Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.1% → 78.5%

Time: 15.8s

Alternatives: 19

Speedup: 1.2×

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: 78.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \frac{\frac{D}{d}}{2}\\ t_2 := {\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)}\\ t_3 := \left(t\_2 \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_4 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ t_5 := \frac{\frac{D}{d}}{-2} \cdot M\\ \mathbf{if}\;t\_3 \leq -1 \cdot 10^{+210}:\\ \;\;\;\;\left(t\_2 \cdot t\_0\right) \cdot \left(1 - \left(t\_1 \cdot \left(M \cdot \frac{h}{\ell}\right)\right) \cdot \left(\left(0.5 \cdot \frac{D}{2}\right) \cdot \frac{M}{d}\right)\right)\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-229}:\\ \;\;\;\;\mathsf{fma}\left(\left(--0.5\right) \cdot \left(\frac{M \cdot D}{-2 \cdot d} \cdot \frac{D \cdot \frac{M}{2}}{d}\right), \frac{h}{\ell}, 1\right) \cdot t\_4\\ \mathbf{elif}\;t\_3 \leq 10^{+291}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5 \cdot {\left(M \cdot t\_1\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot t\_0\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{h}{\ell} \cdot \left(t\_5 \cdot -0.5\right), t\_5, 1\right) \cdot t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l)))
        (t_1 (/ (/ D d) 2.0))
        (t_2 (pow (/ d h) (pow 2.0 -1.0)))
        (t_3
         (*
          (* t_2 (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_4 (/ (fabs d) (sqrt (* l h))))
        (t_5 (* (/ (/ D d) -2.0) M)))
   (if (<= t_3 -1e+210)
     (*
      (* t_2 t_0)
      (- 1.0 (* (* t_1 (* M (/ h l))) (* (* 0.5 (/ D 2.0)) (/ M d)))))
     (if (<= t_3 5e-229)
       (*
        (fma
         (* (- -0.5) (* (/ (* M D) (* -2.0 d)) (/ (* D (/ M 2.0)) d)))
         (/ h l)
         1.0)
        t_4)
       (if (<= t_3 1e+291)
         (*
          (* (fma (* -0.5 (pow (* M t_1) 2.0)) (/ h l) 1.0) t_0)
          (sqrt (/ d h)))
         (* (fma (* (/ h l) (* t_5 -0.5)) t_5 1.0) t_4))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / l));
	double t_1 = (D / d) / 2.0;
	double t_2 = pow((d / h), pow(2.0, -1.0));
	double t_3 = (t_2 * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_4 = fabs(d) / sqrt((l * h));
	double t_5 = ((D / d) / -2.0) * M;
	double tmp;
	if (t_3 <= -1e+210) {
		tmp = (t_2 * t_0) * (1.0 - ((t_1 * (M * (h / l))) * ((0.5 * (D / 2.0)) * (M / d))));
	} else if (t_3 <= 5e-229) {
		tmp = fma((-(-0.5) * (((M * D) / (-2.0 * d)) * ((D * (M / 2.0)) / d))), (h / l), 1.0) * t_4;
	} else if (t_3 <= 1e+291) {
		tmp = (fma((-0.5 * pow((M * t_1), 2.0)), (h / l), 1.0) * t_0) * sqrt((d / h));
	} else {
		tmp = fma(((h / l) * (t_5 * -0.5)), t_5, 1.0) * t_4;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / l))
	t_1 = Float64(Float64(D / d) / 2.0)
	t_2 = Float64(d / h) ^ (2.0 ^ -1.0)
	t_3 = Float64(Float64(t_2 * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_4 = Float64(abs(d) / sqrt(Float64(l * h)))
	t_5 = Float64(Float64(Float64(D / d) / -2.0) * M)
	tmp = 0.0
	if (t_3 <= -1e+210)
		tmp = Float64(Float64(t_2 * t_0) * Float64(1.0 - Float64(Float64(t_1 * Float64(M * Float64(h / l))) * Float64(Float64(0.5 * Float64(D / 2.0)) * Float64(M / d)))));
	elseif (t_3 <= 5e-229)
		tmp = Float64(fma(Float64(Float64(-(-0.5)) * Float64(Float64(Float64(M * D) / Float64(-2.0 * d)) * Float64(Float64(D * Float64(M / 2.0)) / d))), Float64(h / l), 1.0) * t_4);
	elseif (t_3 <= 1e+291)
		tmp = Float64(Float64(fma(Float64(-0.5 * (Float64(M * t_1) ^ 2.0)), Float64(h / l), 1.0) * t_0) * sqrt(Float64(d / h)));
	else
		tmp = Float64(fma(Float64(Float64(h / l) * Float64(t_5 * -0.5)), t_5, 1.0) * t_4);
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(D / d), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$2 * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(D / d), $MachinePrecision] / -2.0), $MachinePrecision] * M), $MachinePrecision]}, If[LessEqual[t$95$3, -1e+210], N[(N[(t$95$2 * t$95$0), $MachinePrecision] * N[(1.0 - N[(N[(t$95$1 * N[(M * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[(D / 2.0), $MachinePrecision]), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e-229], N[(N[(N[((--0.5) * N[(N[(N[(M * D), $MachinePrecision] / N[(-2.0 * d), $MachinePrecision]), $MachinePrecision] * N[(N[(D * N[(M / 2.0), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$4), $MachinePrecision], If[LessEqual[t$95$3, 1e+291], N[(N[(N[(N[(-0.5 * N[Power[N[(M * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(h / l), $MachinePrecision] * N[(t$95$5 * -0.5), $MachinePrecision]), $MachinePrecision] * t$95$5 + 1.0), $MachinePrecision] * t$95$4), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -9.99999999999999927e209

   1. Initial program 89.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 91.9%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 91.9

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 91.9

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

   6. Applied rewrites 91.9%

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

   if -9.99999999999999927e209 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.00000000000000016e-229

   1. Initial program 60.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 43.0

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

   4. Applied rewrites 43.0%

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

   6. Applied rewrites 85.9%

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

      1. unpow1 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. pow-prod-down N/A

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

      5. pow-sqr N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/l/ N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. metadata-eval N/A

