.4%

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \color{blue}{1} \]

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1.9999999999999999e261

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.4%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.4%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. mult-flip N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{\frac{1}{h}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \frac{1}{h} \]

      4. pow1/2 N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot {\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right) \cdot \frac{1}{h} \]

      5. metadata-eval N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{h} \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{h} \]

      7. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{h} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{\color{blue}{1}}{h} \]

      9. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{d \cdot h}\right) \cdot \frac{\color{blue}{1}}{h} \]

      10. associate-*l* N/A

          \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{d \cdot h} \cdot \frac{1}{h}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{d \cdot h} \cdot \frac{1}{h}\right)} \]

      12. lift-pow.f64 N/A

          \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\color{blue}{\sqrt{d \cdot h}} \cdot \frac{1}{h}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{d \cdot h} \cdot \frac{1}{h}\right) \]

      14. metadata-eval N/A

          \[\leadsto {\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left(\sqrt{d \cdot h} \cdot \frac{1}{h}\right) \]

      15. pow1/2 N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\sqrt{d \cdot h}} \cdot \frac{1}{h}\right) \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\sqrt{d \cdot h}} \cdot \frac{1}{h}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{d \cdot h} \cdot \color{blue}{\frac{1}{h}}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{d \cdot h} \cdot \frac{1}{h}\right) \]

      19. *-commutative N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{h \cdot d} \cdot \frac{1}{h}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{h \cdot d} \cdot \frac{1}{h}\right) \]

      21. lower-/.f64 23.4%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{h \cdot d} \cdot \frac{1}{\color{blue}{h}}\right) \]

   6. Applied rewrites 23.4%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\sqrt{h \cdot d} \cdot \frac{1}{h}\right)} \]

   7. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]
   8. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]

      2. lower-/.f64 39.3%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]

   9. Applied rewrites 39.3%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]

   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 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.4%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.4%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   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 12.3%

         \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   7. Applied rewrites 12.3%

      \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 13: 56.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-162}:\\ \;\;\;\;-1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+261}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ (fabs d) (sqrt (* l h))) 1.0))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_1 -5e-162)
     (* -1.0 (/ (* d (sqrt (/ h l))) h))
     (if (<= t_1 0.0)
       t_0
       (if (<= t_1 2e+261) (* (sqrt (/ d l)) (sqrt (/ d h))) t_0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (fabs(d) / sqrt((l * h))) * 1.0;
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= -5e-162) {
		tmp = -1.0 * ((d * sqrt((h / l))) / h);
	} else if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 2e+261) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (abs(d) / sqrt((l * h))) * 1.0d0
    t_1 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_1 <= (-5d-162)) then
        tmp = (-1.0d0) * ((d * sqrt((h / l))) / h)
    else if (t_1 <= 0.0d0) then
        tmp = t_0
    else if (t_1 <= 2d+261) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.abs(d) / Math.sqrt((l * h))) * 1.0;
	double t_1 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= -5e-162) {
		tmp = -1.0 * ((d * Math.sqrt((h / l))) / h);
	} else if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 2e+261) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.fabs(d) / math.sqrt((l * h))) * 1.0
	t_1 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_1 <= -5e-162:
		tmp = -1.0 * ((d * math.sqrt((h / l))) / h)
	elif t_1 <= 0.0:
		tmp = t_0
	elif t_1 <= 2e+261:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	else:
		tmp = t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * 1.0)
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_1 <= -5e-162)
		tmp = Float64(-1.0 * Float64(Float64(d * sqrt(Float64(h / l))) / h));
	elseif (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 2e+261)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (abs(d) / sqrt((l * h))) * 1.0;
	t_1 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_1 <= -5e-162)
		tmp = -1.0 * ((d * sqrt((h / l))) / h);
	elseif (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 2e+261)
		tmp = sqrt((d / l)) * sqrt((d / h));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-162], N[(-1.0 * N[(N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], t$95$0, If[LessEqual[t$95$1, 2e+261], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5.0000000000000001e-162

