Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.1% → 79.3%

Time: 13.2s

Alternatives: 22

Speedup: 3.4×

Specification

Math

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

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: 66.1% accurate, 1.0× speedup

Math

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

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: 79.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{D}{d}}{2} \cdot M\\ t_1 := 1 - \frac{t\_0 \cdot \left(t\_0 \cdot \left(0.5 \cdot h\right)\right)}{\ell}\\ \mathbf{if}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot t\_1\\ \mathbf{elif}\;h \leq 1.5 \cdot 10^{+170}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d} \cdot \left(\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {t\_0}^{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right)}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ (/ D d) 2.0) M))
        (t_1 (- 1.0 (/ (* t_0 (* t_0 (* 0.5 h))) l))))
   (if (<= h -2e-310)
     (* (* (/ (sqrt (- d)) (sqrt (- h))) (pow (/ d l) (/ 1.0 2.0))) t_1)
     (if (<= h 1.5e+170)
       (* (* (sqrt (/ 1.0 (* l h))) d) t_1)
       (/
        (*
         (sqrt d)
         (* (fma (* (/ h l) -0.5) (pow t_0 2.0) 1.0) (sqrt (/ d l))))
        (sqrt h))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = ((D / d) / 2.0) * M;
	double t_1 = 1.0 - ((t_0 * (t_0 * (0.5 * h))) / l);
	double tmp;
	if (h <= -2e-310) {
		tmp = ((sqrt(-d) / sqrt(-h)) * pow((d / l), (1.0 / 2.0))) * t_1;
	} else if (h <= 1.5e+170) {
		tmp = (sqrt((1.0 / (l * h))) * d) * t_1;
	} else {
		tmp = (sqrt(d) * (fma(((h / l) * -0.5), pow(t_0, 2.0), 1.0) * sqrt((d / l)))) / sqrt(h);
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(Float64(D / d) / 2.0) * M)
	t_1 = Float64(1.0 - Float64(Float64(t_0 * Float64(t_0 * Float64(0.5 * h))) / l))
	tmp = 0.0
	if (h <= -2e-310)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * t_1);
	elseif (h <= 1.5e+170)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(l * h))) * d) * t_1);
	else
		tmp = Float64(Float64(sqrt(d) * Float64(fma(Float64(Float64(h / l) * -0.5), (t_0 ^ 2.0), 1.0) * sqrt(Float64(d / l)))) / sqrt(h));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[(D / d), $MachinePrecision] / 2.0), $MachinePrecision] * M), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[(t$95$0 * N[(t$95$0 * N[(0.5 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -2e-310], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[h, 1.5e+170], N[(N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] * N[(N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * N[Power[t$95$0, 2.0], $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.999999999999994e-310

   1. Initial program 70.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. 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

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 69.4

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 69.4

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

   4. Applied rewrites 69.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 72.2%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 72.2

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 80.9

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

   8. Applied rewrites 80.9%

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

   if -1.999999999999994e-310 < h < 1.49999999999999998e170

   1. Initial program 69.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. 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

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 72.6

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 72.6

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

   4. Applied rewrites 72.6%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 76.3%

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

   7. Taylor expanded in d around 0

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

   9. Applied rewrites 92.3%

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

   if 1.49999999999999998e170 < h

   1. Initial program 61.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. 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

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 61.4

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 61.4

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

   4. Applied rewrites 61.4%

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

   6. Applied rewrites 79.3%

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

Alternative 2: 76.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\ell \cdot h}}\\ t_1 := \frac{\frac{D}{d}}{2} \cdot M\\ t_2 := 1 - \frac{t\_1 \cdot \left(t\_1 \cdot \left(0.5 \cdot h\right)\right)}{\ell}\\ t_3 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq -1.52 \cdot 10^{+215}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.125 \cdot \frac{\left(\left(\frac{M}{d} \cdot D\right) \cdot D\right) \cdot M}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot t\_3\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(-d\right) \cdot t\_0\right) \cdot t\_2\\ \mathbf{elif}\;h \leq 1.5 \cdot 10^{+170}:\\ \;\;\;\;\left(t\_0 \cdot d\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d} \cdot \left(\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {t\_1}^{2}, 1\right) \cdot t\_3\right)}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (* l h))))
        (t_1 (* (/ (/ D d) 2.0) M))
        (t_2 (- 1.0 (/ (* t_1 (* t_1 (* 0.5 h))) l)))
        (t_3 (sqrt (/ d l))))
   (if (<= h -1.52e+215)
     (*
      (*
       (fma (* -0.125 (/ (* (* (* (/ M d) D) D) M) d)) (/ h l) 1.0)
       (sqrt (/ d h)))
      t_3)
     (if (<= h -2e-310)
       (* (* (- d) t_0) t_2)
       (if (<= h 1.5e+170)
         (* (* t_0 d) t_2)
         (/
          (* (sqrt d) (* (fma (* (/ h l) -0.5) (pow t_1 2.0) 1.0) t_3))
          (sqrt h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((1.0 / (l * h)));
	double t_1 = ((D / d) / 2.0) * M;
	double t_2 = 1.0 - ((t_1 * (t_1 * (0.5 * h))) / l);
	double t_3 = sqrt((d / l));
	double tmp;
	if (h <= -1.52e+215) {
		tmp = (fma((-0.125 * (((((M / d) * D) * D) * M) / d)), (h / l), 1.0) * sqrt((d / h))) * t_3;
	} else if (h <= -2e-310) {
		tmp = (-d * t_0) * t_2;
	} else if (h <= 1.5e+170) {
		tmp = (t_0 * d) * t_2;
	} else {
		tmp = (sqrt(d) * (fma(((h / l) * -0.5), pow(t_1, 2.0), 1.0) * t_3)) / sqrt(h);
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(1.0 / Float64(l * h)))
	t_1 = Float64(Float64(Float64(D / d) / 2.0) * M)
	t_2 = Float64(1.0 - Float64(Float64(t_1 * Float64(t_1 * Float64(0.5 * h))) / l))
	t_3 = sqrt(Float64(d / l))
	tmp = 0.0
	if (h <= -1.52e+215)
		tmp = Float64(Float64(fma(Float64(-0.125 * Float64(Float64(Float64(Float64(Float64(M / d) * D) * D) * M) / d)), Float64(h / l), 1.0) * sqrt(Float64(d / h))) * t_3);
	elseif (h <= -2e-310)
		tmp = Float64(Float64(Float64(-d) * t_0) * t_2);
	elseif (h <= 1.5e+170)
		tmp = Float64(Float64(t_0 * d) * t_2);
	else
		tmp = Float64(Float64(sqrt(d) * Float64(fma(Float64(Float64(h / l) * -0.5), (t_1 ^ 2.0), 1.0) * t_3)) / sqrt(h));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(D / d), $MachinePrecision] / 2.0), $MachinePrecision] * M), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[(N[(t$95$1 * N[(t$95$1 * N[(0.5 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[h, -1.52e+215], N[(N[(N[(N[(-0.125 * N[(N[(N[(N[(N[(M / d), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision] * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[h, -2e-310], N[(N[((-d) * t$95$0), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[h, 1.5e+170], N[(N[(t$95$0 * d), $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] * N[(N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * N[Power[t$95$1, 2.0], $MachinePrecision] + 1.0), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if h < -1.51999999999999999e215

   1. Initial program 60.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 60.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 60.6

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

   6. Applied rewrites 60.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. times-frac N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{8}} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{8} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)}{d}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      11. lower-/.f64 60.6

          \[\leadsto \left(\mathsf{fma}\left(-0.125 \cdot \color{blue}{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)}{d}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(M \cdot D\right)}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      14. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(D \cdot M\right)}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      15. associate-*r* N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{\color{blue}{\left(\left(\frac{M}{d} \cdot D\right) \cdot D\right) \cdot M}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      16. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{\color{blue}{\left(\left(\frac{M}{d} \cdot D\right) \cdot D\right) \cdot M}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      17. lower-*.f64 60.6

          \[\leadsto \left(\mathsf{fma}\left(-0.125 \cdot \frac{\color{blue}{\left(\left(\frac{M}{d} \cdot D\right) \cdot D\right)} \cdot M}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   8. Applied rewrites 60.6%

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{-0.125 \cdot \frac{\left(\left(\frac{M}{d} \cdot D\right) \cdot D\right) \cdot M}{d}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   if -1.51999999999999999e215 < h < -1.999999999999994e-310

   1. Initial program 71.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. 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

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 71.0

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 71.0

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

   4. Applied rewrites 71.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 74.3%

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

   7. Taylor expanded in h around -inf

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

   9. Applied rewrites 83.0%

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

   if -1.999999999999994e-310 < h < 1.49999999999999998e170

   1. Initial program 69.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. 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

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 72.6

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 72.6

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

   4. Applied rewrites 72.6%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 76.3%

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

   7. Taylor expanded in d around 0

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

   9. Applied rewrites 92.3%

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

   if 1.49999999999999998e170 < h

   1. Initial program 61.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. 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

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 61.4

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 61.4

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

   4. Applied rewrites 61.4%

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

   6. Applied rewrites 79.3%

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

Alternative 3: 76.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{M}{d} \cdot D\\ t_1 := \sqrt{\frac{1}{\ell \cdot h}}\\ t_2 := \frac{\frac{D}{d}}{2}\\ t_3 := t\_2 \cdot M\\ t_4 := 1 - \frac{t\_3 \cdot \left(t\_3 \cdot \left(0.5 \cdot h\right)\right)}{\ell}\\ t_5 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq -1.52 \cdot 10^{+215}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.125 \cdot \frac{\left(t\_0 \cdot D\right) \cdot M}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot t\_5\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(-d\right) \cdot t\_1\right) \cdot t\_4\\ \mathbf{elif}\;h \leq 1.25 \cdot 10^{+170}:\\ \;\;\;\;\left(t\_1 \cdot d\right) \cdot t\_4\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_5 \cdot \sqrt{d}}{\sqrt{h}} \cdot \left(1 - \frac{t\_0 \cdot \left(\left(0.5 \cdot \left(M \cdot t\_2\right)\right) \cdot \frac{h}{\ell}\right)}{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ M d) D))
        (t_1 (sqrt (/ 1.0 (* l h))))
        (t_2 (/ (/ D d) 2.0))
        (t_3 (* t_2 M))
        (t_4 (- 1.0 (/ (* t_3 (* t_3 (* 0.5 h))) l)))
        (t_5 (sqrt (/ d l))))
   (if (<= h -1.52e+215)
     (*
      (* (fma (* -0.125 (/ (* (* t_0 D) M) d)) (/ h l) 1.0) (sqrt (/ d h)))
      t_5)
     (if (<= h -2e-310)
       (* (* (- d) t_1) t_4)
       (if (<= h 1.25e+170)
         (* (* t_1 d) t_4)
         (*
          (/ (* t_5 (sqrt d)) (sqrt h))
          (- 1.0 (/ (* t_0 (* (* 0.5 (* M t_2)) (/ h l))) 2.0))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (M / d) * D;
	double t_1 = sqrt((1.0 / (l * h)));
	double t_2 = (D / d) / 2.0;
	double t_3 = t_2 * M;
	double t_4 = 1.0 - ((t_3 * (t_3 * (0.5 * h))) / l);
	double t_5 = sqrt((d / l));
	double tmp;
	if (h <= -1.52e+215) {
		tmp = (fma((-0.125 * (((t_0 * D) * M) / d)), (h / l), 1.0) * sqrt((d / h))) * t_5;
	} else if (h <= -2e-310) {
		tmp = (-d * t_1) * t_4;
	} else if (h <= 1.25e+170) {
		tmp = (t_1 * d) * t_4;
	} else {
		tmp = ((t_5 * sqrt(d)) / sqrt(h)) * (1.0 - ((t_0 * ((0.5 * (M * t_2)) * (h / l))) / 2.0));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(M / d) * D)
	t_1 = sqrt(Float64(1.0 / Float64(l * h)))
	t_2 = Float64(Float64(D / d) / 2.0)
	t_3 = Float64(t_2 * M)
	t_4 = Float64(1.0 - Float64(Float64(t_3 * Float64(t_3 * Float64(0.5 * h))) / l))
	t_5 = sqrt(Float64(d / l))
	tmp = 0.0
	if (h <= -1.52e+215)
		tmp = Float64(Float64(fma(Float64(-0.125 * Float64(Float64(Float64(t_0 * D) * M) / d)), Float64(h / l), 1.0) * sqrt(Float64(d / h))) * t_5);
	elseif (h <= -2e-310)
		tmp = Float64(Float64(Float64(-d) * t_1) * t_4);
	elseif (h <= 1.25e+170)
		tmp = Float64(Float64(t_1 * d) * t_4);
	else
		tmp = Float64(Float64(Float64(t_5 * sqrt(d)) / sqrt(h)) * Float64(1.0 - Float64(Float64(t_0 * Float64(Float64(0.5 * Float64(M * t_2)) * Float64(h / l))) / 2.0)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(M / d), $MachinePrecision] * D), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(D / d), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * M), $MachinePrecision]}, Block[{t$95$4 = N[(1.0 - N[(N[(t$95$3 * N[(t$95$3 * N[(0.5 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[h, -1.52e+215], N[(N[(N[(N[(-0.125 * N[(N[(N[(t$95$0 * D), $MachinePrecision] * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$5), $MachinePrecision], If[LessEqual[h, -2e-310], N[(N[((-d) * t$95$1), $MachinePrecision] * t$95$4), $MachinePrecision], If[LessEqual[h, 1.25e+170], N[(N[(t$95$1 * d), $MachinePrecision] * t$95$4), $MachinePrecision], N[(N[(N[(t$95$5 * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(t$95$0 * N[(N[(0.5 * N[(M * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if h < -1.51999999999999999e215

   1. Initial program 60.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 60.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 60.6