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

      13. unpow2 N/A

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

      14. associate-/r* N/A

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

      15. associate-/r* N/A

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

      16. frac-times N/A

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

      17. sqr-neg-rev N/A

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

      18. times-frac N/A

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

      19. lower-*.f64 N/A

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

   8. Applied rewrites 87.4%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-prod N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. sqr-neg-rev N/A

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

      6. sqrt-prod N/A

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

      7. rem-square-sqrt 52.6

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r/ N/A

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

      12. lift-/.f64 N/A

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

      13. times-frac N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. count-2-rev N/A

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

      17. rem-square-sqrt N/A

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

      18. sqrt-prod N/A

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

      19. sqr-neg-rev N/A

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

      20. lift-neg.f64 N/A

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

      21. lift-neg.f64 N/A

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

      22. sqrt-prod N/A

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

      23. rem-square-sqrt N/A

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

      24. lift-neg.f64 N/A

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

      25. rem-square-sqrt N/A

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

      26. sqrt-prod N/A

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

      27. sqr-neg-rev N/A

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

      28. lift-neg.f64 N/A

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

      29. lift-neg.f64 N/A

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

      30. sqrt-prod N/A

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

      31. rem-square-sqrt N/A

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

   10. Applied rewrites 87.4%

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

   if 5.00000000000000016e-229 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.9999999999999996e290

   1. Initial program 99.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 47.4

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

   4. Applied rewrites 47.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   6. Applied rewrites 99.2%

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

   if 9.9999999999999996e290 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 21.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 23.5

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

   4. Applied rewrites 23.5%

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

   6. Applied rewrites 54.0%

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

      1. unpow1 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. pow-prod-down N/A

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

      5. pow-sqr N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/l/ N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. metadata-eval N/A

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

      13. unpow2 N/A

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

      14. associate-/r* N/A

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

      15. associate-/r* N/A

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

      16. frac-times N/A

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

      17. sqr-neg-rev N/A

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

      18. times-frac N/A

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

      19. lower-*.f64 N/A

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

   8. Applied rewrites 52.8%

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

   10. Applied rewrites 59.2%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -1 \cdot 10^{+210}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{\frac{D}{d}}{2} \cdot \left(M \cdot \frac{h}{\ell}\right)\right) \cdot \left(\left(0.5 \cdot \frac{D}{2}\right) \cdot \frac{M}{d}\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{-229}:\\ \;\;\;\;\mathsf{fma}\left(\left(--0.5\right) \cdot \left(\frac{M \cdot D}{-2 \cdot d} \cdot \frac{D \cdot \frac{M}{2}}{d}\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 10^{+291}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5 \cdot {\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{h}{\ell} \cdot \left(\left(\frac{\frac{D}{d}}{-2} \cdot M\right) \cdot -0.5\right), \frac{\frac{D}{d}}{-2} \cdot M, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 71.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+71}:\\ \;\;\;\;\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \left(\frac{D}{d} \cdot \frac{\frac{D}{\ell} \cdot h}{d}\right)\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-229}:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{\frac{M \cdot M}{d}}{d}, \frac{h}{\ell}, 1\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+291}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;1 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{h}{\ell \cdot d} \cdot \frac{D \cdot D}{d}\right) \cdot M\right) \cdot \left(-0.125 \cdot M\right)\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (/ (fabs d) (sqrt (* l h)))))
   (if (<= t_0 -1e+71)
     (* (* (* (* M M) -0.125) (* (/ D d) (/ (* (/ D l) h) d))) t_1)
     (if (<= t_0 5e-229)
       (* (fma (* (* -0.125 (* D D)) (/ (/ (* M M) d) d)) (/ h l) 1.0) t_1)
       (if (<= t_0 1e+291)
         (* (sqrt (/ d l)) (sqrt (/ d h)))
         (if (<= t_0 INFINITY)
           (* 1.0 t_1)
           (* (* (* (* (/ h (* l d)) (/ (* D D) d)) M) (* -0.125 M)) t_1)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = fabs(d) / sqrt((l * h));
	double tmp;
	if (t_0 <= -1e+71) {
		tmp = (((M * M) * -0.125) * ((D / d) * (((D / l) * h) / d))) * t_1;
	} else if (t_0 <= 5e-229) {
		tmp = fma(((-0.125 * (D * D)) * (((M * M) / d) / d)), (h / l), 1.0) * t_1;
	} else if (t_0 <= 1e+291) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 1.0 * t_1;
	} else {
		tmp = ((((h / (l * d)) * ((D * D) / d)) * M) * (-0.125 * M)) * t_1;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(abs(d) / sqrt(Float64(l * h)))
	tmp = 0.0
	if (t_0 <= -1e+71)
		tmp = Float64(Float64(Float64(Float64(M * M) * -0.125) * Float64(Float64(D / d) * Float64(Float64(Float64(D / l) * h) / d))) * t_1);
	elseif (t_0 <= 5e-229)
		tmp = Float64(fma(Float64(Float64(-0.125 * Float64(D * D)) * Float64(Float64(Float64(M * M) / d) / d)), Float64(h / l), 1.0) * t_1);
	elseif (t_0 <= 1e+291)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (t_0 <= Inf)
		tmp = Float64(1.0 * t_1);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(h / Float64(l * d)) * Float64(Float64(D * D) / d)) * M) * Float64(-0.125 * M)) * t_1);
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+71], N[(N[(N[(N[(M * M), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[(N[(N[(D / l), $MachinePrecision] * h), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 5e-229], N[(N[(N[(N[(-0.125 * N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 1e+291], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(1.0 * t$95$1), $MachinePrecision], N[(N[(N[(N[(N[(h / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(N[(D * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision] * N[(-0.125 * M), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1e71

   1. Initial program 90.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 48.3

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

   4. Applied rewrites 48.3%

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

   6. Applied rewrites 84.1%

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

   7. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 63.7

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

   9. Applied rewrites 63.7%

      \[\leadsto \color{blue}{\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{h}{d \cdot d}\right)\right)} \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]
   10. Step-by-step derivation

   11. Applied rewrites 79.1%

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

   if -1e71 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.00000000000000016e-229