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. 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. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. mult-flip N/A

          \[\leadsto \sqrt{\color{blue}{\left(d \cdot \frac{1}{h}\right)} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. associate-*l* N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \left(\frac{1}{h} \cdot \frac{d}{\ell}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \left(\frac{1}{h} \cdot \frac{d}{\ell}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{d \cdot \left(\frac{1}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. frac-2neg N/A

          \[\leadsto \sqrt{d \cdot \left(\frac{1}{h} \cdot \color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. frac-2neg N/A

          \[\leadsto \sqrt{d \cdot \left(\frac{1}{h} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. frac-times N/A

          \[\leadsto \sqrt{d \cdot \color{blue}{\frac{1 \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)\right)}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. remove-double-neg N/A

          \[\leadsto \sqrt{d \cdot \frac{1 \cdot \color{blue}{d}}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. *-lft-identity N/A

          \[\leadsto \sqrt{d \cdot \frac{\color{blue}{d}}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. remove-double-neg N/A

          \[\leadsto \sqrt{d \cdot \frac{d}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(h\right)\right)\right)\right)} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. remove-double-neg N/A

          \[\leadsto \sqrt{d \cdot \frac{d}{\color{blue}{h} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. remove-double-neg N/A

          \[\leadsto \sqrt{d \cdot \frac{d}{h \cdot \color{blue}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      22. lower-/.f64 N/A

          \[\leadsto \sqrt{d \cdot \color{blue}{\frac{d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      23. lower-*.f64 50.7%

          \[\leadsto \sqrt{d \cdot \frac{d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 50.7%

      \[\leadsto \color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. lower-*.f64 26.3%

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   6. Applied rewrites 26.3%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   7. Taylor expanded in h around 0

      \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      4. lower-/.f64 14.1%

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   9. Applied rewrites 14.1%

      \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

   if -5.0000000000000001e-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 or 1.9999999999999999e261 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

   3. Applied rewrites 67.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]

   5. Applied rewrites 75.4%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot \left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot h\right)}{\left(d + d\right) \cdot \ell}\right)} \]

   6. Taylor expanded in d around inf

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 42.4%

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \color{blue}{1} \]

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1.9999999999999999e261

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.4%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.4%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. mult-flip N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{\frac{1}{h}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \frac{1}{h} \]

      4. pow1/2 N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot {\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right) \cdot \frac{1}{h} \]

      5. metadata-eval N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{h} \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{h} \]

      7. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{h} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{\color{blue}{1}}{h} \]

      9. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{d \cdot h}\right) \cdot \frac{\color{blue}{1}}{h} \]

      10. associate-*l* N/A

          \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{d \cdot h} \cdot \frac{1}{h}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{d \cdot h} \cdot \frac{1}{h}\right)} \]

      12. lift-pow.f64 N/A

          \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\color{blue}{\sqrt{d \cdot h}} \cdot \frac{1}{h}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{d \cdot h} \cdot \frac{1}{h}\right) \]

      14. metadata-eval N/A

          \[\leadsto {\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left(\sqrt{d \cdot h} \cdot \frac{1}{h}\right) \]

      15. pow1/2 N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\sqrt{d \cdot h}} \cdot \frac{1}{h}\right) \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\sqrt{d \cdot h}} \cdot \frac{1}{h}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{d \cdot h} \cdot \color{blue}{\frac{1}{h}}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{d \cdot h} \cdot \frac{1}{h}\right) \]

      19. *-commutative N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{h \cdot d} \cdot \frac{1}{h}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{h \cdot d} \cdot \frac{1}{h}\right) \]

      21. lower-/.f64 23.4%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{h \cdot d} \cdot \frac{1}{\color{blue}{h}}\right) \]

   6. Applied rewrites 23.4%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\sqrt{h \cdot d} \cdot \frac{1}{h}\right)} \]

   7. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]
   8. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]

      2. lower-/.f64 39.3%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]