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

   6. Applied rewrites 60.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. times-frac N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{8}} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{8} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)}{d}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      11. lower-/.f64 60.6

          \[\leadsto \left(\mathsf{fma}\left(-0.125 \cdot \color{blue}{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)}{d}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(M \cdot D\right)}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      14. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(D \cdot M\right)}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      15. associate-*r* N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{\color{blue}{\left(\left(\frac{M}{d} \cdot D\right) \cdot D\right) \cdot M}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      16. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{\color{blue}{\left(\left(\frac{M}{d} \cdot D\right) \cdot D\right) \cdot M}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      17. lower-*.f64 60.6

          \[\leadsto \left(\mathsf{fma}\left(-0.125 \cdot \frac{\color{blue}{\left(\left(\frac{M}{d} \cdot D\right) \cdot D\right)} \cdot M}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   8. Applied rewrites 60.6%

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{-0.125 \cdot \frac{\left(\left(\frac{M}{d} \cdot D\right) \cdot D\right) \cdot M}{d}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   if -1.51999999999999999e215 < h < -1.999999999999994e-310

   1. Initial program 71.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. 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

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 71.0

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 71.0

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

   4. Applied rewrites 71.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 74.3%

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

   7. Taylor expanded in h around -inf

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

   9. Applied rewrites 83.0%

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

   if -1.999999999999994e-310 < h < 1.24999999999999994e170

   1. Initial program 69.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. 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

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 72.6

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 72.6

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

   4. Applied rewrites 72.6%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 76.3%

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

   7. Taylor expanded in d around 0

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

   9. Applied rewrites 92.3%

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

   if 1.24999999999999994e170 < h

   1. Initial program 61.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. 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

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 61.4

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 61.4

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

   4. Applied rewrites 61.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 64.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. *-commutative N/A

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

      11. associate-*l/ N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-/.f64 N/A

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

   8. Applied rewrites 58.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. pow1/2 N/A

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

      8. sqrt-div N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-pow.f64 N/A

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

      15. pow1/2 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lower-sqrt.f64 74.7

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

   10. Applied rewrites 74.7%

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

Alternative 4: 75.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\ell \cdot h}}\\ t_1 := \frac{\frac{D}{d}}{2} \cdot M\\ t_2 := 1 - \frac{t\_1 \cdot \left(t\_1 \cdot \left(0.5 \cdot h\right)\right)}{\ell}\\ \mathbf{if}\;h \leq -1.52 \cdot 10^{+215}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.125 \cdot \frac{\left(\left(\frac{M}{d} \cdot D\right) \cdot D\right) \cdot M}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(-d\right) \cdot t\_0\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot d\right) \cdot t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (* l h))))
        (t_1 (* (/ (/ D d) 2.0) M))
        (t_2 (- 1.0 (/ (* t_1 (* t_1 (* 0.5 h))) l))))
   (if (<= h -1.52e+215)
     (*
      (*
       (fma (* -0.125 (/ (* (* (* (/ M d) D) D) M) d)) (/ h l) 1.0)
       (sqrt (/ d h)))
      (sqrt (/ d l)))
     (if (<= h -2e-310) (* (* (- d) t_0) t_2) (* (* t_0 d) t_2)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((1.0 / (l * h)));
	double t_1 = ((D / d) / 2.0) * M;
	double t_2 = 1.0 - ((t_1 * (t_1 * (0.5 * h))) / l);
	double tmp;
	if (h <= -1.52e+215) {
		tmp = (fma((-0.125 * (((((M / d) * D) * D) * M) / d)), (h / l), 1.0) * sqrt((d / h))) * sqrt((d / l));
	} else if (h <= -2e-310) {
		tmp = (-d * t_0) * t_2;
	} else {
		tmp = (t_0 * d) * t_2;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(1.0 / Float64(l * h)))
	t_1 = Float64(Float64(Float64(D / d) / 2.0) * M)
	t_2 = Float64(1.0 - Float64(Float64(t_1 * Float64(t_1 * Float64(0.5 * h))) / l))
	tmp = 0.0
	if (h <= -1.52e+215)
		tmp = Float64(Float64(fma(Float64(-0.125 * Float64(Float64(Float64(Float64(Float64(M / d) * D) * D) * M) / d)), Float64(h / l), 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	elseif (h <= -2e-310)
		tmp = Float64(Float64(Float64(-d) * t_0) * t_2);
	else
		tmp = Float64(Float64(t_0 * d) * t_2);
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(D / d), $MachinePrecision] / 2.0), $MachinePrecision] * M), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[(N[(t$95$1 * N[(t$95$1 * N[(0.5 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -1.52e+215], N[(N[(N[(N[(-0.125 * N[(N[(N[(N[(N[(M / d), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision] * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -2e-310], N[(N[((-d) * t$95$0), $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[(t$95$0 * d), $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.51999999999999999e215

   1. Initial program 60.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 60.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 60.6

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

   6. Applied rewrites 60.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. times-frac N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{8}} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{8} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)}{d}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      11. lower-/.f64 60.6

          \[\leadsto \left(\mathsf{fma}\left(-0.125 \cdot \color{blue}{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)}{d}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(M \cdot D\right)}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      14. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(D \cdot M\right)}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      15. associate-*r* N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{\color{blue}{\left(\left(\frac{M}{d} \cdot D\right) \cdot D\right) \cdot M}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      16. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{\color{blue}{\left(\left(\frac{M}{d} \cdot D\right) \cdot D\right) \cdot M}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      17. lower-*.f64 60.6

          \[\leadsto \left(\mathsf{fma}\left(-0.125 \cdot \frac{\color{blue}{\left(\left(\frac{M}{d} \cdot D\right) \cdot D\right)} \cdot M}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   8. Applied rewrites 60.6%

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{-0.125 \cdot \frac{\left(\left(\frac{M}{d} \cdot D\right) \cdot D\right) \cdot M}{d}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   if -1.51999999999999999e215 < h < -1.999999999999994e-310

   1. Initial program 71.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. 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

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 71.0

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 71.0

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

   4. Applied rewrites 71.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 74.3%

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

   7. Taylor expanded in h around -inf

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

   9. Applied rewrites 83.0%

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

   if -1.999999999999994e-310 < h

   1. Initial program 67.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. 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

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 69.9

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 69.9

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

   4. Applied rewrites 69.9%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 73.5%

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

   7. Taylor expanded in d around 0

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

   9. Applied rewrites 83.5%

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

Alternative 5: 72.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{M}{d} \cdot D\\ t_1 := \sqrt{\frac{1}{\ell \cdot h}}\\ t_2 := \frac{\frac{D}{d}}{2}\\ t_3 := t\_2 \cdot M\\ \mathbf{if}\;h \leq -1.52 \cdot 10^{+215}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.125 \cdot \frac{\left(t\_0 \cdot D\right) \cdot M}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(-d\right) \cdot t\_1\right) \cdot \left(1 - \frac{t\_0 \cdot \left(\left(0.5 \cdot \left(M \cdot t\_2\right)\right) \cdot \frac{h}{\ell}\right)}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 \cdot d\right) \cdot \left(1 - \frac{t\_3 \cdot \left(t\_3 \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ M d) D))
        (t_1 (sqrt (/ 1.0 (* l h))))
        (t_2 (/ (/ D d) 2.0))
        (t_3 (* t_2 M)))
   (if (<= h -1.52e+215)
     (*
      (* (fma (* -0.125 (/ (* (* t_0 D) M) d)) (/ h l) 1.0) (sqrt (/ d h)))
      (sqrt (/ d l)))
     (if (<= h -2e-310)
       (* (* (- d) t_1) (- 1.0 (/ (* t_0 (* (* 0.5 (* M t_2)) (/ h l))) 2.0)))
       (* (* t_1 d) (- 1.0 (/ (* t_3 (* t_3 (* 0.5 h))) l)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (M / d) * D;
	double t_1 = sqrt((1.0 / (l * h)));
	double t_2 = (D / d) / 2.0;
	double t_3 = t_2 * M;
	double tmp;
	if (h <= -1.52e+215) {
		tmp = (fma((-0.125 * (((t_0 * D) * M) / d)), (h / l), 1.0) * sqrt((d / h))) * sqrt((d / l));
	} else if (h <= -2e-310) {
		tmp = (-d * t_1) * (1.0 - ((t_0 * ((0.5 * (M * t_2)) * (h / l))) / 2.0));
	} else {
		tmp = (t_1 * d) * (1.0 - ((t_3 * (t_3 * (0.5 * h))) / l));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(M / d) * D)
	t_1 = sqrt(Float64(1.0 / Float64(l * h)))
	t_2 = Float64(Float64(D / d) / 2.0)
	t_3 = Float64(t_2 * M)
	tmp = 0.0
	if (h <= -1.52e+215)
		tmp = Float64(Float64(fma(Float64(-0.125 * Float64(Float64(Float64(t_0 * D) * M) / d)), Float64(h / l), 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	elseif (h <= -2e-310)
		tmp = Float64(Float64(Float64(-d) * t_1) * Float64(1.0 - Float64(Float64(t_0 * Float64(Float64(0.5 * Float64(M * t_2)) * Float64(h / l))) / 2.0)));
	else
		tmp = Float64(Float64(t_1 * d) * Float64(1.0 - Float64(Float64(t_3 * Float64(t_3 * Float64(0.5 * h))) / l)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(M / d), $MachinePrecision] * D), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(D / d), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * M), $MachinePrecision]}, If[LessEqual[h, -1.52e+215], N[(N[(N[(N[(-0.125 * N[(N[(N[(t$95$0 * D), $MachinePrecision] * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -2e-310], N[(N[((-d) * t$95$1), $MachinePrecision] * N[(1.0 - N[(N[(t$95$0 * N[(N[(0.5 * N[(M * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * d), $MachinePrecision] * N[(1.0 - N[(N[(t$95$3 * N[(t$95$3 * N[(0.5 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.51999999999999999e215

   1. Initial program 60.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 60.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 60.6

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

   6. Applied rewrites 60.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. times-frac N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{8}} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{8} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)}{d}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      11. lower-/.f64 60.6

          \[\leadsto \left(\mathsf{fma}\left(-0.125 \cdot \color{blue}{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)}{d}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(M \cdot D\right)}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      14. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(D \cdot M\right)}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      15. associate-*r* N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{\color{blue}{\left(\left(\frac{M}{d} \cdot D\right) \cdot D\right) \cdot M}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      16. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{\color{blue}{\left(\left(\frac{M}{d} \cdot D\right) \cdot D\right) \cdot M}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      17. lower-*.f64 60.6

          \[\leadsto \left(\mathsf{fma}\left(-0.125 \cdot \frac{\color{blue}{\left(\left(\frac{M}{d} \cdot D\right) \cdot D\right)} \cdot M}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   8. Applied rewrites 60.6%

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{-0.125 \cdot \frac{\left(\left(\frac{M}{d} \cdot D\right) \cdot D\right) \cdot M}{d}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   if -1.51999999999999999e215 < h < -1.999999999999994e-310

   1. Initial program 71.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. 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

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 71.0

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 71.0

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

   4. Applied rewrites 71.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 74.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. *-commutative N/A

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

      11. associate-*l/ N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-/.f64 N/A

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

   8. Applied rewrites 74.1%

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

   9. Taylor expanded in h around -inf

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

   11. Applied rewrites 80.1%

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

   if -1.999999999999994e-310 < h

   1. Initial program 67.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. 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

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 69.9

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 69.9

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

   4. Applied rewrites 69.9%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 73.5%

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

   7. Taylor expanded in d around 0

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

   9. Applied rewrites 83.5%

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

Alternative 6: 70.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\ell \cdot h}}\\ t_1 := \frac{\frac{D}{d}}{2} \cdot M\\ \mathbf{if}\;d \leq -4.8 \cdot 10^{-118}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5 \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{M}{d} \cdot D}{4 \cdot d}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -4.6 \cdot 10^{-299}:\\ \;\;\;\;\left(\left(-d\right) \cdot t\_0\right) \cdot \left(1 - \left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot \frac{M}{d}}{d}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot d\right) \cdot \left(1 - \frac{t\_1 \cdot \left(t\_1 \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (* l h)))) (t_1 (* (/ (/ D d) 2.0) M)))
   (if (<= d -4.8e-118)
     (*
      (*
       (fma (* -0.5 (* (* M D) (/ (* (/ M d) D) (* 4.0 d)))) (/ h l) 1.0)
       (sqrt (/ d h)))
      (sqrt (/ d l)))
     (if (<= d -4.6e-299)
       (*
        (* (- d) t_0)
        (- 1.0 (* (* (* 0.125 (* D D)) (/ (* M (/ M d)) d)) (/ h l))))
       (* (* t_0 d) (- 1.0 (/ (* t_1 (* t_1 (* 0.5 h))) l)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((1.0 / (l * h)));
	double t_1 = ((D / d) / 2.0) * M;
	double tmp;
	if (d <= -4.8e-118) {
		tmp = (fma((-0.5 * ((M * D) * (((M / d) * D) / (4.0 * d)))), (h / l), 1.0) * sqrt((d / h))) * sqrt((d / l));
	} else if (d <= -4.6e-299) {
		tmp = (-d * t_0) * (1.0 - (((0.125 * (D * D)) * ((M * (M / d)) / d)) * (h / l)));
	} else {
		tmp = (t_0 * d) * (1.0 - ((t_1 * (t_1 * (0.5 * h))) / l));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(1.0 / Float64(l * h)))
	t_1 = Float64(Float64(Float64(D / d) / 2.0) * M)
	tmp = 0.0
	if (d <= -4.8e-118)
		tmp = Float64(Float64(fma(Float64(-0.5 * Float64(Float64(M * D) * Float64(Float64(Float64(M / d) * D) / Float64(4.0 * d)))), Float64(h / l), 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	elseif (d <= -4.6e-299)
		tmp = Float64(Float64(Float64(-d) * t_0) * Float64(1.0 - Float64(Float64(Float64(0.125 * Float64(D * D)) * Float64(Float64(M * Float64(M / d)) / d)) * Float64(h / l))));
	else
		tmp = Float64(Float64(t_0 * d) * Float64(1.0 - Float64(Float64(t_1 * Float64(t_1 * Float64(0.5 * h))) / l)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(D / d), $MachinePrecision] / 2.0), $MachinePrecision] * M), $MachinePrecision]}, If[LessEqual[d, -4.8e-118], N[(N[(N[(N[(-0.5 * N[(N[(M * D), $MachinePrecision] * N[(N[(N[(M / d), $MachinePrecision] * D), $MachinePrecision] / N[(4.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4.6e-299], N[(N[((-d) * t$95$0), $MachinePrecision] * N[(1.0 - N[(N[(N[(0.125 * N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(M * N[(M / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * d), $MachinePrecision] * N[(1.0 - N[(N[(t$95$1 * N[(t$95$1 * N[(0.5 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.8000000000000003e-118