   1. Initial program 52.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 40.0

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

   4. Applied rewrites 40.0%

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

   6. Applied rewrites 87.1%

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

   7. Taylor expanded in d around 0

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{8} \cdot \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2}}{{d}^{2}}\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \frac{{M}^{2}}{{d}^{2}}}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \frac{{M}^{2}}{{d}^{2}}}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{8} \cdot {D}^{2}\right)} \cdot \frac{{M}^{2}}{{d}^{2}}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{8} \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot \frac{{M}^{2}}{{d}^{2}}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{8} \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot \frac{{M}^{2}}{{d}^{2}}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \frac{{M}^{2}}{\color{blue}{d \cdot d}}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\frac{\frac{{M}^{2}}{d}}{d}}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\frac{\frac{{M}^{2}}{d}}{d}}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \frac{\color{blue}{\frac{{M}^{2}}{d}}}{d}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      11. unpow2 N/A

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

      12. lower-*.f64 54.6

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

   9. Applied rewrites 54.6%

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

   if 5.00000000000000016e-229 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.9999999999999996e290

   1. Initial program 99.2%

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 39.5

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 39.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 39.6%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

   9. Applied rewrites 98.3%

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

   if 9.9999999999999996e290 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 47.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 44.1

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

   4. Applied rewrites 44.1%

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

   6. Applied rewrites 90.5%

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

   7. Taylor expanded in d around inf

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

   9. Applied rewrites 90.5%

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

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

   1. Initial program 0.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 6.5

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

   4. Applied rewrites 6.5%

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

   6. Applied rewrites 23.7%

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

   7. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 19.5

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

   9. Applied rewrites 19.5%

      \[\leadsto \color{blue}{\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{h}{d \cdot d}\right)\right)} \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]
   10. Step-by-step derivation

   11. Applied rewrites 29.4%

       \[\leadsto \left(\left(\left(\frac{h}{\ell \cdot d} \cdot \frac{D \cdot D}{d}\right) \cdot M\right) \cdot \color{blue}{\left(-0.125 \cdot M\right)}\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -1 \cdot 10^{+71}:\\ \;\;\;\;\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \left(\frac{D}{d} \cdot \frac{\frac{D}{\ell} \cdot h}{d}\right)\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{-229}:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{\frac{M \cdot M}{d}}{d}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 10^{+291}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;1 \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{h}{\ell \cdot d} \cdot \frac{D \cdot D}{d}\right) \cdot M\right) \cdot \left(-0.125 \cdot M\right)\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ t_2 := 1 \cdot t\_1\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-111}:\\ \;\;\;\;\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \left(\frac{D}{d} \cdot \frac{\frac{D}{\ell} \cdot h}{d}\right)\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-229}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 10^{+291}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{h}{\ell \cdot d} \cdot \frac{D \cdot D}{d}\right) \cdot M\right) \cdot \left(-0.125 \cdot M\right)\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (/ (fabs d) (sqrt (* l h))))
        (t_2 (* 1.0 t_1)))
   (if (<= t_0 -5e-111)
     (* (* (* (* M M) -0.125) (* (/ D d) (/ (* (/ D l) h) d))) t_1)
     (if (<= t_0 5e-229)
       t_2
       (if (<= t_0 1e+291)
         (* (sqrt (/ d l)) (sqrt (/ d h)))
         (if (<= t_0 INFINITY)
           t_2
           (* (* (* (* (/ h (* l d)) (/ (* D D) d)) M) (* -0.125 M)) t_1)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = fabs(d) / sqrt((l * h));
	double t_2 = 1.0 * t_1;
	double tmp;
	if (t_0 <= -5e-111) {
		tmp = (((M * M) * -0.125) * ((D / d) * (((D / l) * h) / d))) * t_1;
	} else if (t_0 <= 5e-229) {
		tmp = t_2;
	} else if (t_0 <= 1e+291) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = ((((h / (l * d)) * ((D * D) / d)) * M) * (-0.125 * M)) * t_1;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = Math.abs(d) / Math.sqrt((l * h));
	double t_2 = 1.0 * t_1;
	double tmp;
	if (t_0 <= -5e-111) {
		tmp = (((M * M) * -0.125) * ((D / d) * (((D / l) * h) / d))) * t_1;
	} else if (t_0 <= 5e-229) {
		tmp = t_2;
	} else if (t_0 <= 1e+291) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else {
		tmp = ((((h / (l * d)) * ((D * D) / d)) * M) * (-0.125 * M)) * t_1;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_1 = math.fabs(d) / math.sqrt((l * h))
	t_2 = 1.0 * t_1
	tmp = 0
	if t_0 <= -5e-111:
		tmp = (((M * M) * -0.125) * ((D / d) * (((D / l) * h) / d))) * t_1
	elif t_0 <= 5e-229:
		tmp = t_2
	elif t_0 <= 1e+291:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	elif t_0 <= math.inf:
		tmp = t_2
	else:
		tmp = ((((h / (l * d)) * ((D * D) / d)) * M) * (-0.125 * M)) * t_1
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(abs(d) / sqrt(Float64(l * h)))
	t_2 = Float64(1.0 * t_1)
	tmp = 0.0
	if (t_0 <= -5e-111)
		tmp = Float64(Float64(Float64(Float64(M * M) * -0.125) * Float64(Float64(D / d) * Float64(Float64(Float64(D / l) * h) / d))) * t_1);
	elseif (t_0 <= 5e-229)
		tmp = t_2;
	elseif (t_0 <= 1e+291)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (t_0 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(Float64(Float64(Float64(Float64(h / Float64(l * d)) * Float64(Float64(D * D) / d)) * M) * Float64(-0.125 * M)) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_1 = abs(d) / sqrt((l * h));
	t_2 = 1.0 * t_1;
	tmp = 0.0;
	if (t_0 <= -5e-111)
		tmp = (((M * M) * -0.125) * ((D / d) * (((D / l) * h) / d))) * t_1;
	elseif (t_0 <= 5e-229)
		tmp = t_2;
	elseif (t_0 <= 1e+291)
		tmp = sqrt((d / l)) * sqrt((d / h));
	elseif (t_0 <= Inf)
		tmp = t_2;
	else
		tmp = ((((h / (l * d)) * ((D * D) / d)) * M) * (-0.125 * M)) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-111], N[(N[(N[(N[(M * M), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[(N[(N[(D / l), $MachinePrecision] * h), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 5e-229], t$95$2, If[LessEqual[t$95$0, 1e+291], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], t$95$2, N[(N[(N[(N[(N[(h / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(N[(D * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision] * N[(-0.125 * M), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5.0000000000000003e-111