   9. Applied rewrites 39.3%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 56.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-162}:\\ \;\;\;\;\left(-d\right) \cdot \frac{\sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+261}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ (fabs d) (sqrt (* l h))) 1.0))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_1 -5e-162)
     (* (- d) (/ (sqrt (/ h l)) h))
     (if (<= t_1 0.0)
       t_0
       (if (<= t_1 2e+261) (* (sqrt (/ d l)) (sqrt (/ d h))) t_0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (fabs(d) / sqrt((l * h))) * 1.0;
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= -5e-162) {
		tmp = -d * (sqrt((h / l)) / h);
	} else if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 2e+261) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (abs(d) / sqrt((l * h))) * 1.0d0
    t_1 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_1 <= (-5d-162)) then
        tmp = -d * (sqrt((h / l)) / h)
    else if (t_1 <= 0.0d0) then
        tmp = t_0
    else if (t_1 <= 2d+261) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.abs(d) / Math.sqrt((l * h))) * 1.0;
	double t_1 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= -5e-162) {
		tmp = -d * (Math.sqrt((h / l)) / h);
	} else if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 2e+261) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.fabs(d) / math.sqrt((l * h))) * 1.0
	t_1 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_1 <= -5e-162:
		tmp = -d * (math.sqrt((h / l)) / h)
	elif t_1 <= 0.0:
		tmp = t_0
	elif t_1 <= 2e+261:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	else:
		tmp = t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * 1.0)
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_1 <= -5e-162)
		tmp = Float64(Float64(-d) * Float64(sqrt(Float64(h / l)) / h));
	elseif (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 2e+261)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (abs(d) / sqrt((l * h))) * 1.0;
	t_1 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_1 <= -5e-162)
		tmp = -d * (sqrt((h / l)) / h);
	elseif (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 2e+261)
		tmp = sqrt((d / l)) * sqrt((d / h));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-162], N[((-d) * N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], t$95$0, If[LessEqual[t$95$1, 2e+261], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5.0000000000000001e-162

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. 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. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. mult-flip N/A

          \[\leadsto \sqrt{\color{blue}{\left(d \cdot \frac{1}{h}\right)} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. associate-*l* N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \left(\frac{1}{h} \cdot \frac{d}{\ell}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \left(\frac{1}{h} \cdot \frac{d}{\ell}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{d \cdot \left(\frac{1}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. frac-2neg N/A

          \[\leadsto \sqrt{d \cdot \left(\frac{1}{h} \cdot \color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. frac-2neg N/A

          \[\leadsto \sqrt{d \cdot \left(\frac{1}{h} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. frac-times N/A

          \[\leadsto \sqrt{d \cdot \color{blue}{\frac{1 \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)\right)}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. remove-double-neg N/A

          \[\leadsto \sqrt{d \cdot \frac{1 \cdot \color{blue}{d}}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. *-lft-identity N/A

          \[\leadsto \sqrt{d \cdot \frac{\color{blue}{d}}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. remove-double-neg N/A

          \[\leadsto \sqrt{d \cdot \frac{d}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(h\right)\right)\right)\right)} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. remove-double-neg N/A

          \[\leadsto \sqrt{d \cdot \frac{d}{\color{blue}{h} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. remove-double-neg N/A

          \[\leadsto \sqrt{d \cdot \frac{d}{h \cdot \color{blue}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      22. lower-/.f64 N/A

          \[\leadsto \sqrt{d \cdot \color{blue}{\frac{d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      23. lower-*.f64 50.7%

          \[\leadsto \sqrt{d \cdot \frac{d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 50.7%

      \[\leadsto \color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. lower-*.f64 26.3%

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   6. Applied rewrites 26.3%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. Step-by-step derivation

      1. pow1/2 26.3%

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. lift-*.f64 26.3%

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. frac-times 26.3%

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      8. metadata-eval N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      9. pow-prod-down N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      11. lift-/.f64 26.3%

          \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   8. Applied rewrites 26.3%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   9. Taylor expanded in h around 0

      \[\leadsto \left(-d\right) \cdot \frac{\sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(-d\right) \cdot \frac{\sqrt{\frac{h}{\ell}}}{h} \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \left(-d\right) \cdot \frac{\sqrt{\frac{h}{\ell}}}{h} \]

       3. lower-/.f64 13.6%

          \[\leadsto \left(-d\right) \cdot \frac{\sqrt{\frac{h}{\ell}}}{h} \]

   11. Applied rewrites 13.6%

       \[\leadsto \left(-d\right) \cdot \frac{\sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

   if -5.0000000000000001e-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 or 1.9999999999999999e261 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

   3. Applied rewrites 67.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]

   5. Applied rewrites 75.4%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot \left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot h\right)}{\left(d + d\right) \cdot \ell}\right)} \]