   1. Initial program 81.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 80.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 78.0

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

   6. Applied rewrites 78.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 80.4

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 80.4

          \[\leadsto \left(\mathsf{fma}\left(-0.5 \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{M}{d} \cdot D}{\color{blue}{4} \cdot d}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   8. Applied rewrites 80.4%

      \[\leadsto \left(\mathsf{fma}\left(-0.5 \cdot \color{blue}{\left(\left(M \cdot D\right) \cdot \frac{\frac{M}{d} \cdot D}{4 \cdot d}\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   if -4.8000000000000003e-118 < d < -4.6000000000000001e-299

   1. Initial program 42.2%

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

   3. Taylor expanded in d around 0

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

   5. Applied rewrites 3.9%

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

   6. Taylor expanded in d around 0

      \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}\right)} \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 0.5%

      \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot \frac{M}{d}}{d}\right)} \cdot \frac{h}{\ell}\right) \]

   9. Taylor expanded in h around -inf

      \[\leadsto \left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \cdot \left(1 - \left(\left(\frac{1}{8} \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot \frac{M}{d}}{d}\right) \cdot \frac{h}{\ell}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 46.9%

       \[\leadsto \left(\left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{\ell \cdot h}}}\right) \cdot \left(1 - \left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot \frac{M}{d}}{d}\right) \cdot \frac{h}{\ell}\right) \]

   if -4.6000000000000001e-299 < d

   1. Initial program 67.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. 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

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 70.1

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 70.1

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

   4. Applied rewrites 70.1%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 73.6%

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

   7. Taylor expanded in d around 0

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

   9. Applied rewrites 83.6%

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

Alternative 7: 68.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\ell \cdot h}}\\ t_1 := \left(-d\right) \cdot t\_0\\ \mathbf{if}\;d \leq -2.1 \cdot 10^{+115}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -4.8 \cdot 10^{-118}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5 \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d \cdot \left(4 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -4.6 \cdot 10^{-299}:\\ \;\;\;\;t\_1 \cdot \left(1 - \left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot \frac{M}{d}}{d}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot d\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{\left(M \cdot h\right) \cdot D}{d} \cdot 0.25\right)}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (* l h)))) (t_1 (* (- d) t_0)))
   (if (<= d -2.1e+115)
     t_1
     (if (<= d -4.8e-118)
       (*
        (*
         (fma (* -0.5 (/ (* (* M D) (* M D)) (* d (* 4.0 d)))) (/ h l) 1.0)
         (sqrt (/ d h)))
        (sqrt (/ d l)))
       (if (<= d -4.6e-299)
         (* t_1 (- 1.0 (* (* (* 0.125 (* D D)) (/ (* M (/ M d)) d)) (/ h l))))
         (*
          (* t_0 d)
          (-
           1.0
           (/ (* (* (/ (/ D d) 2.0) M) (* (/ (* (* M h) D) d) 0.25)) l))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((1.0 / (l * h)));
	double t_1 = -d * t_0;
	double tmp;
	if (d <= -2.1e+115) {
		tmp = t_1;
	} else if (d <= -4.8e-118) {
		tmp = (fma((-0.5 * (((M * D) * (M * D)) / (d * (4.0 * d)))), (h / l), 1.0) * sqrt((d / h))) * sqrt((d / l));
	} else if (d <= -4.6e-299) {
		tmp = t_1 * (1.0 - (((0.125 * (D * D)) * ((M * (M / d)) / d)) * (h / l)));
	} else {
		tmp = (t_0 * d) * (1.0 - (((((D / d) / 2.0) * M) * ((((M * h) * D) / d) * 0.25)) / l));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(1.0 / Float64(l * h)))
	t_1 = Float64(Float64(-d) * t_0)
	tmp = 0.0
	if (d <= -2.1e+115)
		tmp = t_1;
	elseif (d <= -4.8e-118)
		tmp = Float64(Float64(fma(Float64(-0.5 * Float64(Float64(Float64(M * D) * Float64(M * D)) / Float64(d * Float64(4.0 * d)))), Float64(h / l), 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	elseif (d <= -4.6e-299)
		tmp = Float64(t_1 * Float64(1.0 - Float64(Float64(Float64(0.125 * Float64(D * D)) * Float64(Float64(M * Float64(M / d)) / d)) * Float64(h / l))));
	else
		tmp = Float64(Float64(t_0 * d) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(D / d) / 2.0) * M) * Float64(Float64(Float64(Float64(M * h) * D) / d) * 0.25)) / l)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[((-d) * t$95$0), $MachinePrecision]}, If[LessEqual[d, -2.1e+115], t$95$1, If[LessEqual[d, -4.8e-118], N[(N[(N[(N[(-0.5 * N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / N[(d * N[(4.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4.6e-299], N[(t$95$1 * N[(1.0 - N[(N[(N[(0.125 * N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(M * N[(M / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * d), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(D / d), $MachinePrecision] / 2.0), $MachinePrecision] * M), $MachinePrecision] * N[(N[(N[(N[(M * h), $MachinePrecision] * D), $MachinePrecision] / d), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.10000000000000003e115

   1. Initial program 80.7%

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

   3. Taylor expanded in l around -inf

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

   5. Applied rewrites 70.3%

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

   if -2.10000000000000003e115 < d < -4.8000000000000003e-118

   1. Initial program 82.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 80.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 80.1

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

   6. Applied rewrites 80.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. frac-times N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 82.2

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval 82.2

          \[\leadsto \left(\mathsf{fma}\left(-0.5 \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d \cdot \left(\color{blue}{4} \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   8. Applied rewrites 82.2%

      \[\leadsto \left(\mathsf{fma}\left(-0.5 \cdot \color{blue}{\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d \cdot \left(4 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   if -4.8000000000000003e-118 < d < -4.6000000000000001e-299

   1. Initial program 42.2%

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

   3. Taylor expanded in d around 0

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

   5. Applied rewrites 3.9%

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

   6. Taylor expanded in d around 0

      \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}\right)} \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 0.5%

      \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot \frac{M}{d}}{d}\right)} \cdot \frac{h}{\ell}\right) \]

   9. Taylor expanded in h around -inf

      \[\leadsto \left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \cdot \left(1 - \left(\left(\frac{1}{8} \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot \frac{M}{d}}{d}\right) \cdot \frac{h}{\ell}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 46.9%

       \[\leadsto \left(\left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{\ell \cdot h}}}\right) \cdot \left(1 - \left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot \frac{M}{d}}{d}\right) \cdot \frac{h}{\ell}\right) \]

   if -4.6000000000000001e-299 < d

   1. Initial program 67.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. 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

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 70.1

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 70.1

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

   4. Applied rewrites 70.1%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 73.6%

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

   7. Taylor expanded in d around 0

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

   9. Applied rewrites 83.6%

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

   10. Taylor expanded in d around 0

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

   12. Applied rewrites 81.6%

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

Alternative 8: 65.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{if}\;\ell \leq -4.6 \cdot 10^{+139}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq -1.05 \cdot 10^{-302}:\\ \;\;\;\;\left(\left(-d\right) \cdot t\_0\right) \cdot \left(1 - \left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot \frac{M}{d}}{d}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{+63}:\\ \;\;\;\;\left(t\_0 \cdot d\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{\left(M \cdot h\right) \cdot D}{d} \cdot 0.25\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (* l h)))))
   (if (<= l -4.6e+139)
     (* (sqrt (/ d h)) (sqrt (/ d l)))
     (if (<= l -1.05e-302)
       (*
        (* (- d) t_0)
        (- 1.0 (* (* (* 0.125 (* D D)) (/ (* M (/ M d)) d)) (/ h l))))
       (if (<= l 8e+63)
         (*
          (* t_0 d)
          (- 1.0 (/ (* (* (/ (/ D d) 2.0) M) (* (/ (* (* M h) D) d) 0.25)) l)))
         (* (/ 1.0 (* (sqrt l) (sqrt h))) d))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((1.0 / (l * h)));
	double tmp;
	if (l <= -4.6e+139) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (l <= -1.05e-302) {
		tmp = (-d * t_0) * (1.0 - (((0.125 * (D * D)) * ((M * (M / d)) / d)) * (h / l)));
	} else if (l <= 8e+63) {
		tmp = (t_0 * d) * (1.0 - (((((D / d) / 2.0) * M) * ((((M * h) * D) / d) * 0.25)) / l));
	} else {
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 / (l * h)))
    if (l <= (-4.6d+139)) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else if (l <= (-1.05d-302)) then
        tmp = (-d * t_0) * (1.0d0 - (((0.125d0 * (d_1 * d_1)) * ((m * (m / d)) / d)) * (h / l)))
    else if (l <= 8d+63) then
        tmp = (t_0 * d) * (1.0d0 - (((((d_1 / d) / 2.0d0) * m) * ((((m * h) * d_1) / d) * 0.25d0)) / l))
    else
        tmp = (1.0d0 / (sqrt(l) * sqrt(h))) * d
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((1.0 / (l * h)));
	double tmp;
	if (l <= -4.6e+139) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (l <= -1.05e-302) {
		tmp = (-d * t_0) * (1.0 - (((0.125 * (D * D)) * ((M * (M / d)) / d)) * (h / l)));
	} else if (l <= 8e+63) {
		tmp = (t_0 * d) * (1.0 - (((((D / d) / 2.0) * M) * ((((M * h) * D) / d) * 0.25)) / l));
	} else {
		tmp = (1.0 / (Math.sqrt(l) * Math.sqrt(h))) * d;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((1.0 / (l * h)))
	tmp = 0
	if l <= -4.6e+139:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	elif l <= -1.05e-302:
		tmp = (-d * t_0) * (1.0 - (((0.125 * (D * D)) * ((M * (M / d)) / d)) * (h / l)))
	elif l <= 8e+63:
		tmp = (t_0 * d) * (1.0 - (((((D / d) / 2.0) * M) * ((((M * h) * D) / d) * 0.25)) / l))
	else:
		tmp = (1.0 / (math.sqrt(l) * math.sqrt(h))) * d
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(1.0 / Float64(l * h)))
	tmp = 0.0
	if (l <= -4.6e+139)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (l <= -1.05e-302)
		tmp = Float64(Float64(Float64(-d) * t_0) * Float64(1.0 - Float64(Float64(Float64(0.125 * Float64(D * D)) * Float64(Float64(M * Float64(M / d)) / d)) * Float64(h / l))));
	elseif (l <= 8e+63)
		tmp = Float64(Float64(t_0 * d) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(D / d) / 2.0) * M) * Float64(Float64(Float64(Float64(M * h) * D) / d) * 0.25)) / l)));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(l) * sqrt(h))) * d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((1.0 / (l * h)));
	tmp = 0.0;
	if (l <= -4.6e+139)
		tmp = sqrt((d / h)) * sqrt((d / l));
	elseif (l <= -1.05e-302)
		tmp = (-d * t_0) * (1.0 - (((0.125 * (D * D)) * ((M * (M / d)) / d)) * (h / l)));
	elseif (l <= 8e+63)
		tmp = (t_0 * d) * (1.0 - (((((D / d) / 2.0) * M) * ((((M * h) * D) / d) * 0.25)) / l));
	else
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -4.6e+139], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1.05e-302], N[(N[((-d) * t$95$0), $MachinePrecision] * N[(1.0 - N[(N[(N[(0.125 * N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(M * N[(M / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 8e+63], N[(N[(t$95$0 * d), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(D / d), $MachinePrecision] / 2.0), $MachinePrecision] * M), $MachinePrecision] * N[(N[(N[(N[(M * h), $MachinePrecision] * D), $MachinePrecision] / d), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -4.6e139

   1. Initial program 56.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 56.4%

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

   5. Taylor expanded in d around inf

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

   7. Applied rewrites 55.0%

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

   if -4.6e139 < l < -1.05000000000000006e-302

   1. Initial program 75.3%

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

   3. Taylor expanded in d around 0

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

   5. Applied rewrites 0.5%

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

   6. Taylor expanded in d around 0

      \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}\right)} \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 0.3%

      \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot \frac{M}{d}}{d}\right)} \cdot \frac{h}{\ell}\right) \]

   9. Taylor expanded in h around -inf

      \[\leadsto \left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \cdot \left(1 - \left(\left(\frac{1}{8} \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot \frac{M}{d}}{d}\right) \cdot \frac{h}{\ell}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 69.0%

       \[\leadsto \left(\left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{\ell \cdot h}}}\right) \cdot \left(1 - \left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot \frac{M}{d}}{d}\right) \cdot \frac{h}{\ell}\right) \]

   if -1.05000000000000006e-302 < l < 8.00000000000000046e63

   1. Initial program 76.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. 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

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 76.9

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{\frac{1}{2}} \cdot h\right)}{\ell}\right) \]

      19. metadata-eval 76.9

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot h\right)}{\ell}\right) \]

   4. Applied rewrites 76.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      6. frac-times N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D \cdot M}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

   6. Applied rewrites 79.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}}{\ell}\right) \]

   7. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 91.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

   10. Taylor expanded in d around 0

       \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \color{blue}{\left(\frac{1}{4} \cdot \frac{D \cdot \left(M \cdot h\right)}{d}\right)}}{\ell}\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 88.1%

       \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \color{blue}{\left(\frac{\left(M \cdot h\right) \cdot D}{d} \cdot 0.25\right)}}{\ell}\right) \]

   if 8.00000000000000046e63 < l

   1. Initial program 51.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

   5. Applied rewrites 55.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 55.7%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