   1. Initial program 90.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 49.5

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

   4. Applied rewrites 49.5%

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

   6. Applied rewrites 85.2%

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

   7. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 60.4

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

   9. Applied rewrites 60.4%

      \[\leadsto \color{blue}{\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{h}{d \cdot d}\right)\right)} \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]
   10. Step-by-step derivation

   11. Applied rewrites 76.0%

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

   if -5.0000000000000003e-111 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.00000000000000016e-229 or 9.9999999999999996e290 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 44.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 40.1

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

   4. Applied rewrites 40.1%

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

   6. Applied rewrites 88.1%

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

   7. Taylor expanded in d around inf

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

   9. Applied rewrites 87.2%

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

   if 5.00000000000000016e-229 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.9999999999999996e290

   1. Initial program 99.2%

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 39.5

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 39.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 39.6%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

   9. Applied rewrites 98.3%

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

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

   1. Initial program 0.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 6.5

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

   4. Applied rewrites 6.5%

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

   6. Applied rewrites 23.7%

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

   7. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 19.5

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

   9. Applied rewrites 19.5%

      \[\leadsto \color{blue}{\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{h}{d \cdot d}\right)\right)} \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]
   10. Step-by-step derivation

   11. Applied rewrites 29.4%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-111}:\\ \;\;\;\;\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \left(\frac{D}{d} \cdot \frac{\frac{D}{\ell} \cdot h}{d}\right)\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{-229}:\\ \;\;\;\;1 \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 10^{+291}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;1 \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{h}{\ell \cdot d} \cdot \frac{D \cdot D}{d}\right) \cdot M\right) \cdot \left(-0.125 \cdot M\right)\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ t_2 := 1 \cdot t\_1\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-111}:\\ \;\;\;\;\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \left(\frac{D}{\ell \cdot d} \cdot \frac{D \cdot h}{d}\right)\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-229}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 10^{+291}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{h}{\ell \cdot d} \cdot \frac{D \cdot D}{d}\right) \cdot M\right) \cdot \left(-0.125 \cdot M\right)\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (/ (fabs d) (sqrt (* l h))))
        (t_2 (* 1.0 t_1)))
   (if (<= t_0 -5e-111)
     (* (* (* (* M M) -0.125) (* (/ D (* l d)) (/ (* D h) d))) t_1)
     (if (<= t_0 5e-229)
       t_2
       (if (<= t_0 1e+291)
         (* (sqrt (/ d l)) (sqrt (/ d h)))
         (if (<= t_0 INFINITY)
           t_2
           (* (* (* (* (/ h (* l d)) (/ (* D D) d)) M) (* -0.125 M)) t_1)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = fabs(d) / sqrt((l * h));
	double t_2 = 1.0 * t_1;
	double tmp;
	if (t_0 <= -5e-111) {
		tmp = (((M * M) * -0.125) * ((D / (l * d)) * ((D * h) / d))) * t_1;
	} else if (t_0 <= 5e-229) {
		tmp = t_2;
	} else if (t_0 <= 1e+291) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = ((((h / (l * d)) * ((D * D) / d)) * M) * (-0.125 * M)) * t_1;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = Math.abs(d) / Math.sqrt((l * h));
	double t_2 = 1.0 * t_1;
	double tmp;
	if (t_0 <= -5e-111) {
		tmp = (((M * M) * -0.125) * ((D / (l * d)) * ((D * h) / d))) * t_1;
	} else if (t_0 <= 5e-229) {
		tmp = t_2;
	} else if (t_0 <= 1e+291) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else {
		tmp = ((((h / (l * d)) * ((D * D) / d)) * M) * (-0.125 * M)) * t_1;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_1 = math.fabs(d) / math.sqrt((l * h))
	t_2 = 1.0 * t_1
	tmp = 0
	if t_0 <= -5e-111:
		tmp = (((M * M) * -0.125) * ((D / (l * d)) * ((D * h) / d))) * t_1
	elif t_0 <= 5e-229:
		tmp = t_2
	elif t_0 <= 1e+291:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	elif t_0 <= math.inf:
		tmp = t_2
	else:
		tmp = ((((h / (l * d)) * ((D * D) / d)) * M) * (-0.125 * M)) * t_1
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(abs(d) / sqrt(Float64(l * h)))
	t_2 = Float64(1.0 * t_1)
	tmp = 0.0
	if (t_0 <= -5e-111)
		tmp = Float64(Float64(Float64(Float64(M * M) * -0.125) * Float64(Float64(D / Float64(l * d)) * Float64(Float64(D * h) / d))) * t_1);
	elseif (t_0 <= 5e-229)
		tmp = t_2;
	elseif (t_0 <= 1e+291)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (t_0 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(Float64(Float64(Float64(Float64(h / Float64(l * d)) * Float64(Float64(D * D) / d)) * M) * Float64(-0.125 * M)) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_1 = abs(d) / sqrt((l * h));
	t_2 = 1.0 * t_1;
	tmp = 0.0;
	if (t_0 <= -5e-111)
		tmp = (((M * M) * -0.125) * ((D / (l * d)) * ((D * h) / d))) * t_1;
	elseif (t_0 <= 5e-229)
		tmp = t_2;
	elseif (t_0 <= 1e+291)
		tmp = sqrt((d / l)) * sqrt((d / h));
	elseif (t_0 <= Inf)
		tmp = t_2;
	else
		tmp = ((((h / (l * d)) * ((D * D) / d)) * M) * (-0.125 * M)) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-111], N[(N[(N[(N[(M * M), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(D / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(N[(D * h), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 5e-229], t$95$2, If[LessEqual[t$95$0, 1e+291], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], t$95$2, N[(N[(N[(N[(N[(h / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(N[(D * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision] * N[(-0.125 * M), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5.0000000000000003e-111