   6. Taylor expanded in d around inf

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 42.4%

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \color{blue}{1} \]

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1.9999999999999999e261

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.4%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.4%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. mult-flip N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{\frac{1}{h}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \frac{1}{h} \]

      4. pow1/2 N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot {\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right) \cdot \frac{1}{h} \]

      5. metadata-eval N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{h} \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{h} \]

      7. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{h} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{\color{blue}{1}}{h} \]

      9. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{d \cdot h}\right) \cdot \frac{\color{blue}{1}}{h} \]

      10. associate-*l* N/A

          \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{d \cdot h} \cdot \frac{1}{h}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{d \cdot h} \cdot \frac{1}{h}\right)} \]

      12. lift-pow.f64 N/A

          \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\color{blue}{\sqrt{d \cdot h}} \cdot \frac{1}{h}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{d \cdot h} \cdot \frac{1}{h}\right) \]

      14. metadata-eval N/A

          \[\leadsto {\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left(\sqrt{d \cdot h} \cdot \frac{1}{h}\right) \]

      15. pow1/2 N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\sqrt{d \cdot h}} \cdot \frac{1}{h}\right) \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\sqrt{d \cdot h}} \cdot \frac{1}{h}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{d \cdot h} \cdot \color{blue}{\frac{1}{h}}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{d \cdot h} \cdot \frac{1}{h}\right) \]

      19. *-commutative N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{h \cdot d} \cdot \frac{1}{h}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{h \cdot d} \cdot \frac{1}{h}\right) \]

      21. lower-/.f64 23.4%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{h \cdot d} \cdot \frac{1}{\color{blue}{h}}\right) \]

   6. Applied rewrites 23.4%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\sqrt{h \cdot d} \cdot \frac{1}{h}\right)} \]

   7. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]
   8. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]

      2. lower-/.f64 39.3%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]

   9. Applied rewrites 39.3%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 49.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-162}:\\ \;\;\;\;\left(-d\right) \cdot \frac{\sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
       (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))
      -5e-162)
   (* (- d) (/ (sqrt (/ h l)) h))
   (* (/ (fabs d) (sqrt (* l h))) 1.0)))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e-162) {
		tmp = -d * (sqrt((h / l)) / h);
	} else {
		tmp = (fabs(d) / sqrt((l * h))) * 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))) <= (-5d-162)) then
        tmp = -d * (sqrt((h / l)) / h)
    else
        tmp = (abs(d) / sqrt((l * h))) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e-162) {
		tmp = -d * (Math.sqrt((h / l)) / h);
	} else {
		tmp = (Math.abs(d) / Math.sqrt((l * h))) * 1.0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if ((math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e-162:
		tmp = -d * (math.sqrt((h / l)) / h)
	else:
		tmp = (math.fabs(d) / math.sqrt((l * h))) * 1.0
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= -5e-162)
		tmp = Float64(Float64(-d) * Float64(sqrt(Float64(h / l)) / h));
	else
		tmp = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (((((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)))) <= -5e-162)
		tmp = -d * (sqrt((h / l)) / h);
	else
		tmp = (abs(d) / sqrt((l * h))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-162], N[((-d) * N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5.0000000000000001e-162

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. 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. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. mult-flip N/A

          \[\leadsto \sqrt{\color{blue}{\left(d \cdot \frac{1}{h}\right)} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. associate-*l* N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \left(\frac{1}{h} \cdot \frac{d}{\ell}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \left(\frac{1}{h} \cdot \frac{d}{\ell}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{d \cdot \left(\frac{1}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. frac-2neg N/A

          \[\leadsto \sqrt{d \cdot \left(\frac{1}{h} \cdot \color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. frac-2neg N/A

          \[\leadsto \sqrt{d \cdot \left(\frac{1}{h} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. frac-times N/A

          \[\leadsto \sqrt{d \cdot \color{blue}{\frac{1 \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)\right)}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. remove-double-neg N/A

          \[\leadsto \sqrt{d \cdot \frac{1 \cdot \color{blue}{d}}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. *-lft-identity N/A

          \[\leadsto \sqrt{d \cdot \frac{\color{blue}{d}}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. remove-double-neg N/A