   9. Applied rewrites 71.4%

      \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 9: 70.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{if}\;d \leq -4.8 \cdot 10^{-118}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5 \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{M}{d} \cdot D}{4 \cdot d}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -4.6 \cdot 10^{-299}:\\ \;\;\;\;\left(\left(-d\right) \cdot t\_0\right) \cdot \left(1 - \left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot \frac{M}{d}}{d}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot d\right) \cdot \left(1 - \frac{\frac{\left(\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot h\right) \cdot 0.5\right) \cdot \left(D \cdot M\right)}{2 \cdot d}}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (* l h)))))
   (if (<= d -4.8e-118)
     (*
      (*
       (fma (* -0.5 (* (* M D) (/ (* (/ M d) D) (* 4.0 d)))) (/ h l) 1.0)
       (sqrt (/ d h)))
      (sqrt (/ d l)))
     (if (<= d -4.6e-299)
       (*
        (* (- d) t_0)
        (- 1.0 (* (* (* 0.125 (* D D)) (/ (* M (/ M d)) d)) (/ h l))))
       (*
        (* t_0 d)
        (-
         1.0
         (/
          (/ (* (* (* (* (/ (/ D d) 2.0) M) h) 0.5) (* D M)) (* 2.0 d))
          l)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((1.0 / (l * h)));
	double tmp;
	if (d <= -4.8e-118) {
		tmp = (fma((-0.5 * ((M * D) * (((M / d) * D) / (4.0 * d)))), (h / l), 1.0) * sqrt((d / h))) * sqrt((d / l));
	} else if (d <= -4.6e-299) {
		tmp = (-d * t_0) * (1.0 - (((0.125 * (D * D)) * ((M * (M / d)) / d)) * (h / l)));
	} else {
		tmp = (t_0 * d) * (1.0 - ((((((((D / d) / 2.0) * M) * h) * 0.5) * (D * M)) / (2.0 * d)) / l));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(1.0 / Float64(l * h)))
	tmp = 0.0
	if (d <= -4.8e-118)
		tmp = Float64(Float64(fma(Float64(-0.5 * Float64(Float64(M * D) * Float64(Float64(Float64(M / d) * D) / Float64(4.0 * d)))), Float64(h / l), 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	elseif (d <= -4.6e-299)
		tmp = Float64(Float64(Float64(-d) * t_0) * Float64(1.0 - Float64(Float64(Float64(0.125 * Float64(D * D)) * Float64(Float64(M * Float64(M / d)) / d)) * Float64(h / l))));
	else
		tmp = Float64(Float64(t_0 * d) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(D / d) / 2.0) * M) * h) * 0.5) * Float64(D * M)) / Float64(2.0 * d)) / l)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -4.8e-118], N[(N[(N[(N[(-0.5 * N[(N[(M * D), $MachinePrecision] * N[(N[(N[(M / d), $MachinePrecision] * D), $MachinePrecision] / N[(4.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4.6e-299], N[(N[((-d) * t$95$0), $MachinePrecision] * N[(1.0 - N[(N[(N[(0.125 * N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(M * N[(M / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * d), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(N[(N[(N[(D / d), $MachinePrecision] / 2.0), $MachinePrecision] * M), $MachinePrecision] * h), $MachinePrecision] * 0.5), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.8000000000000003e-118

   1. Initial program 81.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

   4. Applied rewrites 80.4%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.5 \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\left(\left(\frac{D}{2} \cdot \frac{M}{d}\right) \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)} \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right) \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      5. associate-*l/ N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\color{blue}{\frac{D \cdot \frac{M}{d}}{2}} \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      8. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      9. frac-times N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \color{blue}{\frac{D \cdot M}{2 \cdot d}}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      10. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \frac{\color{blue}{M \cdot D}}{2 \cdot d}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \frac{\color{blue}{M \cdot D}}{2 \cdot d}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \frac{M \cdot D}{\color{blue}{2 \cdot d}}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      13. frac-times N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      14. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(D \cdot \frac{M}{d}\right) \cdot \left(M \cdot D\right)}}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      16. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      18. lower-*.f64 78.0

          \[\leadsto \left(\mathsf{fma}\left(-0.5 \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)}{\color{blue}{2 \cdot \left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   6. Applied rewrites 78.0%

      \[\leadsto \left(\mathsf{fma}\left(-0.5 \cdot \color{blue}{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)}}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      3. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(M \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      4. associate-/l* N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\left(\left(M \cdot D\right) \cdot \frac{\frac{M}{d} \cdot D}{2 \cdot \left(2 \cdot d\right)}\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\left(\left(M \cdot D\right) \cdot \frac{\frac{M}{d} \cdot D}{2 \cdot \left(2 \cdot d\right)}\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      6. lower-/.f64 80.4

         \[\leadsto \left(\mathsf{fma}\left(-0.5 \cdot \left(\left(M \cdot D\right) \cdot \color{blue}{\frac{\frac{M}{d} \cdot D}{2 \cdot \left(2 \cdot d\right)}}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{M}{d} \cdot D}{\color{blue}{2 \cdot \left(2 \cdot d\right)}}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{M}{d} \cdot D}{2 \cdot \color{blue}{\left(2 \cdot d\right)}}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      9. associate-*r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{M}{d} \cdot D}{\color{blue}{\left(2 \cdot 2\right) \cdot d}}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{M}{d} \cdot D}{\color{blue}{\left(2 \cdot 2\right) \cdot d}}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      11. metadata-eval 80.4

          \[\leadsto \left(\mathsf{fma}\left(-0.5 \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{M}{d} \cdot D}{\color{blue}{4} \cdot d}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   8. Applied rewrites 80.4%

      \[\leadsto \left(\mathsf{fma}\left(-0.5 \cdot \color{blue}{\left(\left(M \cdot D\right) \cdot \frac{\frac{M}{d} \cdot D}{4 \cdot d}\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   if -4.8000000000000003e-118 < d < -4.6000000000000001e-299

   1. Initial program 42.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 3.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Taylor expanded in d around 0

      \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}\right)} \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 0.5%

      \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot \frac{M}{d}}{d}\right)} \cdot \frac{h}{\ell}\right) \]

   9. Taylor expanded in h around -inf

      \[\leadsto \left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \cdot \left(1 - \left(\left(\frac{1}{8} \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot \frac{M}{d}}{d}\right) \cdot \frac{h}{\ell}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 46.9%

       \[\leadsto \left(\left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{\ell \cdot h}}}\right) \cdot \left(1 - \left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot \frac{M}{d}}{d}\right) \cdot \frac{h}{\ell}\right) \]

   if -4.6000000000000001e-299 < d

   1. Initial program 67.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. 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

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot h}{\ell}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 70.1

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{\frac{1}{2}} \cdot h\right)}{\ell}\right) \]

      19. metadata-eval 70.1

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot h\right)}{\ell}\right) \]

   4. Applied rewrites 70.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      6. frac-times N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D \cdot M}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

   6. Applied rewrites 73.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}}{\ell}\right) \]

   7. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 83.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \frac{\color{blue}{\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right) \cdot \left(\frac{\frac{D}{d}}{2} \cdot M\right)}}{\ell}\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \frac{\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right) \cdot \color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right)}}{\ell}\right) \]

       4. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \frac{\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right) \cdot \color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}{\ell}\right) \]

       5. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \frac{\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right) \cdot \left(M \cdot \color{blue}{\frac{\frac{D}{d}}{2}}\right)}{\ell}\right) \]

       6. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \frac{\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right) \cdot \left(M \cdot \frac{\color{blue}{\frac{D}{d}}}{2}\right)}{\ell}\right) \]

       7. associate-/l/ N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \frac{\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right) \cdot \left(M \cdot \color{blue}{\frac{D}{d \cdot 2}}\right)}{\ell}\right) \]

       8. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \frac{\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right) \cdot \left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}{\ell}\right) \]

       9. associate-/l* N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \frac{\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right) \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}}{\ell}\right) \]

       10. associate-*r/ N/A

           \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \frac{\color{blue}{\frac{\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right) \cdot \left(M \cdot D\right)}{2 \cdot d}}}{\ell}\right) \]

       11. lower-/.f64 N/A

           \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \frac{\color{blue}{\frac{\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right) \cdot \left(M \cdot D\right)}{2 \cdot d}}}{\ell}\right) \]

   11. Applied rewrites 82.3%

       \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \frac{\color{blue}{\frac{\left(\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot h\right) \cdot 0.5\right) \cdot \left(D \cdot M\right)}{2 \cdot d}}}{\ell}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 69.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{if}\;d \leq -4.8 \cdot 10^{-118}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.125 \cdot \frac{\left(\left(\frac{M}{d} \cdot D\right) \cdot D\right) \cdot M}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -4.6 \cdot 10^{-299}:\\ \;\;\;\;\left(\left(-d\right) \cdot t\_0\right) \cdot \left(1 - \left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot \frac{M}{d}}{d}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot d\right) \cdot \left(1 - \frac{\frac{\left(\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot h\right) \cdot 0.5\right) \cdot \left(D \cdot M\right)}{2 \cdot d}}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (* l h)))))
   (if (<= d -4.8e-118)
     (*
      (*
       (fma (* -0.125 (/ (* (* (* (/ M d) D) D) M) d)) (/ h l) 1.0)
       (sqrt (/ d h)))
      (sqrt (/ d l)))
     (if (<= d -4.6e-299)
       (*
        (* (- d) t_0)
        (- 1.0 (* (* (* 0.125 (* D D)) (/ (* M (/ M d)) d)) (/ h l))))
       (*
        (* t_0 d)
        (-
         1.0
         (/
          (/ (* (* (* (* (/ (/ D d) 2.0) M) h) 0.5) (* D M)) (* 2.0 d))
          l)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((1.0 / (l * h)));
	double tmp;
	if (d <= -4.8e-118) {
		tmp = (fma((-0.125 * (((((M / d) * D) * D) * M) / d)), (h / l), 1.0) * sqrt((d / h))) * sqrt((d / l));
	} else if (d <= -4.6e-299) {
		tmp = (-d * t_0) * (1.0 - (((0.125 * (D * D)) * ((M * (M / d)) / d)) * (h / l)));
	} else {
		tmp = (t_0 * d) * (1.0 - ((((((((D / d) / 2.0) * M) * h) * 0.5) * (D * M)) / (2.0 * d)) / l));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(1.0 / Float64(l * h)))
	tmp = 0.0
	if (d <= -4.8e-118)
		tmp = Float64(Float64(fma(Float64(-0.125 * Float64(Float64(Float64(Float64(Float64(M / d) * D) * D) * M) / d)), Float64(h / l), 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	elseif (d <= -4.6e-299)
		tmp = Float64(Float64(Float64(-d) * t_0) * Float64(1.0 - Float64(Float64(Float64(0.125 * Float64(D * D)) * Float64(Float64(M * Float64(M / d)) / d)) * Float64(h / l))));
	else
		tmp = Float64(Float64(t_0 * d) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(D / d) / 2.0) * M) * h) * 0.5) * Float64(D * M)) / Float64(2.0 * d)) / l)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -4.8e-118], N[(N[(N[(N[(-0.125 * N[(N[(N[(N[(N[(M / d), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision] * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4.6e-299], N[(N[((-d) * t$95$0), $MachinePrecision] * N[(1.0 - N[(N[(N[(0.125 * N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(M * N[(M / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * d), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(N[(N[(N[(D / d), $MachinePrecision] / 2.0), $MachinePrecision] * M), $MachinePrecision] * h), $MachinePrecision] * 0.5), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.8000000000000003e-118

   1. Initial program 81.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

   4. Applied rewrites 80.4%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.5 \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\left(\left(\frac{D}{2} \cdot \frac{M}{d}\right) \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)} \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right) \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      5. associate-*l/ N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\color{blue}{\frac{D \cdot \frac{M}{d}}{2}} \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      8. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      9. frac-times N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \color{blue}{\frac{D \cdot M}{2 \cdot d}}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      10. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \frac{\color{blue}{M \cdot D}}{2 \cdot d}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \frac{\color{blue}{M \cdot D}}{2 \cdot d}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \frac{M \cdot D}{\color{blue}{2 \cdot d}}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      13. frac-times N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      14. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(D \cdot \frac{M}{d}\right) \cdot \left(M \cdot D\right)}}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      16. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      18. lower-*.f64 78.0

          \[\leadsto \left(\mathsf{fma}\left(-0.5 \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)}{\color{blue}{2 \cdot \left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   6. Applied rewrites 78.0%

      \[\leadsto \left(\mathsf{fma}\left(-0.5 \cdot \color{blue}{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      3. associate-*r/ N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{2} \cdot \left(\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)\right)}{2 \cdot \left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\frac{-1}{2} \cdot \left(\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)\right)}{\color{blue}{2 \cdot \left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\frac{-1}{2} \cdot \left(\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)\right)}{2 \cdot \color{blue}{\left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      6. associate-*r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\frac{-1}{2} \cdot \left(\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)\right)}{\color{blue}{\left(2 \cdot 2\right) \cdot d}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      7. times-frac N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{2}}{2 \cdot 2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)}{d}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      8. metadata-eval N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\frac{-1}{2}}{\color{blue}{4}} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      9. metadata-eval N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{8}} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{8} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)}{d}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      11. lower-/.f64 78.0

          \[\leadsto \left(\mathsf{fma}\left(-0.125 \cdot \color{blue}{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)}{d}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(M \cdot D\right)}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      14. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(D \cdot M\right)}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      15. associate-*r* N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{\color{blue}{\left(\left(\frac{M}{d} \cdot D\right) \cdot D\right) \cdot M}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      16. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{\color{blue}{\left(\left(\frac{M}{d} \cdot D\right) \cdot D\right) \cdot M}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      17. lower-*.f64 78.0

          \[\leadsto \left(\mathsf{fma}\left(-0.125 \cdot \frac{\color{blue}{\left(\left(\frac{M}{d} \cdot D\right) \cdot D\right)} \cdot M}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   8. Applied rewrites 78.0%

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{-0.125 \cdot \frac{\left(\left(\frac{M}{d} \cdot D\right) \cdot D\right) \cdot M}{d}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   if -4.8000000000000003e-118 < d < -4.6000000000000001e-299

   1. Initial program 42.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 3.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Taylor expanded in d around 0

      \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}\right)} \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 0.5%

      \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot \frac{M}{d}}{d}\right)} \cdot \frac{h}{\ell}\right) \]