   1. Initial program 90.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 49.5

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

   4. Applied rewrites 49.5%

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

   6. Applied rewrites 85.2%

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

   7. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 60.4

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

   9. Applied rewrites 60.4%

      \[\leadsto \color{blue}{\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{h}{d \cdot d}\right)\right)} \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]
   10. Step-by-step derivation

   11. Applied rewrites 73.5%

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

   if -5.0000000000000003e-111 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.00000000000000016e-229 or 9.9999999999999996e290 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 44.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 40.1

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

   4. Applied rewrites 40.1%

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

   6. Applied rewrites 88.1%

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

   7. Taylor expanded in d around inf

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

   9. Applied rewrites 87.2%

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

   if 5.00000000000000016e-229 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.9999999999999996e290

   1. Initial program 99.2%

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 39.5

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 39.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 39.6%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

   9. Applied rewrites 98.3%

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

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

   1. Initial program 0.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 6.5

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

   4. Applied rewrites 6.5%

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

   6. Applied rewrites 23.7%

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

   7. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 19.5

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

   9. Applied rewrites 19.5%

      \[\leadsto \color{blue}{\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{h}{d \cdot d}\right)\right)} \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]
   10. Step-by-step derivation

   11. Applied rewrites 29.4%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-111}:\\ \;\;\;\;\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \left(\frac{D}{\ell \cdot d} \cdot \frac{D \cdot h}{d}\right)\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{-229}:\\ \;\;\;\;1 \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 10^{+291}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;1 \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{h}{\ell \cdot d} \cdot \frac{D \cdot D}{d}\right) \cdot M\right) \cdot \left(-0.125 \cdot M\right)\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ t_2 := 1 \cdot t\_1\\ t_3 := \left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \left(\frac{D}{\ell \cdot d} \cdot \frac{D \cdot h}{d}\right)\right) \cdot t\_1\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-111}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-229}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 10^{+291}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (/ (fabs d) (sqrt (* l h))))
        (t_2 (* 1.0 t_1))
        (t_3 (* (* (* (* M M) -0.125) (* (/ D (* l d)) (/ (* D h) d))) t_1)))
   (if (<= t_0 -5e-111)
     t_3
     (if (<= t_0 5e-229)
       t_2
       (if (<= t_0 1e+291)
         (* (sqrt (/ d l)) (sqrt (/ d h)))
         (if (<= t_0 INFINITY) t_2 t_3))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = fabs(d) / sqrt((l * h));
	double t_2 = 1.0 * t_1;
	double t_3 = (((M * M) * -0.125) * ((D / (l * d)) * ((D * h) / d))) * t_1;
	double tmp;
	if (t_0 <= -5e-111) {
		tmp = t_3;
	} else if (t_0 <= 5e-229) {
		tmp = t_2;
	} else if (t_0 <= 1e+291) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = Math.abs(d) / Math.sqrt((l * h));
	double t_2 = 1.0 * t_1;
	double t_3 = (((M * M) * -0.125) * ((D / (l * d)) * ((D * h) / d))) * t_1;
	double tmp;
	if (t_0 <= -5e-111) {
		tmp = t_3;
	} else if (t_0 <= 5e-229) {
		tmp = t_2;
	} else if (t_0 <= 1e+291) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_1 = math.fabs(d) / math.sqrt((l * h))
	t_2 = 1.0 * t_1
	t_3 = (((M * M) * -0.125) * ((D / (l * d)) * ((D * h) / d))) * t_1
	tmp = 0
	if t_0 <= -5e-111:
		tmp = t_3
	elif t_0 <= 5e-229:
		tmp = t_2
	elif t_0 <= 1e+291:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	elif t_0 <= math.inf:
		tmp = t_2
	else:
		tmp = t_3
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(abs(d) / sqrt(Float64(l * h)))
	t_2 = Float64(1.0 * t_1)
	t_3 = Float64(Float64(Float64(Float64(M * M) * -0.125) * Float64(Float64(D / Float64(l * d)) * Float64(Float64(D * h) / d))) * t_1)
	tmp = 0.0
	if (t_0 <= -5e-111)
		tmp = t_3;
	elseif (t_0 <= 5e-229)
		tmp = t_2;
	elseif (t_0 <= 1e+291)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (t_0 <= Inf)
		tmp = t_2;
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_1 = abs(d) / sqrt((l * h));
	t_2 = 1.0 * t_1;
	t_3 = (((M * M) * -0.125) * ((D / (l * d)) * ((D * h) / d))) * t_1;
	tmp = 0.0;
	if (t_0 <= -5e-111)
		tmp = t_3;
	elseif (t_0 <= 5e-229)
		tmp = t_2;
	elseif (t_0 <= 1e+291)
		tmp = sqrt((d / l)) * sqrt((d / h));
	elseif (t_0 <= Inf)
		tmp = t_2;
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(M * M), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(D / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(N[(D * h), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-111], t$95$3, If[LessEqual[t$95$0, 5e-229], t$95$2, If[LessEqual[t$95$0, 1e+291], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], t$95$2, t$95$3]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5.0000000000000003e-111 or +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 58.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 34.2

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

   4. Applied rewrites 34.2%

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

   6. Applied rewrites 63.3%

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

   7. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 45.9

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

   9. Applied rewrites 45.9%

      \[\leadsto \color{blue}{\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{h}{d \cdot d}\right)\right)} \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]
   10. Step-by-step derivation

   11. Applied rewrites 56.8%

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

   if -5.0000000000000003e-111 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.00000000000000016e-229 or 9.9999999999999996e290 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 44.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 40.1

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

   4. Applied rewrites 40.1%

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

   6. Applied rewrites 88.1%

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

   7. Taylor expanded in d around inf

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

   9. Applied rewrites 87.2%

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

   if 5.00000000000000016e-229 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.9999999999999996e290

   1. Initial program 99.2%

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 39.5

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 39.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 39.6%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