          \[\leadsto \sqrt{d \cdot \frac{d}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(h\right)\right)\right)\right)} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. remove-double-neg N/A

          \[\leadsto \sqrt{d \cdot \frac{d}{\color{blue}{h} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. remove-double-neg N/A

          \[\leadsto \sqrt{d \cdot \frac{d}{h \cdot \color{blue}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      22. lower-/.f64 N/A

          \[\leadsto \sqrt{d \cdot \color{blue}{\frac{d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      23. lower-*.f64 50.7%

          \[\leadsto \sqrt{d \cdot \frac{d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 50.7%

      \[\leadsto \color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. lower-*.f64 26.3%

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   6. Applied rewrites 26.3%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. Step-by-step derivation

      1. pow1/2 26.3%

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. lift-*.f64 26.3%

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. frac-times 26.3%

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      8. metadata-eval N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      9. pow-prod-down N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      11. lift-/.f64 26.3%

          \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   8. Applied rewrites 26.3%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   9. Taylor expanded in h around 0

      \[\leadsto \left(-d\right) \cdot \frac{\sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(-d\right) \cdot \frac{\sqrt{\frac{h}{\ell}}}{h} \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \left(-d\right) \cdot \frac{\sqrt{\frac{h}{\ell}}}{h} \]

       3. lower-/.f64 13.6%

          \[\leadsto \left(-d\right) \cdot \frac{\sqrt{\frac{h}{\ell}}}{h} \]

   11. Applied rewrites 13.6%

       \[\leadsto \left(-d\right) \cdot \frac{\sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

   if -5.0000000000000001e-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. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

   3. Applied rewrites 67.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]

   5. Applied rewrites 75.4%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot \left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot h\right)}{\left(d + d\right) \cdot \ell}\right)} \]

   6. Taylor expanded in d around inf

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 42.4%

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 45.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-162}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
       (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))
      -5e-162)
   (* (- d) (sqrt (/ 1.0 (* h l))))
   (* (/ (fabs d) (sqrt (* l h))) 1.0)))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e-162) {
		tmp = -d * sqrt((1.0 / (h * l)));
	} else {
		tmp = (fabs(d) / sqrt((l * h))) * 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))) <= (-5d-162)) then
        tmp = -d * sqrt((1.0d0 / (h * l)))
    else
        tmp = (abs(d) / sqrt((l * h))) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e-162) {
		tmp = -d * Math.sqrt((1.0 / (h * l)));
	} else {
		tmp = (Math.abs(d) / Math.sqrt((l * h))) * 1.0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if ((math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e-162:
		tmp = -d * math.sqrt((1.0 / (h * l)))
	else:
		tmp = (math.fabs(d) / math.sqrt((l * h))) * 1.0
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= -5e-162)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l))));
	else
		tmp = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (((((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)))) <= -5e-162)
		tmp = -d * sqrt((1.0 / (h * l)));
	else
		tmp = (abs(d) / sqrt((l * h))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-162], N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5.0000000000000001e-162

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. 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. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. mult-flip N/A

          \[\leadsto \sqrt{\color{blue}{\left(d \cdot \frac{1}{h}\right)} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. associate-*l* N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \left(\frac{1}{h} \cdot \frac{d}{\ell}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \left(\frac{1}{h} \cdot \frac{d}{\ell}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{d \cdot \left(\frac{1}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. frac-2neg N/A

          \[\leadsto \sqrt{d \cdot \left(\frac{1}{h} \cdot \color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. frac-2neg N/A

          \[\leadsto \sqrt{d \cdot \left(\frac{1}{h} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. frac-times N/A

          \[\leadsto \sqrt{d \cdot \color{blue}{\frac{1 \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)\right)}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. remove-double-neg N/A

          \[\leadsto \sqrt{d \cdot \frac{1 \cdot \color{blue}{d}}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. *-lft-identity N/A

          \[\leadsto \sqrt{d \cdot \frac{\color{blue}{d}}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. remove-double-neg N/A

          \[\leadsto \sqrt{d \cdot \frac{d}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(h\right)\right)\right)\right)} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. remove-double-neg N/A

          \[\leadsto \sqrt{d \cdot \frac{d}{\color{blue}{h} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. remove-double-neg N/A

          \[\leadsto \sqrt{d \cdot \frac{d}{h \cdot \color{blue}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      22. lower-/.f64 N/A

          \[\leadsto \sqrt{d \cdot \color{blue}{\frac{d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      23. lower-*.f64 50.7%

          \[\leadsto \sqrt{d \cdot \frac{d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 50.7%

      \[\leadsto \color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. lower-*.f64 26.3%