   9. Taylor expanded in h around -inf

      \[\leadsto \left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \cdot \left(1 - \left(\left(\frac{1}{8} \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot \frac{M}{d}}{d}\right) \cdot \frac{h}{\ell}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 46.9%

       \[\leadsto \left(\left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{\ell \cdot h}}}\right) \cdot \left(1 - \left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot \frac{M}{d}}{d}\right) \cdot \frac{h}{\ell}\right) \]

   if -4.6000000000000001e-299 < d

   1. Initial program 67.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. 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

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot h}{\ell}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 70.1

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{\frac{1}{2}} \cdot h\right)}{\ell}\right) \]

      19. metadata-eval 70.1

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot h\right)}{\ell}\right) \]

   4. Applied rewrites 70.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      6. frac-times N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D \cdot M}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

   6. Applied rewrites 73.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}}{\ell}\right) \]

   7. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 83.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \frac{\color{blue}{\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right) \cdot \left(\frac{\frac{D}{d}}{2} \cdot M\right)}}{\ell}\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \frac{\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right) \cdot \color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right)}}{\ell}\right) \]

       4. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \frac{\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right) \cdot \color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}{\ell}\right) \]

       5. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \frac{\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right) \cdot \left(M \cdot \color{blue}{\frac{\frac{D}{d}}{2}}\right)}{\ell}\right) \]

       6. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \frac{\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right) \cdot \left(M \cdot \frac{\color{blue}{\frac{D}{d}}}{2}\right)}{\ell}\right) \]

       7. associate-/l/ N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \frac{\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right) \cdot \left(M \cdot \color{blue}{\frac{D}{d \cdot 2}}\right)}{\ell}\right) \]

       8. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \frac{\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right) \cdot \left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}{\ell}\right) \]

       9. associate-/l* N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \frac{\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right) \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}}{\ell}\right) \]

       10. associate-*r/ N/A

           \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \frac{\color{blue}{\frac{\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right) \cdot \left(M \cdot D\right)}{2 \cdot d}}}{\ell}\right) \]

       11. lower-/.f64 N/A

           \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \frac{\color{blue}{\frac{\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right) \cdot \left(M \cdot D\right)}{2 \cdot d}}}{\ell}\right) \]

   11. Applied rewrites 82.3%

       \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \frac{\color{blue}{\frac{\left(\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot h\right) \cdot 0.5\right) \cdot \left(D \cdot M\right)}{2 \cdot d}}}{\ell}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 52.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\ell \cdot h}}\\ t_1 := t\_0 \cdot d\\ \mathbf{if}\;d \leq -4.7 \cdot 10^{+38}:\\ \;\;\;\;\left(-d\right) \cdot t\_0\\ \mathbf{elif}\;d \leq -4.6 \cdot 10^{-299}:\\ \;\;\;\;\left(\left(-0.125 \cdot \sqrt{\frac{h}{\left(d \cdot d\right) \cdot d}}\right) \cdot \frac{\left(D \cdot D\right) \cdot \left(M \cdot M\right)}{-\ell}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 1.6 \cdot 10^{-149}:\\ \;\;\;\;t\_1 \cdot \left(1 - \left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot \frac{M}{d}}{d}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;d \leq 8.2 \cdot 10^{+197}:\\ \;\;\;\;t\_1 \cdot \left(1 - \frac{\left(\frac{\left(M \cdot M\right) \cdot h}{d} \cdot \left(D \cdot \frac{D}{d}\right)\right) \cdot 0.125}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (* l h)))) (t_1 (* t_0 d)))
   (if (<= d -4.7e+38)
     (* (- d) t_0)
     (if (<= d -4.6e-299)
       (*
        (* (* -0.125 (sqrt (/ h (* (* d d) d)))) (/ (* (* D D) (* M M)) (- l)))
        (sqrt (/ d l)))
       (if (<= d 1.6e-149)
         (* t_1 (- 1.0 (* (* (* 0.125 (* D D)) (/ (* M (/ M d)) d)) (/ h l))))
         (if (<= d 8.2e+197)
           (*
            t_1
            (- 1.0 (/ (* (* (/ (* (* M M) h) d) (* D (/ D d))) 0.125) l)))
           (/ d (* (sqrt l) (sqrt h)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((1.0 / (l * h)));
	double t_1 = t_0 * d;
	double tmp;
	if (d <= -4.7e+38) {
		tmp = -d * t_0;
	} else if (d <= -4.6e-299) {
		tmp = ((-0.125 * sqrt((h / ((d * d) * d)))) * (((D * D) * (M * M)) / -l)) * sqrt((d / l));
	} else if (d <= 1.6e-149) {
		tmp = t_1 * (1.0 - (((0.125 * (D * D)) * ((M * (M / d)) / d)) * (h / l)));
	} else if (d <= 8.2e+197) {
		tmp = t_1 * (1.0 - ((((((M * M) * h) / d) * (D * (D / d))) * 0.125) / l));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((1.0d0 / (l * h)))
    t_1 = t_0 * d
    if (d <= (-4.7d+38)) then
        tmp = -d * t_0
    else if (d <= (-4.6d-299)) then
        tmp = (((-0.125d0) * sqrt((h / ((d * d) * d)))) * (((d_1 * d_1) * (m * m)) / -l)) * sqrt((d / l))
    else if (d <= 1.6d-149) then
        tmp = t_1 * (1.0d0 - (((0.125d0 * (d_1 * d_1)) * ((m * (m / d)) / d)) * (h / l)))
    else if (d <= 8.2d+197) then
        tmp = t_1 * (1.0d0 - ((((((m * m) * h) / d) * (d_1 * (d_1 / d))) * 0.125d0) / l))
    else
        tmp = d / (sqrt(l) * sqrt(h))
    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 / (l * h)));
	double t_1 = t_0 * d;
	double tmp;
	if (d <= -4.7e+38) {
		tmp = -d * t_0;
	} else if (d <= -4.6e-299) {
		tmp = ((-0.125 * Math.sqrt((h / ((d * d) * d)))) * (((D * D) * (M * M)) / -l)) * Math.sqrt((d / l));
	} else if (d <= 1.6e-149) {
		tmp = t_1 * (1.0 - (((0.125 * (D * D)) * ((M * (M / d)) / d)) * (h / l)));
	} else if (d <= 8.2e+197) {
		tmp = t_1 * (1.0 - ((((((M * M) * h) / d) * (D * (D / d))) * 0.125) / l));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((1.0 / (l * h)))
	t_1 = t_0 * d
	tmp = 0
	if d <= -4.7e+38:
		tmp = -d * t_0
	elif d <= -4.6e-299:
		tmp = ((-0.125 * math.sqrt((h / ((d * d) * d)))) * (((D * D) * (M * M)) / -l)) * math.sqrt((d / l))
	elif d <= 1.6e-149:
		tmp = t_1 * (1.0 - (((0.125 * (D * D)) * ((M * (M / d)) / d)) * (h / l)))
	elif d <= 8.2e+197:
		tmp = t_1 * (1.0 - ((((((M * M) * h) / d) * (D * (D / d))) * 0.125) / l))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(1.0 / Float64(l * h)))
	t_1 = Float64(t_0 * d)
	tmp = 0.0
	if (d <= -4.7e+38)
		tmp = Float64(Float64(-d) * t_0);
	elseif (d <= -4.6e-299)
		tmp = Float64(Float64(Float64(-0.125 * sqrt(Float64(h / Float64(Float64(d * d) * d)))) * Float64(Float64(Float64(D * D) * Float64(M * M)) / Float64(-l))) * sqrt(Float64(d / l)));
	elseif (d <= 1.6e-149)
		tmp = Float64(t_1 * Float64(1.0 - Float64(Float64(Float64(0.125 * Float64(D * D)) * Float64(Float64(M * Float64(M / d)) / d)) * Float64(h / l))));
	elseif (d <= 8.2e+197)
		tmp = Float64(t_1 * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(M * M) * h) / d) * Float64(D * Float64(D / d))) * 0.125) / l)));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((1.0 / (l * h)));
	t_1 = t_0 * d;
	tmp = 0.0;
	if (d <= -4.7e+38)
		tmp = -d * t_0;
	elseif (d <= -4.6e-299)
		tmp = ((-0.125 * sqrt((h / ((d * d) * d)))) * (((D * D) * (M * M)) / -l)) * sqrt((d / l));
	elseif (d <= 1.6e-149)
		tmp = t_1 * (1.0 - (((0.125 * (D * D)) * ((M * (M / d)) / d)) * (h / l)));
	elseif (d <= 8.2e+197)
		tmp = t_1 * (1.0 - ((((((M * M) * h) / d) * (D * (D / d))) * 0.125) / l));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * d), $MachinePrecision]}, If[LessEqual[d, -4.7e+38], N[((-d) * t$95$0), $MachinePrecision], If[LessEqual[d, -4.6e-299], N[(N[(N[(-0.125 * N[Sqrt[N[(h / N[(N[(d * d), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(D * D), $MachinePrecision] * N[(M * M), $MachinePrecision]), $MachinePrecision] / (-l)), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.6e-149], N[(t$95$1 * N[(1.0 - N[(N[(N[(0.125 * N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(M * N[(M / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 8.2e+197], N[(t$95$1 * N[(1.0 - N[(N[(N[(N[(N[(N[(M * M), $MachinePrecision] * h), $MachinePrecision] / d), $MachinePrecision] * N[(D * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -4.6999999999999999e38

   1. Initial program 80.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

   5. Applied rewrites 64.0%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if -4.6999999999999999e38 < d < -4.6000000000000001e-299

   1. Initial program 60.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

   4. Applied rewrites 59.9%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.5 \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]

   5. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{\ell} \cdot \sqrt{\frac{h}{{d}^{3}}}\right)\right)} \cdot \sqrt{\frac{d}{\ell}} \]
   6. Step-by-step derivation

   7. Applied rewrites 42.0%

      \[\leadsto \color{blue}{\left(\left(-0.125 \cdot \sqrt{\frac{h}{{d}^{3}}}\right) \cdot \frac{\left(\left(-D\right) \cdot D\right) \cdot \left(M \cdot M\right)}{\ell}\right)} \cdot \sqrt{\frac{d}{\ell}} \]
   8. Step-by-step derivation

   9. Applied rewrites 42.0%

      \[\leadsto \left(\left(-0.125 \cdot \sqrt{\frac{h}{\left(d \cdot d\right) \cdot d}}\right) \cdot \frac{\left(\left(-D\right) \cdot D\right) \cdot \left(M \cdot M\right)}{\ell}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   if -4.6000000000000001e-299 < d < 1.60000000000000001e-149

   1. Initial program 48.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 79.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Taylor expanded in d around 0

      \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}\right)} \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 75.4%

      \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot \frac{M}{d}}{d}\right)} \cdot \frac{h}{\ell}\right) \]

   if 1.60000000000000001e-149 < d < 8.2000000000000006e197

   1. Initial program 78.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. 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

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot h}{\ell}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 81.3

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{\frac{1}{2}} \cdot h\right)}{\ell}\right) \]

      19. metadata-eval 81.3

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot h\right)}{\ell}\right) \]

   4. Applied rewrites 81.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      6. frac-times N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D \cdot M}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

   6. Applied rewrites 86.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}}{\ell}\right) \]

   7. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 86.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

   10. Taylor expanded in d around 0

       \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}}{\ell}\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 74.7%

       \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{\left(M \cdot M\right) \cdot h}{d} \cdot \left(D \cdot \frac{D}{d}\right)\right) \cdot 0.125}}{\ell}\right) \]

   if 8.2000000000000006e197 < d

   1. Initial program 65.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

   5. Applied rewrites 56.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 56.0%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

   9. Applied rewrites 56.1%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 77.5%

       \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.7 \cdot 10^{+38}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;d \leq -4.6 \cdot 10^{-299}:\\ \;\;\;\;\left(\left(-0.125 \cdot \sqrt{\frac{h}{\left(d \cdot d\right) \cdot d}}\right) \cdot \frac{\left(D \cdot D\right) \cdot \left(M \cdot M\right)}{-\ell}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 1.6 \cdot 10^{-149}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot \frac{M}{d}}{d}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;d \leq 8.2 \cdot 10^{+197}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \frac{\left(\frac{\left(M \cdot M\right) \cdot h}{d} \cdot \left(D \cdot \frac{D}{d}\right)\right) \cdot 0.125}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 68.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{if}\;d \leq -4.8 \cdot 10^{-118}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.125 \cdot \frac{\left(\left(\frac{M}{d} \cdot D\right) \cdot D\right) \cdot M}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -4.6 \cdot 10^{-299}:\\ \;\;\;\;\left(\left(-d\right) \cdot t\_0\right) \cdot \left(1 - \left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot \frac{M}{d}}{d}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot d\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{\left(M \cdot h\right) \cdot D}{d} \cdot 0.25\right)}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (* l h)))))
   (if (<= d -4.8e-118)
     (*
      (*
       (fma (* -0.125 (/ (* (* (* (/ M d) D) D) M) d)) (/ h l) 1.0)
       (sqrt (/ d h)))
      (sqrt (/ d l)))
     (if (<= d -4.6e-299)
       (*
        (* (- d) t_0)
        (- 1.0 (* (* (* 0.125 (* D D)) (/ (* M (/ M d)) d)) (/ h l))))
       (*
        (* t_0 d)
        (-
         1.0
         (/ (* (* (/ (/ D d) 2.0) M) (* (/ (* (* M h) D) d) 0.25)) l)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((1.0 / (l * h)));
	double tmp;
	if (d <= -4.8e-118) {
		tmp = (fma((-0.125 * (((((M / d) * D) * D) * M) / d)), (h / l), 1.0) * sqrt((d / h))) * sqrt((d / l));
	} else if (d <= -4.6e-299) {
		tmp = (-d * t_0) * (1.0 - (((0.125 * (D * D)) * ((M * (M / d)) / d)) * (h / l)));
	} else {
		tmp = (t_0 * d) * (1.0 - (((((D / d) / 2.0) * M) * ((((M * h) * D) / d) * 0.25)) / l));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(1.0 / Float64(l * h)))
	tmp = 0.0
	if (d <= -4.8e-118)
		tmp = Float64(Float64(fma(Float64(-0.125 * Float64(Float64(Float64(Float64(Float64(M / d) * D) * D) * M) / d)), Float64(h / l), 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	elseif (d <= -4.6e-299)
		tmp = Float64(Float64(Float64(-d) * t_0) * Float64(1.0 - Float64(Float64(Float64(0.125 * Float64(D * D)) * Float64(Float64(M * Float64(M / d)) / d)) * Float64(h / l))));
	else
		tmp = Float64(Float64(t_0 * d) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(D / d) / 2.0) * M) * Float64(Float64(Float64(Float64(M * h) * D) / d) * 0.25)) / l)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -4.8e-118], N[(N[(N[(N[(-0.125 * N[(N[(N[(N[(N[(M / d), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision] * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4.6e-299], N[(N[((-d) * t$95$0), $MachinePrecision] * N[(1.0 - N[(N[(N[(0.125 * N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(M * N[(M / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * d), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(D / d), $MachinePrecision] / 2.0), $MachinePrecision] * M), $MachinePrecision] * N[(N[(N[(N[(M * h), $MachinePrecision] * D), $MachinePrecision] / d), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.8000000000000003e-118