   9. Applied rewrites 98.3%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-111}:\\ \;\;\;\;\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \left(\frac{D}{\ell \cdot d} \cdot \frac{D \cdot h}{d}\right)\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{-229}:\\ \;\;\;\;1 \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 10^{+291}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;1 \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \left(\frac{D}{\ell \cdot d} \cdot \frac{D \cdot h}{d}\right)\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ t_2 := 1 \cdot t\_1\\ t_3 := \frac{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(\left(-0.125 \cdot M\right) \cdot M\right)}{\left(\ell \cdot d\right) \cdot d} \cdot t\_1\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-111}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-229}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 10^{+291}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (/ (fabs d) (sqrt (* l h))))
        (t_2 (* 1.0 t_1))
        (t_3 (* (/ (* (* (* D D) h) (* (* -0.125 M) M)) (* (* l d) d)) t_1)))
   (if (<= t_0 -5e-111)
     t_3
     (if (<= t_0 5e-229)
       t_2
       (if (<= t_0 1e+291)
         (* (sqrt (/ d l)) (sqrt (/ d h)))
         (if (<= t_0 INFINITY) t_2 t_3))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = fabs(d) / sqrt((l * h));
	double t_2 = 1.0 * t_1;
	double t_3 = ((((D * D) * h) * ((-0.125 * M) * M)) / ((l * d) * d)) * t_1;
	double tmp;
	if (t_0 <= -5e-111) {
		tmp = t_3;
	} else if (t_0 <= 5e-229) {
		tmp = t_2;
	} else if (t_0 <= 1e+291) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = Math.abs(d) / Math.sqrt((l * h));
	double t_2 = 1.0 * t_1;
	double t_3 = ((((D * D) * h) * ((-0.125 * M) * M)) / ((l * d) * d)) * t_1;
	double tmp;
	if (t_0 <= -5e-111) {
		tmp = t_3;
	} else if (t_0 <= 5e-229) {
		tmp = t_2;
	} else if (t_0 <= 1e+291) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_1 = math.fabs(d) / math.sqrt((l * h))
	t_2 = 1.0 * t_1
	t_3 = ((((D * D) * h) * ((-0.125 * M) * M)) / ((l * d) * d)) * t_1
	tmp = 0
	if t_0 <= -5e-111:
		tmp = t_3
	elif t_0 <= 5e-229:
		tmp = t_2
	elif t_0 <= 1e+291:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	elif t_0 <= math.inf:
		tmp = t_2
	else:
		tmp = t_3
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(abs(d) / sqrt(Float64(l * h)))
	t_2 = Float64(1.0 * t_1)
	t_3 = Float64(Float64(Float64(Float64(Float64(D * D) * h) * Float64(Float64(-0.125 * M) * M)) / Float64(Float64(l * d) * d)) * t_1)
	tmp = 0.0
	if (t_0 <= -5e-111)
		tmp = t_3;
	elseif (t_0 <= 5e-229)
		tmp = t_2;
	elseif (t_0 <= 1e+291)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (t_0 <= Inf)
		tmp = t_2;
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_1 = abs(d) / sqrt((l * h));
	t_2 = 1.0 * t_1;
	t_3 = ((((D * D) * h) * ((-0.125 * M) * M)) / ((l * d) * d)) * t_1;
	tmp = 0.0;
	if (t_0 <= -5e-111)
		tmp = t_3;
	elseif (t_0 <= 5e-229)
		tmp = t_2;
	elseif (t_0 <= 1e+291)
		tmp = sqrt((d / l)) * sqrt((d / h));
	elseif (t_0 <= Inf)
		tmp = t_2;
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(D * D), $MachinePrecision] * h), $MachinePrecision] * N[(N[(-0.125 * M), $MachinePrecision] * M), $MachinePrecision]), $MachinePrecision] / N[(N[(l * d), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-111], t$95$3, If[LessEqual[t$95$0, 5e-229], t$95$2, If[LessEqual[t$95$0, 1e+291], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], t$95$2, t$95$3]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5.0000000000000003e-111 or +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 58.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 34.2

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

   4. Applied rewrites 34.2%

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

   6. Applied rewrites 63.3%

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

   7. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 45.9

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

   9. Applied rewrites 45.9%

      \[\leadsto \color{blue}{\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{h}{d \cdot d}\right)\right)} \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]
   10. Step-by-step derivation

   11. Applied rewrites 50.0%

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

   if -5.0000000000000003e-111 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.00000000000000016e-229 or 9.9999999999999996e290 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 44.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 40.1

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

   4. Applied rewrites 40.1%

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

   6. Applied rewrites 88.1%

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

   7. Taylor expanded in d around inf

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

   9. Applied rewrites 87.2%

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

   if 5.00000000000000016e-229 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.9999999999999996e290

   1. Initial program 99.2%

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 39.5

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 39.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 39.6%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