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   6. Applied rewrites 26.3%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. Step-by-step derivation

      1. pow1/2 26.3%

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. lift-*.f64 26.3%

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. frac-times 26.3%

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      8. metadata-eval N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      9. pow-prod-down N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      11. lift-/.f64 26.3%

          \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   8. Applied rewrites 26.3%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   if -5.0000000000000001e-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. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

   3. Applied rewrites 67.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]

   5. Applied rewrites 75.4%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot \left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot h\right)}{\left(d + d\right) \cdot \ell}\right)} \]

   6. Taylor expanded in d around inf

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 42.4%

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 40.8% accurate, 5.6× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{if}\;\ell \leq 3.2 \cdot 10^{-134}:\\ \;\;\;\;\left(-d\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot t\_0\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (* h l)))))
   (if (<= l 3.2e-134) (* (- d) t_0) (* d t_0))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((1.0 / (h * l)));
	double tmp;
	if (l <= 3.2e-134) {
		tmp = -d * t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 / (h * l)))
    if (l <= 3.2d-134) then
        tmp = -d * t_0
    else
        tmp = d * t_0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((1.0 / (h * l)));
	double tmp;
	if (l <= 3.2e-134) {
		tmp = -d * t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((1.0 / (h * l)))
	tmp = 0
	if l <= 3.2e-134:
		tmp = -d * t_0
	else:
		tmp = d * t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(1.0 / Float64(h * l)))
	tmp = 0.0
	if (l <= 3.2e-134)
		tmp = Float64(Float64(-d) * t_0);
	else
		tmp = Float64(d * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((1.0 / (h * l)));
	tmp = 0.0;
	if (l <= 3.2e-134)
		tmp = -d * t_0;
	else
		tmp = d * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, 3.2e-134], N[((-d) * t$95$0), $MachinePrecision], N[(d * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 3.2000000000000001e-134

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. 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. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. mult-flip N/A

          \[\leadsto \sqrt{\color{blue}{\left(d \cdot \frac{1}{h}\right)} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. associate-*l* N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \left(\frac{1}{h} \cdot \frac{d}{\ell}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \left(\frac{1}{h} \cdot \frac{d}{\ell}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{d \cdot \left(\frac{1}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. frac-2neg N/A

          \[\leadsto \sqrt{d \cdot \left(\frac{1}{h} \cdot \color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. frac-2neg N/A

          \[\leadsto \sqrt{d \cdot \left(\frac{1}{h} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. frac-times N/A

          \[\leadsto \sqrt{d \cdot \color{blue}{\frac{1 \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)\right)}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. remove-double-neg N/A

          \[\leadsto \sqrt{d \cdot \frac{1 \cdot \color{blue}{d}}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. *-lft-identity N/A

          \[\leadsto \sqrt{d \cdot \frac{\color{blue}{d}}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. remove-double-neg N/A

          \[\leadsto \sqrt{d \cdot \frac{d}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(h\right)\right)\right)\right)} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. remove-double-neg N/A

          \[\leadsto \sqrt{d \cdot \frac{d}{\color{blue}{h} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. remove-double-neg N/A

          \[\leadsto \sqrt{d \cdot \frac{d}{h \cdot \color{blue}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      22. lower-/.f64 N/A

          \[\leadsto \sqrt{d \cdot \color{blue}{\frac{d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      23. lower-*.f64 50.7%

          \[\leadsto \sqrt{d \cdot \frac{d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 50.7%

      \[\leadsto \color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. lower-*.f64 26.3%

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   6. Applied rewrites 26.3%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. Step-by-step derivation

      1. pow1/2 26.3%

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. lift-*.f64 26.3%

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. frac-times 26.3%

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      8. metadata-eval N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      9. pow-prod-down N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      11. lift-/.f64 26.3%