   1. Initial program 81.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

   4. Applied rewrites 80.4%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.5 \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\left(\left(\frac{D}{2} \cdot \frac{M}{d}\right) \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)} \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right) \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      5. associate-*l/ N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\color{blue}{\frac{D \cdot \frac{M}{d}}{2}} \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      8. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      9. frac-times N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \color{blue}{\frac{D \cdot M}{2 \cdot d}}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      10. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \frac{\color{blue}{M \cdot D}}{2 \cdot d}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \frac{\color{blue}{M \cdot D}}{2 \cdot d}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \frac{M \cdot D}{\color{blue}{2 \cdot d}}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      13. frac-times N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      14. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(D \cdot \frac{M}{d}\right) \cdot \left(M \cdot D\right)}}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      16. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      18. lower-*.f64 78.0

          \[\leadsto \left(\mathsf{fma}\left(-0.5 \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)}{\color{blue}{2 \cdot \left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   6. Applied rewrites 78.0%

      \[\leadsto \left(\mathsf{fma}\left(-0.5 \cdot \color{blue}{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      3. associate-*r/ N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{2} \cdot \left(\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)\right)}{2 \cdot \left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\frac{-1}{2} \cdot \left(\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)\right)}{\color{blue}{2 \cdot \left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\frac{-1}{2} \cdot \left(\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)\right)}{2 \cdot \color{blue}{\left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      6. associate-*r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\frac{-1}{2} \cdot \left(\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)\right)}{\color{blue}{\left(2 \cdot 2\right) \cdot d}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      7. times-frac N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{2}}{2 \cdot 2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)}{d}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      8. metadata-eval N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\frac{-1}{2}}{\color{blue}{4}} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      9. metadata-eval N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{8}} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{8} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)}{d}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      11. lower-/.f64 78.0

          \[\leadsto \left(\mathsf{fma}\left(-0.125 \cdot \color{blue}{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)}{d}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot D\right)}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(M \cdot D\right)}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      14. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(D \cdot M\right)}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      15. associate-*r* N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{\color{blue}{\left(\left(\frac{M}{d} \cdot D\right) \cdot D\right) \cdot M}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      16. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{\color{blue}{\left(\left(\frac{M}{d} \cdot D\right) \cdot D\right) \cdot M}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      17. lower-*.f64 78.0

          \[\leadsto \left(\mathsf{fma}\left(-0.125 \cdot \frac{\color{blue}{\left(\left(\frac{M}{d} \cdot D\right) \cdot D\right)} \cdot M}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   8. Applied rewrites 78.0%

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{-0.125 \cdot \frac{\left(\left(\frac{M}{d} \cdot D\right) \cdot D\right) \cdot M}{d}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   if -4.8000000000000003e-118 < d < -4.6000000000000001e-299

   1. Initial program 42.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 3.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Taylor expanded in d around 0

      \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}\right)} \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 0.5%

      \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot \frac{M}{d}}{d}\right)} \cdot \frac{h}{\ell}\right) \]

   9. Taylor expanded in h around -inf

      \[\leadsto \left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \cdot \left(1 - \left(\left(\frac{1}{8} \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot \frac{M}{d}}{d}\right) \cdot \frac{h}{\ell}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 46.9%

       \[\leadsto \left(\left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{\ell \cdot h}}}\right) \cdot \left(1 - \left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot \frac{M}{d}}{d}\right) \cdot \frac{h}{\ell}\right) \]

   if -4.6000000000000001e-299 < d

   1. Initial program 67.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. 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

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot h}{\ell}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 70.1

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{\frac{1}{2}} \cdot h\right)}{\ell}\right) \]

      19. metadata-eval 70.1

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot h\right)}{\ell}\right) \]

   4. Applied rewrites 70.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      6. frac-times N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D \cdot M}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

   6. Applied rewrites 73.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}}{\ell}\right) \]

   7. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 83.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

   10. Taylor expanded in d around 0

       \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \color{blue}{\left(\frac{1}{4} \cdot \frac{D \cdot \left(M \cdot h\right)}{d}\right)}}{\ell}\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 81.6%

       \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \color{blue}{\left(\frac{\left(M \cdot h\right) \cdot D}{d} \cdot 0.25\right)}}{\ell}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 60.4% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{if}\;\ell \leq -4.6 \cdot 10^{+139}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq -2.05 \cdot 10^{-275}:\\ \;\;\;\;\left(\left(-d\right) \cdot t\_0\right) \cdot \left(1 - \left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot \frac{M}{d}}{d}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{+57}:\\ \;\;\;\;\left(t\_0 \cdot d\right) \cdot \left(1 - \frac{\left(\left(\left(D \cdot D\right) \cdot 0.125\right) \cdot M\right) \cdot \frac{M}{d}}{d} \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (* l h)))))
   (if (<= l -4.6e+139)
     (* (sqrt (/ d h)) (sqrt (/ d l)))
     (if (<= l -2.05e-275)
       (*
        (* (- d) t_0)
        (- 1.0 (* (* (* 0.125 (* D D)) (/ (* M (/ M d)) d)) (/ h l))))
       (if (<= l 1.95e+57)
         (*
          (* t_0 d)
          (- 1.0 (* (/ (* (* (* (* D D) 0.125) M) (/ M d)) d) (/ h l))))
         (* (/ 1.0 (* (sqrt l) (sqrt h))) d))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((1.0 / (l * h)));
	double tmp;
	if (l <= -4.6e+139) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (l <= -2.05e-275) {
		tmp = (-d * t_0) * (1.0 - (((0.125 * (D * D)) * ((M * (M / d)) / d)) * (h / l)));
	} else if (l <= 1.95e+57) {
		tmp = (t_0 * d) * (1.0 - ((((((D * D) * 0.125) * M) * (M / d)) / d) * (h / l)));
	} else {
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 / (l * h)))
    if (l <= (-4.6d+139)) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else if (l <= (-2.05d-275)) then
        tmp = (-d * t_0) * (1.0d0 - (((0.125d0 * (d_1 * d_1)) * ((m * (m / d)) / d)) * (h / l)))
    else if (l <= 1.95d+57) then
        tmp = (t_0 * d) * (1.0d0 - ((((((d_1 * d_1) * 0.125d0) * m) * (m / d)) / d) * (h / l)))
    else
        tmp = (1.0d0 / (sqrt(l) * sqrt(h))) * d
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((1.0 / (l * h)));
	double tmp;
	if (l <= -4.6e+139) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (l <= -2.05e-275) {
		tmp = (-d * t_0) * (1.0 - (((0.125 * (D * D)) * ((M * (M / d)) / d)) * (h / l)));
	} else if (l <= 1.95e+57) {
		tmp = (t_0 * d) * (1.0 - ((((((D * D) * 0.125) * M) * (M / d)) / d) * (h / l)));
	} else {
		tmp = (1.0 / (Math.sqrt(l) * Math.sqrt(h))) * d;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((1.0 / (l * h)))
	tmp = 0
	if l <= -4.6e+139:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	elif l <= -2.05e-275:
		tmp = (-d * t_0) * (1.0 - (((0.125 * (D * D)) * ((M * (M / d)) / d)) * (h / l)))
	elif l <= 1.95e+57:
		tmp = (t_0 * d) * (1.0 - ((((((D * D) * 0.125) * M) * (M / d)) / d) * (h / l)))
	else:
		tmp = (1.0 / (math.sqrt(l) * math.sqrt(h))) * d
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(1.0 / Float64(l * h)))
	tmp = 0.0
	if (l <= -4.6e+139)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (l <= -2.05e-275)
		tmp = Float64(Float64(Float64(-d) * t_0) * Float64(1.0 - Float64(Float64(Float64(0.125 * Float64(D * D)) * Float64(Float64(M * Float64(M / d)) / d)) * Float64(h / l))));
	elseif (l <= 1.95e+57)
		tmp = Float64(Float64(t_0 * d) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(D * D) * 0.125) * M) * Float64(M / d)) / d) * Float64(h / l))));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(l) * sqrt(h))) * d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((1.0 / (l * h)));
	tmp = 0.0;
	if (l <= -4.6e+139)
		tmp = sqrt((d / h)) * sqrt((d / l));
	elseif (l <= -2.05e-275)
		tmp = (-d * t_0) * (1.0 - (((0.125 * (D * D)) * ((M * (M / d)) / d)) * (h / l)));
	elseif (l <= 1.95e+57)
		tmp = (t_0 * d) * (1.0 - ((((((D * D) * 0.125) * M) * (M / d)) / d) * (h / l)));
	else
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -4.6e+139], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2.05e-275], N[(N[((-d) * t$95$0), $MachinePrecision] * N[(1.0 - N[(N[(N[(0.125 * N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(M * N[(M / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.95e+57], N[(N[(t$95$0 * d), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(N[(D * D), $MachinePrecision] * 0.125), $MachinePrecision] * M), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -4.6e139

   1. Initial program 56.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

   4. Applied rewrites 56.4%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.5 \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}} \]
   6. Step-by-step derivation

   7. Applied rewrites 55.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}} \]

   if -4.6e139 < l < -2.04999999999999987e-275

   1. Initial program 74.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 0.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Taylor expanded in d around 0

      \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}\right)} \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 0.3%

      \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot \frac{M}{d}}{d}\right)} \cdot \frac{h}{\ell}\right) \]

   9. Taylor expanded in h around -inf

      \[\leadsto \left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \cdot \left(1 - \left(\left(\frac{1}{8} \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot \frac{M}{d}}{d}\right) \cdot \frac{h}{\ell}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 68.7%

       \[\leadsto \left(\left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{\ell \cdot h}}}\right) \cdot \left(1 - \left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot \frac{M}{d}}{d}\right) \cdot \frac{h}{\ell}\right) \]

   if -2.04999999999999987e-275 < l < 1.94999999999999984e57

   1. Initial program 79.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 83.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Taylor expanded in d around 0

      \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}\right)} \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 73.3%

      \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot \frac{M}{d}}{d}\right)} \cdot \frac{h}{\ell}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 77.1%

       \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \frac{\left(\left(\left(D \cdot D\right) \cdot 0.125\right) \cdot M\right) \cdot \frac{M}{d}}{\color{blue}{d}} \cdot \frac{h}{\ell}\right) \]

   if 1.94999999999999984e57 < l

   1. Initial program 50.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

   5. Applied rewrites 54.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 54.5%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

   9. Applied rewrites 69.2%

      \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 14: 53.3% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{if}\;d \leq -4.7 \cdot 10^{+38}:\\ \;\;\;\;\left(-d\right) \cdot t\_0\\ \mathbf{elif}\;d \leq -4.6 \cdot 10^{-299}:\\ \;\;\;\;\left(\left(-0.125 \cdot \sqrt{\frac{h}{\left(d \cdot d\right) \cdot d}}\right) \cdot \frac{\left(D \cdot D\right) \cdot \left(M \cdot M\right)}{-\ell}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 5.7 \cdot 10^{+197}:\\ \;\;\;\;\left(t\_0 \cdot d\right) \cdot \left(1 - \frac{\left(\left(\left(D \cdot D\right) \cdot 0.125\right) \cdot M\right) \cdot \frac{M}{d}}{d} \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (* l h)))))
   (if (<= d -4.7e+38)
     (* (- d) t_0)
     (if (<= d -4.6e-299)
       (*
        (* (* -0.125 (sqrt (/ h (* (* d d) d)))) (/ (* (* D D) (* M M)) (- l)))
        (sqrt (/ d l)))
       (if (<= d 5.7e+197)
         (*
          (* t_0 d)
          (- 1.0 (* (/ (* (* (* (* D D) 0.125) M) (/ M d)) d) (/ h l))))
         (/ d (* (sqrt l) (sqrt h))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((1.0 / (l * h)));
	double tmp;
	if (d <= -4.7e+38) {
		tmp = -d * t_0;
	} else if (d <= -4.6e-299) {
		tmp = ((-0.125 * sqrt((h / ((d * d) * d)))) * (((D * D) * (M * M)) / -l)) * sqrt((d / l));
	} else if (d <= 5.7e+197) {
		tmp = (t_0 * d) * (1.0 - ((((((D * D) * 0.125) * M) * (M / d)) / d) * (h / l)));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 / (l * h)))
    if (d <= (-4.7d+38)) then
        tmp = -d * t_0
    else if (d <= (-4.6d-299)) then
        tmp = (((-0.125d0) * sqrt((h / ((d * d) * d)))) * (((d_1 * d_1) * (m * m)) / -l)) * sqrt((d / l))
    else if (d <= 5.7d+197) then
        tmp = (t_0 * d) * (1.0d0 - ((((((d_1 * d_1) * 0.125d0) * m) * (m / d)) / d) * (h / l)))
    else
        tmp = d / (sqrt(l) * sqrt(h))
    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 / (l * h)));
	double tmp;
	if (d <= -4.7e+38) {
		tmp = -d * t_0;
	} else if (d <= -4.6e-299) {
		tmp = ((-0.125 * Math.sqrt((h / ((d * d) * d)))) * (((D * D) * (M * M)) / -l)) * Math.sqrt((d / l));
	} else if (d <= 5.7e+197) {
		tmp = (t_0 * d) * (1.0 - ((((((D * D) * 0.125) * M) * (M / d)) / d) * (h / l)));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((1.0 / (l * h)))
	tmp = 0
	if d <= -4.7e+38:
		tmp = -d * t_0
	elif d <= -4.6e-299:
		tmp = ((-0.125 * math.sqrt((h / ((d * d) * d)))) * (((D * D) * (M * M)) / -l)) * math.sqrt((d / l))
	elif d <= 5.7e+197:
		tmp = (t_0 * d) * (1.0 - ((((((D * D) * 0.125) * M) * (M / d)) / d) * (h / l)))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(1.0 / Float64(l * h)))
	tmp = 0.0
	if (d <= -4.7e+38)
		tmp = Float64(Float64(-d) * t_0);
	elseif (d <= -4.6e-299)
		tmp = Float64(Float64(Float64(-0.125 * sqrt(Float64(h / Float64(Float64(d * d) * d)))) * Float64(Float64(Float64(D * D) * Float64(M * M)) / Float64(-l))) * sqrt(Float64(d / l)));
	elseif (d <= 5.7e+197)
		tmp = Float64(Float64(t_0 * d) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(D * D) * 0.125) * M) * Float64(M / d)) / d) * Float64(h / l))));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((1.0 / (l * h)));
	tmp = 0.0;
	if (d <= -4.7e+38)
		tmp = -d * t_0;
	elseif (d <= -4.6e-299)
		tmp = ((-0.125 * sqrt((h / ((d * d) * d)))) * (((D * D) * (M * M)) / -l)) * sqrt((d / l));
	elseif (d <= 5.7e+197)
		tmp = (t_0 * d) * (1.0 - ((((((D * D) * 0.125) * M) * (M / d)) / d) * (h / l)));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -4.7e+38], N[((-d) * t$95$0), $MachinePrecision], If[LessEqual[d, -4.6e-299], N[(N[(N[(-0.125 * N[Sqrt[N[(h / N[(N[(d * d), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(D * D), $MachinePrecision] * N[(M * M), $MachinePrecision]), $MachinePrecision] / (-l)), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5.7e+197], N[(N[(t$95$0 * d), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(N[(D * D), $MachinePrecision] * 0.125), $MachinePrecision] * M), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -4.6999999999999999e38