   9. Applied rewrites 98.3%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-111}:\\ \;\;\;\;\frac{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(\left(-0.125 \cdot M\right) \cdot M\right)}{\left(\ell \cdot d\right) \cdot d} \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{-229}:\\ \;\;\;\;1 \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 10^{+291}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;1 \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(\left(-0.125 \cdot M\right) \cdot M\right)}{\left(\ell \cdot d\right) \cdot d} \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ t_2 := 1 \cdot t\_1\\ t_3 := \left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \left(\left(D \cdot D\right) \cdot \frac{h}{\left(\ell \cdot d\right) \cdot d}\right)\right) \cdot t\_1\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-111}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-229}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 10^{+291}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (/ (fabs d) (sqrt (* l h))))
        (t_2 (* 1.0 t_1))
        (t_3 (* (* (* (* M M) -0.125) (* (* D D) (/ h (* (* l d) d)))) t_1)))
   (if (<= t_0 -5e-111)
     t_3
     (if (<= t_0 5e-229)
       t_2
       (if (<= t_0 1e+291)
         (* (sqrt (/ d l)) (sqrt (/ d h)))
         (if (<= t_0 INFINITY) t_2 t_3))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = fabs(d) / sqrt((l * h));
	double t_2 = 1.0 * t_1;
	double t_3 = (((M * M) * -0.125) * ((D * D) * (h / ((l * d) * d)))) * t_1;
	double tmp;
	if (t_0 <= -5e-111) {
		tmp = t_3;
	} else if (t_0 <= 5e-229) {
		tmp = t_2;
	} else if (t_0 <= 1e+291) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = Math.abs(d) / Math.sqrt((l * h));
	double t_2 = 1.0 * t_1;
	double t_3 = (((M * M) * -0.125) * ((D * D) * (h / ((l * d) * d)))) * t_1;
	double tmp;
	if (t_0 <= -5e-111) {
		tmp = t_3;
	} else if (t_0 <= 5e-229) {
		tmp = t_2;
	} else if (t_0 <= 1e+291) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_1 = math.fabs(d) / math.sqrt((l * h))
	t_2 = 1.0 * t_1
	t_3 = (((M * M) * -0.125) * ((D * D) * (h / ((l * d) * d)))) * t_1
	tmp = 0
	if t_0 <= -5e-111:
		tmp = t_3
	elif t_0 <= 5e-229:
		tmp = t_2
	elif t_0 <= 1e+291:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	elif t_0 <= math.inf:
		tmp = t_2
	else:
		tmp = t_3
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(abs(d) / sqrt(Float64(l * h)))
	t_2 = Float64(1.0 * t_1)
	t_3 = Float64(Float64(Float64(Float64(M * M) * -0.125) * Float64(Float64(D * D) * Float64(h / Float64(Float64(l * d) * d)))) * t_1)
	tmp = 0.0
	if (t_0 <= -5e-111)
		tmp = t_3;
	elseif (t_0 <= 5e-229)
		tmp = t_2;
	elseif (t_0 <= 1e+291)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (t_0 <= Inf)
		tmp = t_2;
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_1 = abs(d) / sqrt((l * h));
	t_2 = 1.0 * t_1;
	t_3 = (((M * M) * -0.125) * ((D * D) * (h / ((l * d) * d)))) * t_1;
	tmp = 0.0;
	if (t_0 <= -5e-111)
		tmp = t_3;
	elseif (t_0 <= 5e-229)
		tmp = t_2;
	elseif (t_0 <= 1e+291)
		tmp = sqrt((d / l)) * sqrt((d / h));
	elseif (t_0 <= Inf)
		tmp = t_2;
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(M * M), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(D * D), $MachinePrecision] * N[(h / N[(N[(l * d), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-111], t$95$3, If[LessEqual[t$95$0, 5e-229], t$95$2, If[LessEqual[t$95$0, 1e+291], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], t$95$2, t$95$3]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5.0000000000000003e-111 or +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 58.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 34.2

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

   4. Applied rewrites 34.2%

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

   6. Applied rewrites 63.3%

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

   7. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 45.9

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

   9. Applied rewrites 45.9%

      \[\leadsto \color{blue}{\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{h}{d \cdot d}\right)\right)} \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]
   10. Step-by-step derivation

   11. Applied rewrites 49.8%

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

   if -5.0000000000000003e-111 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.00000000000000016e-229 or 9.9999999999999996e290 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 44.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 40.1

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

   4. Applied rewrites 40.1%

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

   6. Applied rewrites 88.1%

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

   7. Taylor expanded in d around inf

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

   9. Applied rewrites 87.2%

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

   if 5.00000000000000016e-229 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.9999999999999996e290

   1. Initial program 99.2%

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 39.5

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 39.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 39.6%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

   9. Applied rewrites 98.3%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-111}:\\ \;\;\;\;\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \left(\left(D \cdot D\right) \cdot \frac{h}{\left(\ell \cdot d\right) \cdot d}\right)\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{-229}:\\ \;\;\;\;1 \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 10^{+291}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;1 \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \left(\left(D \cdot D\right) \cdot \frac{h}{\left(\ell \cdot d\right) \cdot d}\right)\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{\ell \cdot h}\\ t_2 := 1 \cdot \frac{\left|d\right|}{t\_1}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-111}:\\ \;\;\;\;\frac{\left(\left(D \cdot D\right) \cdot -0.125\right) \cdot \left(\frac{h}{d} \cdot \frac{M \cdot M}{\ell}\right)}{t\_1}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-229}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 10^{+291}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-d\right) \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (sqrt (* l h)))
        (t_2 (* 1.0 (/ (fabs d) t_1))))
   (if (<= t_0 -5e-111)
     (/ (* (* (* D D) -0.125) (* (/ h d) (/ (* M M) l))) t_1)
     (if (<= t_0 5e-229)
       t_2
       (if (<= t_0 1e+291)
         (* (sqrt (/ d l)) (sqrt (/ d h)))
         (if (<= t_0 INFINITY) t_2 (/ (* (- d) (sqrt (/ h l))) h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = sqrt((l * h));
	double t_2 = 1.0 * (fabs(d) / t_1);
	double tmp;
	if (t_0 <= -5e-111) {
		tmp = (((D * D) * -0.125) * ((h / d) * ((M * M) / l))) / t_1;
	} else if (t_0 <= 5e-229) {
		tmp = t_2;
	} else if (t_0 <= 1e+291) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = (-d * sqrt((h / l))) / h;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = Math.sqrt((l * h));
	double t_2 = 1.0 * (Math.abs(d) / t_1);
	double tmp;
	if (t_0 <= -5e-111) {
		tmp = (((D * D) * -0.125) * ((h / d) * ((M * M) / l))) / t_1;
	} else if (t_0 <= 5e-229) {
		tmp = t_2;
	} else if (t_0 <= 1e+291) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else {
		tmp = (-d * Math.sqrt((h / l))) / h;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_1 = math.sqrt((l * h))
	t_2 = 1.0 * (math.fabs(d) / t_1)
	tmp = 0
	if t_0 <= -5e-111:
		tmp = (((D * D) * -0.125) * ((h / d) * ((M * M) / l))) / t_1
	elif t_0 <= 5e-229:
		tmp = t_2
	elif t_0 <= 1e+291:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	elif t_0 <= math.inf:
		tmp = t_2
	else:
		tmp = (-d * math.sqrt((h / l))) / h
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = sqrt(Float64(l * h))
	t_2 = Float64(1.0 * Float64(abs(d) / t_1))
	tmp = 0.0
	if (t_0 <= -5e-111)
		tmp = Float64(Float64(Float64(Float64(D * D) * -0.125) * Float64(Float64(h / d) * Float64(Float64(M * M) / l))) / t_1);
	elseif (t_0 <= 5e-229)
		tmp = t_2;
	elseif (t_0 <= 1e+291)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (t_0 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(Float64(Float64(-d) * sqrt(Float64(h / l))) / h);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_1 = sqrt((l * h));
	t_2 = 1.0 * (abs(d) / t_1);
	tmp = 0.0;
	if (t_0 <= -5e-111)
		tmp = (((D * D) * -0.125) * ((h / d) * ((M * M) / l))) / t_1;
	elseif (t_0 <= 5e-229)
		tmp = t_2;
	elseif (t_0 <= 1e+291)
		tmp = sqrt((d / l)) * sqrt((d / h));
	elseif (t_0 <= Inf)
		tmp = t_2;
	else
		tmp = (-d * sqrt((h / l))) / h;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 * N[(N[Abs[d], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-111], N[(N[(N[(N[(D * D), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(h / d), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 5e-229], t$95$2, If[LessEqual[t$95$0, 1e+291], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], t$95$2, N[(N[((-d) * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5.0000000000000003e-111