          \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   8. Applied rewrites 26.3%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   if 3.2000000000000001e-134 < l

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      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. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. mult-flip N/A

          \[\leadsto \sqrt{\color{blue}{\left(d \cdot \frac{1}{h}\right)} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. associate-*l* N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \left(\frac{1}{h} \cdot \frac{d}{\ell}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \left(\frac{1}{h} \cdot \frac{d}{\ell}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{d \cdot \left(\frac{1}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. frac-2neg N/A

          \[\leadsto \sqrt{d \cdot \left(\frac{1}{h} \cdot \color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. frac-2neg N/A

          \[\leadsto \sqrt{d \cdot \left(\frac{1}{h} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. frac-times N/A

          \[\leadsto \sqrt{d \cdot \color{blue}{\frac{1 \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)\right)}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. remove-double-neg N/A

          \[\leadsto \sqrt{d \cdot \frac{1 \cdot \color{blue}{d}}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. *-lft-identity N/A

          \[\leadsto \sqrt{d \cdot \frac{\color{blue}{d}}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. remove-double-neg N/A

          \[\leadsto \sqrt{d \cdot \frac{d}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(h\right)\right)\right)\right)} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. remove-double-neg N/A

          \[\leadsto \sqrt{d \cdot \frac{d}{\color{blue}{h} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. remove-double-neg N/A

          \[\leadsto \sqrt{d \cdot \frac{d}{h \cdot \color{blue}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      22. lower-/.f64 N/A

          \[\leadsto \sqrt{d \cdot \color{blue}{\frac{d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      23. lower-*.f64 50.7%

          \[\leadsto \sqrt{d \cdot \frac{d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 50.7%

      \[\leadsto \color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. lower-*.f64 26.3%

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   6. Applied rewrites 26.3%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      2. sqrt-fabs-rev N/A

         \[\leadsto -1 \cdot \left(d \cdot \left|\sqrt{\frac{1}{h \cdot \ell}}\right|\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \left|\sqrt{\frac{1}{h \cdot \ell}}\right|\right) \]

      4. rem-sqrt-square-rev N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\sqrt{\frac{1}{h \cdot \ell}} \cdot \sqrt{\frac{1}{h \cdot \ell}}}\right) \]

      5. sqrt-prod N/A

         \[\leadsto -1 \cdot \left(d \cdot \left(\sqrt{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \color{blue}{\sqrt{\sqrt{\frac{1}{h \cdot \ell}}}}\right)\right) \]

      6. lower-unsound-*.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \left(\sqrt{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \color{blue}{\sqrt{\sqrt{\frac{1}{h \cdot \ell}}}}\right)\right) \]

      7. lower-unsound-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \left(\sqrt{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \sqrt{\color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}}\right)\right) \]

      8. lower-unsound-sqrt.f64 26.3%

         \[\leadsto -1 \cdot \left(d \cdot \left(\sqrt{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \sqrt{\sqrt{\frac{1}{h \cdot \ell}}}\right)\right) \]

   8. Applied rewrites 26.3%

      \[\leadsto -1 \cdot \left(d \cdot \left(\sqrt{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \color{blue}{\sqrt{\sqrt{\frac{1}{h \cdot \ell}}}}\right)\right) \]

   9. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

       2. lower-sqrt.f64 N/A

          \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       3. lower-/.f64 N/A

          \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       4. lower-*.f64 26.4%

          \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   11. Applied rewrites 26.4%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 26.4% accurate, 7.8× speedup

Math

\[d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

FPCore

(FPCore (d h l M D) :precision binary64 (* d (sqrt (/ 1.0 (* h l)))))

C

double code(double d, double h, double l, double M, double D) {
	return d * sqrt((1.0 / (h * l)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = d * sqrt((1.0d0 / (h * l)))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return d * Math.sqrt((1.0 / (h * l)));
}

Python

def code(d, h, l, M, D):
	return d * math.sqrt((1.0 / (h * l)))