   1. Initial program 80.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

   5. Applied rewrites 64.0%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if -4.6999999999999999e38 < d < -4.6000000000000001e-299

   1. Initial program 60.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

   4. Applied rewrites 59.9%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.5 \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]

   5. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{\ell} \cdot \sqrt{\frac{h}{{d}^{3}}}\right)\right)} \cdot \sqrt{\frac{d}{\ell}} \]
   6. Step-by-step derivation

   7. Applied rewrites 42.0%

      \[\leadsto \color{blue}{\left(\left(-0.125 \cdot \sqrt{\frac{h}{{d}^{3}}}\right) \cdot \frac{\left(\left(-D\right) \cdot D\right) \cdot \left(M \cdot M\right)}{\ell}\right)} \cdot \sqrt{\frac{d}{\ell}} \]
   8. Step-by-step derivation

   9. Applied rewrites 42.0%

      \[\leadsto \left(\left(-0.125 \cdot \sqrt{\frac{h}{\left(d \cdot d\right) \cdot d}}\right) \cdot \frac{\left(\left(-D\right) \cdot D\right) \cdot \left(M \cdot M\right)}{\ell}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   if -4.6000000000000001e-299 < d < 5.70000000000000022e197

   1. Initial program 68.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 77.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Taylor expanded in d around 0

      \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}\right)} \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 65.0%

      \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot \frac{M}{d}}{d}\right)} \cdot \frac{h}{\ell}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 69.9%

       \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \frac{\left(\left(\left(D \cdot D\right) \cdot 0.125\right) \cdot M\right) \cdot \frac{M}{d}}{\color{blue}{d}} \cdot \frac{h}{\ell}\right) \]

   if 5.70000000000000022e197 < d

   1. Initial program 65.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

   5. Applied rewrites 56.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 56.0%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

   9. Applied rewrites 56.1%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 77.5%

       \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.7 \cdot 10^{+38}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;d \leq -4.6 \cdot 10^{-299}:\\ \;\;\;\;\left(\left(-0.125 \cdot \sqrt{\frac{h}{\left(d \cdot d\right) \cdot d}}\right) \cdot \frac{\left(D \cdot D\right) \cdot \left(M \cdot M\right)}{-\ell}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 5.7 \cdot 10^{+197}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \frac{\left(\left(\left(D \cdot D\right) \cdot 0.125\right) \cdot M\right) \cdot \frac{M}{d}}{d} \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 51.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{if}\;d \leq -4.7 \cdot 10^{+38}:\\ \;\;\;\;\left(-d\right) \cdot t\_0\\ \mathbf{elif}\;d \leq -4.6 \cdot 10^{-299}:\\ \;\;\;\;\left(\left(-0.125 \cdot \sqrt{\frac{h}{\left(d \cdot d\right) \cdot d}}\right) \cdot \frac{\left(D \cdot D\right) \cdot \left(M \cdot M\right)}{-\ell}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 1.26 \cdot 10^{+197}:\\ \;\;\;\;\left(t\_0 \cdot d\right) \cdot \left(1 - \left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot \frac{M}{d}}{d}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (* l h)))))
   (if (<= d -4.7e+38)
     (* (- d) t_0)
     (if (<= d -4.6e-299)
       (*
        (* (* -0.125 (sqrt (/ h (* (* d d) d)))) (/ (* (* D D) (* M M)) (- l)))
        (sqrt (/ d l)))
       (if (<= d 1.26e+197)
         (*
          (* t_0 d)
          (- 1.0 (* (* (* 0.125 (* D D)) (/ (* M (/ M d)) d)) (/ h l))))
         (/ d (* (sqrt l) (sqrt h))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((1.0 / (l * h)));
	double tmp;
	if (d <= -4.7e+38) {
		tmp = -d * t_0;
	} else if (d <= -4.6e-299) {
		tmp = ((-0.125 * sqrt((h / ((d * d) * d)))) * (((D * D) * (M * M)) / -l)) * sqrt((d / l));
	} else if (d <= 1.26e+197) {
		tmp = (t_0 * d) * (1.0 - (((0.125 * (D * D)) * ((M * (M / d)) / d)) * (h / l)));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 / (l * h)))
    if (d <= (-4.7d+38)) then
        tmp = -d * t_0
    else if (d <= (-4.6d-299)) then
        tmp = (((-0.125d0) * sqrt((h / ((d * d) * d)))) * (((d_1 * d_1) * (m * m)) / -l)) * sqrt((d / l))
    else if (d <= 1.26d+197) then
        tmp = (t_0 * d) * (1.0d0 - (((0.125d0 * (d_1 * d_1)) * ((m * (m / d)) / d)) * (h / l)))
    else
        tmp = d / (sqrt(l) * sqrt(h))
    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 / (l * h)));
	double tmp;
	if (d <= -4.7e+38) {
		tmp = -d * t_0;
	} else if (d <= -4.6e-299) {
		tmp = ((-0.125 * Math.sqrt((h / ((d * d) * d)))) * (((D * D) * (M * M)) / -l)) * Math.sqrt((d / l));
	} else if (d <= 1.26e+197) {
		tmp = (t_0 * d) * (1.0 - (((0.125 * (D * D)) * ((M * (M / d)) / d)) * (h / l)));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((1.0 / (l * h)))
	tmp = 0
	if d <= -4.7e+38:
		tmp = -d * t_0
	elif d <= -4.6e-299:
		tmp = ((-0.125 * math.sqrt((h / ((d * d) * d)))) * (((D * D) * (M * M)) / -l)) * math.sqrt((d / l))
	elif d <= 1.26e+197:
		tmp = (t_0 * d) * (1.0 - (((0.125 * (D * D)) * ((M * (M / d)) / d)) * (h / l)))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(1.0 / Float64(l * h)))
	tmp = 0.0
	if (d <= -4.7e+38)
		tmp = Float64(Float64(-d) * t_0);
	elseif (d <= -4.6e-299)
		tmp = Float64(Float64(Float64(-0.125 * sqrt(Float64(h / Float64(Float64(d * d) * d)))) * Float64(Float64(Float64(D * D) * Float64(M * M)) / Float64(-l))) * sqrt(Float64(d / l)));
	elseif (d <= 1.26e+197)
		tmp = Float64(Float64(t_0 * d) * Float64(1.0 - Float64(Float64(Float64(0.125 * Float64(D * D)) * Float64(Float64(M * Float64(M / d)) / d)) * Float64(h / l))));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((1.0 / (l * h)));
	tmp = 0.0;
	if (d <= -4.7e+38)
		tmp = -d * t_0;
	elseif (d <= -4.6e-299)
		tmp = ((-0.125 * sqrt((h / ((d * d) * d)))) * (((D * D) * (M * M)) / -l)) * sqrt((d / l));
	elseif (d <= 1.26e+197)
		tmp = (t_0 * d) * (1.0 - (((0.125 * (D * D)) * ((M * (M / d)) / d)) * (h / l)));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -4.7e+38], N[((-d) * t$95$0), $MachinePrecision], If[LessEqual[d, -4.6e-299], N[(N[(N[(-0.125 * N[Sqrt[N[(h / N[(N[(d * d), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(D * D), $MachinePrecision] * N[(M * M), $MachinePrecision]), $MachinePrecision] / (-l)), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.26e+197], N[(N[(t$95$0 * d), $MachinePrecision] * N[(1.0 - N[(N[(N[(0.125 * N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(M * N[(M / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -4.6999999999999999e38

   1. Initial program 80.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

   5. Applied rewrites 64.0%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if -4.6999999999999999e38 < d < -4.6000000000000001e-299

   1. Initial program 60.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

   4. Applied rewrites 59.9%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.5 \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]

   5. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{\ell} \cdot \sqrt{\frac{h}{{d}^{3}}}\right)\right)} \cdot \sqrt{\frac{d}{\ell}} \]
   6. Step-by-step derivation

   7. Applied rewrites 42.0%

      \[\leadsto \color{blue}{\left(\left(-0.125 \cdot \sqrt{\frac{h}{{d}^{3}}}\right) \cdot \frac{\left(\left(-D\right) \cdot D\right) \cdot \left(M \cdot M\right)}{\ell}\right)} \cdot \sqrt{\frac{d}{\ell}} \]
   8. Step-by-step derivation

   9. Applied rewrites 42.0%

      \[\leadsto \left(\left(-0.125 \cdot \sqrt{\frac{h}{\left(d \cdot d\right) \cdot d}}\right) \cdot \frac{\left(\left(-D\right) \cdot D\right) \cdot \left(M \cdot M\right)}{\ell}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   if -4.6000000000000001e-299 < d < 1.26e197

   1. Initial program 68.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 77.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Taylor expanded in d around 0

      \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}\right)} \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 65.0%

      \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot \frac{M}{d}}{d}\right)} \cdot \frac{h}{\ell}\right) \]

   if 1.26e197 < d

   1. Initial program 65.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

   5. Applied rewrites 56.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 56.0%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

   9. Applied rewrites 56.1%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 77.5%

       \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.7 \cdot 10^{+38}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;d \leq -4.6 \cdot 10^{-299}:\\ \;\;\;\;\left(\left(-0.125 \cdot \sqrt{\frac{h}{\left(d \cdot d\right) \cdot d}}\right) \cdot \frac{\left(D \cdot D\right) \cdot \left(M \cdot M\right)}{-\ell}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 1.26 \cdot 10^{+197}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot \frac{M}{d}}{d}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 49.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{if}\;\ell \leq -3.2 \cdot 10^{-102}:\\ \;\;\;\;\left(-d\right) \cdot t\_0\\ \mathbf{elif}\;\ell \leq -1.05 \cdot 10^{-302}:\\ \;\;\;\;\left(\left(-0.125 \cdot \sqrt{\frac{h}{\left(d \cdot d\right) \cdot d}}\right) \cdot \frac{\left(D \cdot D\right) \cdot \left(M \cdot M\right)}{-\ell}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{+56}:\\ \;\;\;\;\left(t\_0 \cdot d\right) \cdot \left(1 - \left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot M}{d \cdot d}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (* l h)))))
   (if (<= l -3.2e-102)
     (* (- d) t_0)
     (if (<= l -1.05e-302)
       (*
        (* (* -0.125 (sqrt (/ h (* (* d d) d)))) (/ (* (* D D) (* M M)) (- l)))
        (sqrt (/ d l)))
       (if (<= l 7.5e+56)
         (*
          (* t_0 d)
          (- 1.0 (* (* (* 0.125 (* D D)) (/ (* M M) (* d d))) (/ h l))))
         (* (/ 1.0 (* (sqrt l) (sqrt h))) d))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((1.0 / (l * h)));
	double tmp;
	if (l <= -3.2e-102) {
		tmp = -d * t_0;
	} else if (l <= -1.05e-302) {
		tmp = ((-0.125 * sqrt((h / ((d * d) * d)))) * (((D * D) * (M * M)) / -l)) * sqrt((d / l));
	} else if (l <= 7.5e+56) {
		tmp = (t_0 * d) * (1.0 - (((0.125 * (D * D)) * ((M * M) / (d * d))) * (h / l)));
	} else {
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 / (l * h)))
    if (l <= (-3.2d-102)) then
        tmp = -d * t_0
    else if (l <= (-1.05d-302)) then
        tmp = (((-0.125d0) * sqrt((h / ((d * d) * d)))) * (((d_1 * d_1) * (m * m)) / -l)) * sqrt((d / l))
    else if (l <= 7.5d+56) then
        tmp = (t_0 * d) * (1.0d0 - (((0.125d0 * (d_1 * d_1)) * ((m * m) / (d * d))) * (h / l)))
    else
        tmp = (1.0d0 / (sqrt(l) * sqrt(h))) * d
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((1.0 / (l * h)));
	double tmp;
	if (l <= -3.2e-102) {
		tmp = -d * t_0;
	} else if (l <= -1.05e-302) {
		tmp = ((-0.125 * Math.sqrt((h / ((d * d) * d)))) * (((D * D) * (M * M)) / -l)) * Math.sqrt((d / l));
	} else if (l <= 7.5e+56) {
		tmp = (t_0 * d) * (1.0 - (((0.125 * (D * D)) * ((M * M) / (d * d))) * (h / l)));
	} else {
		tmp = (1.0 / (Math.sqrt(l) * Math.sqrt(h))) * d;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((1.0 / (l * h)))
	tmp = 0
	if l <= -3.2e-102:
		tmp = -d * t_0
	elif l <= -1.05e-302:
		tmp = ((-0.125 * math.sqrt((h / ((d * d) * d)))) * (((D * D) * (M * M)) / -l)) * math.sqrt((d / l))
	elif l <= 7.5e+56:
		tmp = (t_0 * d) * (1.0 - (((0.125 * (D * D)) * ((M * M) / (d * d))) * (h / l)))
	else:
		tmp = (1.0 / (math.sqrt(l) * math.sqrt(h))) * d
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(1.0 / Float64(l * h)))
	tmp = 0.0
	if (l <= -3.2e-102)
		tmp = Float64(Float64(-d) * t_0);
	elseif (l <= -1.05e-302)
		tmp = Float64(Float64(Float64(-0.125 * sqrt(Float64(h / Float64(Float64(d * d) * d)))) * Float64(Float64(Float64(D * D) * Float64(M * M)) / Float64(-l))) * sqrt(Float64(d / l)));
	elseif (l <= 7.5e+56)
		tmp = Float64(Float64(t_0 * d) * Float64(1.0 - Float64(Float64(Float64(0.125 * Float64(D * D)) * Float64(Float64(M * M) / Float64(d * d))) * Float64(h / l))));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(l) * sqrt(h))) * d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((1.0 / (l * h)));
	tmp = 0.0;
	if (l <= -3.2e-102)
		tmp = -d * t_0;
	elseif (l <= -1.05e-302)
		tmp = ((-0.125 * sqrt((h / ((d * d) * d)))) * (((D * D) * (M * M)) / -l)) * sqrt((d / l));
	elseif (l <= 7.5e+56)
		tmp = (t_0 * d) * (1.0 - (((0.125 * (D * D)) * ((M * M) / (d * d))) * (h / l)));
	else
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -3.2e-102], N[((-d) * t$95$0), $MachinePrecision], If[LessEqual[l, -1.05e-302], N[(N[(N[(-0.125 * N[Sqrt[N[(h / N[(N[(d * d), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(D * D), $MachinePrecision] * N[(M * M), $MachinePrecision]), $MachinePrecision] / (-l)), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 7.5e+56], N[(N[(t$95$0 * d), $MachinePrecision] * N[(1.0 - N[(N[(N[(0.125 * N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -3.19999999999999986e-102