Julia

function code(d, h, l, M, D)
	return Float64(d * sqrt(Float64(1.0 / Float64(h * l))))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = d * sqrt((1.0 / (h * l)));
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.3%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
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. lower-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   9. lift-/.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   10. mult-flip N/A

       \[\leadsto \sqrt{\color{blue}{\left(d \cdot \frac{1}{h}\right)} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   11. associate-*l* N/A

       \[\leadsto \sqrt{\color{blue}{d \cdot \left(\frac{1}{h} \cdot \frac{d}{\ell}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   12. lower-*.f64 N/A

       \[\leadsto \sqrt{\color{blue}{d \cdot \left(\frac{1}{h} \cdot \frac{d}{\ell}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   13. lift-/.f64 N/A

       \[\leadsto \sqrt{d \cdot \left(\frac{1}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   14. frac-2neg N/A

       \[\leadsto \sqrt{d \cdot \left(\frac{1}{h} \cdot \color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   15. frac-2neg N/A

       \[\leadsto \sqrt{d \cdot \left(\frac{1}{h} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   16. frac-times N/A

       \[\leadsto \sqrt{d \cdot \color{blue}{\frac{1 \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)\right)}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   17. remove-double-neg N/A

       \[\leadsto \sqrt{d \cdot \frac{1 \cdot \color{blue}{d}}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   18. *-lft-identity N/A

       \[\leadsto \sqrt{d \cdot \frac{\color{blue}{d}}{h \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   19. remove-double-neg N/A

       \[\leadsto \sqrt{d \cdot \frac{d}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(h\right)\right)\right)\right)} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   20. remove-double-neg N/A

       \[\leadsto \sqrt{d \cdot \frac{d}{\color{blue}{h} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   21. remove-double-neg N/A

       \[\leadsto \sqrt{d \cdot \frac{d}{h \cdot \color{blue}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   22. lower-/.f64 N/A

       \[\leadsto \sqrt{d \cdot \color{blue}{\frac{d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   23. lower-*.f64 50.7%

       \[\leadsto \sqrt{d \cdot \frac{d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Applied rewrites 50.7%

   \[\leadsto \color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

4. Taylor expanded in d around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
5. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto -1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \]

   3. lower-sqrt.f64 N/A

      \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   4. lower-/.f64 N/A

      \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   5. lower-*.f64 26.3%

      \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

6. Applied rewrites 26.3%

   \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
7. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   2. sqrt-fabs-rev N/A

      \[\leadsto -1 \cdot \left(d \cdot \left|\sqrt{\frac{1}{h \cdot \ell}}\right|\right) \]

   3. lift-sqrt.f64 N/A

      \[\leadsto -1 \cdot \left(d \cdot \left|\sqrt{\frac{1}{h \cdot \ell}}\right|\right) \]

   4. rem-sqrt-square-rev N/A

      \[\leadsto -1 \cdot \left(d \cdot \sqrt{\sqrt{\frac{1}{h \cdot \ell}} \cdot \sqrt{\frac{1}{h \cdot \ell}}}\right) \]

   5. sqrt-prod N/A

      \[\leadsto -1 \cdot \left(d \cdot \left(\sqrt{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \color{blue}{\sqrt{\sqrt{\frac{1}{h \cdot \ell}}}}\right)\right) \]

   6. lower-unsound-*.f64 N/A

      \[\leadsto -1 \cdot \left(d \cdot \left(\sqrt{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \color{blue}{\sqrt{\sqrt{\frac{1}{h \cdot \ell}}}}\right)\right) \]

   7. lower-unsound-sqrt.f64 N/A

      \[\leadsto -1 \cdot \left(d \cdot \left(\sqrt{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \sqrt{\color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}}\right)\right) \]

   8. lower-unsound-sqrt.f64 26.3%

      \[\leadsto -1 \cdot \left(d \cdot \left(\sqrt{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \sqrt{\sqrt{\frac{1}{h \cdot \ell}}}\right)\right) \]

8. Applied rewrites 26.3%

   \[\leadsto -1 \cdot \left(d \cdot \left(\sqrt{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \color{blue}{\sqrt{\sqrt{\frac{1}{h \cdot \ell}}}}\right)\right) \]

9. Taylor expanded in d around inf

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
10. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

    2. lower-sqrt.f64 N/A

       \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

    3. lower-/.f64 N/A

       \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

    4. lower-*.f64 26.4%

       \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

11. Applied rewrites 26.4%

    \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025197 
(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)))))