   1. Initial program 68.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

   5. Applied rewrites 48.3%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if -3.19999999999999986e-102 < l < -1.05000000000000006e-302

   1. Initial program 74.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

   4. Applied rewrites 75.6%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.5 \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]

   5. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{\ell} \cdot \sqrt{\frac{h}{{d}^{3}}}\right)\right)} \cdot \sqrt{\frac{d}{\ell}} \]
   6. Step-by-step derivation

   7. Applied rewrites 42.3%

      \[\leadsto \color{blue}{\left(\left(-0.125 \cdot \sqrt{\frac{h}{{d}^{3}}}\right) \cdot \frac{\left(\left(-D\right) \cdot D\right) \cdot \left(M \cdot M\right)}{\ell}\right)} \cdot \sqrt{\frac{d}{\ell}} \]
   8. Step-by-step derivation

   9. Applied rewrites 42.3%

      \[\leadsto \left(\left(-0.125 \cdot \sqrt{\frac{h}{\left(d \cdot d\right) \cdot d}}\right) \cdot \frac{\left(\left(-D\right) \cdot D\right) \cdot \left(M \cdot M\right)}{\ell}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   if -1.05000000000000006e-302 < l < 7.4999999999999999e56

   1. Initial program 78.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 87.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Taylor expanded in d around 0

      \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}\right)} \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 76.8%

      \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot \frac{M}{d}}{d}\right)} \cdot \frac{h}{\ell}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 61.5%

       \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot M}{\color{blue}{d \cdot d}}\right) \cdot \frac{h}{\ell}\right) \]

   if 7.4999999999999999e56 < l

   1. Initial program 50.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

   5. Applied rewrites 54.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 54.5%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

   9. Applied rewrites 69.2%

      \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]
3. Recombined 4 regimes into one program.

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.2 \cdot 10^{-102}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;\ell \leq -1.05 \cdot 10^{-302}:\\ \;\;\;\;\left(\left(-0.125 \cdot \sqrt{\frac{h}{\left(d \cdot d\right) \cdot d}}\right) \cdot \frac{\left(D \cdot D\right) \cdot \left(M \cdot M\right)}{-\ell}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{+56}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot M}{d \cdot d}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 49.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{if}\;\ell \leq 2 \cdot 10^{-310}:\\ \;\;\;\;\left(-d\right) \cdot t\_0\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{+56}:\\ \;\;\;\;\left(t\_0 \cdot d\right) \cdot \left(1 - \left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot M}{d \cdot d}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (* l h)))))
   (if (<= l 2e-310)
     (* (- d) t_0)
     (if (<= l 7.5e+56)
       (*
        (* t_0 d)
        (- 1.0 (* (* (* 0.125 (* D D)) (/ (* M M) (* d d))) (/ h l))))
       (* (/ 1.0 (* (sqrt l) (sqrt h))) d)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((1.0 / (l * h)));
	double tmp;
	if (l <= 2e-310) {
		tmp = -d * t_0;
	} else if (l <= 7.5e+56) {
		tmp = (t_0 * d) * (1.0 - (((0.125 * (D * D)) * ((M * M) / (d * d))) * (h / l)));
	} else {
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 / (l * h)))
    if (l <= 2d-310) then
        tmp = -d * t_0
    else if (l <= 7.5d+56) then
        tmp = (t_0 * d) * (1.0d0 - (((0.125d0 * (d_1 * d_1)) * ((m * m) / (d * d))) * (h / l)))
    else
        tmp = (1.0d0 / (sqrt(l) * sqrt(h))) * d
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((1.0 / (l * h)));
	double tmp;
	if (l <= 2e-310) {
		tmp = -d * t_0;
	} else if (l <= 7.5e+56) {
		tmp = (t_0 * d) * (1.0 - (((0.125 * (D * D)) * ((M * M) / (d * d))) * (h / l)));
	} else {
		tmp = (1.0 / (Math.sqrt(l) * Math.sqrt(h))) * d;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((1.0 / (l * h)))
	tmp = 0
	if l <= 2e-310:
		tmp = -d * t_0
	elif l <= 7.5e+56:
		tmp = (t_0 * d) * (1.0 - (((0.125 * (D * D)) * ((M * M) / (d * d))) * (h / l)))
	else:
		tmp = (1.0 / (math.sqrt(l) * math.sqrt(h))) * d
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(1.0 / Float64(l * h)))
	tmp = 0.0
	if (l <= 2e-310)
		tmp = Float64(Float64(-d) * t_0);
	elseif (l <= 7.5e+56)
		tmp = Float64(Float64(t_0 * d) * Float64(1.0 - Float64(Float64(Float64(0.125 * Float64(D * D)) * Float64(Float64(M * M) / Float64(d * d))) * Float64(h / l))));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(l) * sqrt(h))) * d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((1.0 / (l * h)));
	tmp = 0.0;
	if (l <= 2e-310)
		tmp = -d * t_0;
	elseif (l <= 7.5e+56)
		tmp = (t_0 * d) * (1.0 - (((0.125 * (D * D)) * ((M * M) / (d * d))) * (h / l)));
	else
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, 2e-310], N[((-d) * t$95$0), $MachinePrecision], If[LessEqual[l, 7.5e+56], N[(N[(t$95$0 * d), $MachinePrecision] * N[(1.0 - N[(N[(N[(0.125 * N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.999999999999994e-310

   1. Initial program 70.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

   5. Applied rewrites 42.7%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if 1.999999999999994e-310 < l < 7.4999999999999999e56

   1. Initial program 78.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 88.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Taylor expanded in d around 0

      \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}\right)} \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 77.7%

      \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot \frac{M}{d}}{d}\right)} \cdot \frac{h}{\ell}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 62.3%

       \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot M}{\color{blue}{d \cdot d}}\right) \cdot \frac{h}{\ell}\right) \]

   if 7.4999999999999999e56 < l

   1. Initial program 50.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

   5. Applied rewrites 54.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 54.5%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

   9. Applied rewrites 69.2%

      \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 44.6% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.1 \cdot 10^{-171}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l 2.1e-171)
   (* (- d) (sqrt (/ 1.0 (* l h))))
   (* (/ 1.0 (* (sqrt l) (sqrt h))) d)))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 2.1e-171) {
		tmp = -d * sqrt((1.0 / (l * h)));
	} else {
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= 2.1d-171) then
        tmp = -d * sqrt((1.0d0 / (l * h)))
    else
        tmp = (1.0d0 / (sqrt(l) * sqrt(h))) * d
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 2.1e-171) {
		tmp = -d * Math.sqrt((1.0 / (l * h)));
	} else {
		tmp = (1.0 / (Math.sqrt(l) * Math.sqrt(h))) * d;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= 2.1e-171:
		tmp = -d * math.sqrt((1.0 / (l * h)))
	else:
		tmp = (1.0 / (math.sqrt(l) * math.sqrt(h))) * d
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 2.1e-171)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(l * h))));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(l) * sqrt(h))) * d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= 2.1e-171)
		tmp = -d * sqrt((1.0 / (l * h)));
	else
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, 2.1e-171], N[((-d) * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.1e-171

   1. Initial program 73.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

   5. Applied rewrites 41.8%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if 2.1e-171 < l

   1. Initial program 62.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

   5. Applied rewrites 51.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 51.4%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

   9. Applied rewrites 60.7%

      \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 44.6% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.1 \cdot 10^{-171}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l 2.1e-171)
   (* (- d) (sqrt (/ 1.0 (* l h))))
   (/ d (* (sqrt l) (sqrt h)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 2.1e-171) {
		tmp = -d * sqrt((1.0 / (l * h)));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= 2.1d-171) then
        tmp = -d * sqrt((1.0d0 / (l * h)))
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 2.1e-171) {
		tmp = -d * Math.sqrt((1.0 / (l * h)));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= 2.1e-171:
		tmp = -d * math.sqrt((1.0 / (l * h)))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 2.1e-171)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(l * h))));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= 2.1e-171)
		tmp = -d * sqrt((1.0 / (l * h)));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, 2.1e-171], N[((-d) * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.1e-171

   1. Initial program 73.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

   5. Applied rewrites 41.8%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if 2.1e-171 < l

   1. Initial program 62.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

   5. Applied rewrites 51.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 51.4%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

   9. Applied rewrites 51.4%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 60.7%

       \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 42.0% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{if}\;\ell \leq 2.8 \cdot 10^{-171}:\\ \;\;\;\;\left(-d\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot d\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (* l h)))))
   (if (<= l 2.8e-171) (* (- d) t_0) (* t_0 d))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((1.0 / (l * h)));
	double tmp;
	if (l <= 2.8e-171) {
		tmp = -d * t_0;
	} else {
		tmp = t_0 * d;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 / (l * h)))
    if (l <= 2.8d-171) then
        tmp = -d * t_0
    else
        tmp = t_0 * d
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((1.0 / (l * h)));
	double tmp;
	if (l <= 2.8e-171) {
		tmp = -d * t_0;
	} else {
		tmp = t_0 * d;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((1.0 / (l * h)))
	tmp = 0
	if l <= 2.8e-171:
		tmp = -d * t_0
	else:
		tmp = t_0 * d
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(1.0 / Float64(l * h)))
	tmp = 0.0
	if (l <= 2.8e-171)
		tmp = Float64(Float64(-d) * t_0);
	else
		tmp = Float64(t_0 * d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((1.0 / (l * h)));
	tmp = 0.0;
	if (l <= 2.8e-171)
		tmp = -d * t_0;
	else
		tmp = t_0 * d;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, 2.8e-171], N[((-d) * t$95$0), $MachinePrecision], N[(t$95$0 * d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 2.80000000000000023e-171

   1. Initial program 73.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

   5. Applied rewrites 41.8%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if 2.80000000000000023e-171 < l

   1. Initial program 62.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

   5. Applied rewrites 51.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 26.3% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{\ell \cdot h}} \cdot d \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (* (sqrt (/ 1.0 (* l h))) d))

C

double code(double d, double h, double l, double M, double D) {
	return sqrt((1.0 / (l * h))) * d;
}

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 = sqrt((1.0d0 / (l * h))) * d
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return Math.sqrt((1.0 / (l * h))) * d;
}

Python

def code(d, h, l, M, D):
	return math.sqrt((1.0 / (l * h))) * d

Julia

function code(d, h, l, M, D)
	return Float64(sqrt(Float64(1.0 / Float64(l * h))) * d)
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = sqrt((1.0 / (l * h))) * d;
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]

Derivation

1. Initial program 68.9%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Add Preprocessing

3. Taylor expanded in d around inf

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
4. Step-by-step derivation

5. Applied rewrites 26.6%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
6. Add Preprocessing

Alternative 22: 26.2% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \frac{d}{\sqrt{\ell \cdot h}} \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (/ d (sqrt (* l h))))

C

double code(double d, double h, double l, double M, double D) {
	return d / sqrt((l * h));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = d / sqrt((l * h))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return d / Math.sqrt((l * h));
}

Python

def code(d, h, l, M, D):
	return d / math.sqrt((l * h))

Julia

function code(d, h, l, M, D)
	return Float64(d / sqrt(Float64(l * h)))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = d / sqrt((l * h));
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.9%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Add Preprocessing

3. Taylor expanded in d around inf

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
4. Step-by-step derivation

5. Applied rewrites 26.6%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
6. Step-by-step derivation

7. Applied rewrites 26.2%

   \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
8. Step-by-step derivation

9. Applied rewrites 26.3%

   \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025018 
(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)))))