   1. Initial program 90.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 49.5

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

   4. Applied rewrites 49.5%

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

   6. Applied rewrites 85.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   8. Applied rewrites 47.4%

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

   9. Taylor expanded in d around 0

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

       1. *-commutative N/A

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

       2. associate-/l* N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. *-commutative N/A

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

       11. times-frac N/A

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

       12. lower-*.f64 N/A

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

       13. lower-/.f64 N/A

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

       14. lower-/.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 40.3

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

   11. Applied rewrites 40.3%

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

   if -5.0000000000000003e-111 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.00000000000000016e-229 or 9.9999999999999996e290 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 44.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 40.1

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

   4. Applied rewrites 40.1%

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

   6. Applied rewrites 88.1%

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

   7. Taylor expanded in d around inf

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

   9. Applied rewrites 87.2%

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

   if 5.00000000000000016e-229 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.9999999999999996e290

   1. Initial program 99.2%

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 39.5

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 39.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 39.6%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

   9. Applied rewrites 98.3%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   5. Applied rewrites 24.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(-0.125 \cdot \left(M \cdot M\right)\right) \cdot \frac{D \cdot D}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   6. Taylor expanded in l around -inf

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

   8. Applied rewrites 21.2%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-111}:\\ \;\;\;\;\frac{\left(\left(D \cdot D\right) \cdot -0.125\right) \cdot \left(\frac{h}{d} \cdot \frac{M \cdot M}{\ell}\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{-229}:\\ \;\;\;\;1 \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 10^{+291}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;1 \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-d\right) \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 54.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \frac{\left(-d\right) \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-111}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-229}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+291}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* 1.0 (/ (fabs d) (sqrt (* l h)))))
        (t_1
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_2 (/ (* (- d) (sqrt (/ h l))) h)))
   (if (<= t_1 -5e-111)
     t_2
     (if (<= t_1 5e-229)
       t_0
       (if (<= t_1 1e+291)
         (* (sqrt (/ d l)) (sqrt (/ d h)))
         (if (<= t_1 INFINITY) t_0 t_2))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 * (fabs(d) / sqrt((l * h)));
	double t_1 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = (-d * sqrt((h / l))) / h;
	double tmp;
	if (t_1 <= -5e-111) {
		tmp = t_2;
	} else if (t_1 <= 5e-229) {
		tmp = t_0;
	} else if (t_1 <= 1e+291) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 * (Math.abs(d) / Math.sqrt((l * h)));
	double t_1 = (Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = (-d * Math.sqrt((h / l))) / h;
	double tmp;
	if (t_1 <= -5e-111) {
		tmp = t_2;
	} else if (t_1 <= 5e-229) {
		tmp = t_0;
	} else if (t_1 <= 1e+291) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = 1.0 * (math.fabs(d) / math.sqrt((l * h)))
	t_1 = (math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_2 = (-d * math.sqrt((h / l))) / h
	tmp = 0
	if t_1 <= -5e-111:
		tmp = t_2
	elif t_1 <= 5e-229:
		tmp = t_0
	elif t_1 <= 1e+291:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	elif t_1 <= math.inf:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(1.0 * Float64(abs(d) / sqrt(Float64(l * h))))
	t_1 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_2 = Float64(Float64(Float64(-d) * sqrt(Float64(h / l))) / h)
	tmp = 0.0
	if (t_1 <= -5e-111)
		tmp = t_2;
	elseif (t_1 <= 5e-229)
		tmp = t_0;
	elseif (t_1 <= 1e+291)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (t_1 <= Inf)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = 1.0 * (abs(d) / sqrt((l * h)));
	t_1 = (((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_2 = (-d * sqrt((h / l))) / h;
	tmp = 0.0;
	if (t_1 <= -5e-111)
		tmp = t_2;
	elseif (t_1 <= 5e-229)
		tmp = t_0;
	elseif (t_1 <= 1e+291)
		tmp = sqrt((d / l)) * sqrt((d / h));
	elseif (t_1 <= Inf)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(1.0 * N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-d) * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-111], t$95$2, If[LessEqual[t$95$1, 5e-229], t$95$0, If[LessEqual[t$95$1, 1e+291], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$0, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5.0000000000000003e-111 or +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 58.5%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   5. Applied rewrites 40.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(-0.125 \cdot \left(M \cdot M\right)\right) \cdot \frac{D \cdot D}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   6. Taylor expanded in l around -inf

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

   8. Applied rewrites 27.1%

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

   if -5.0000000000000003e-111 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.00000000000000016e-229 or 9.9999999999999996e290 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 44.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 40.1

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

   4. Applied rewrites 40.1%

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

   6. Applied rewrites 88.1%

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

   7. Taylor expanded in d around inf

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

   9. Applied rewrites 87.2%

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

   if 5.00000000000000016e-229 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.9999999999999996e290

   1. Initial program 99.2%

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 39.5

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 39.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 39.6%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

   9. Applied rewrites 98.3%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-111}:\\ \;\;\;\;\frac{\left(-d\right) \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{-229}:\\ \;\;\;\;1 \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 10^{+291}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;1 \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-d\right) \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 52.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \frac{\left(-d\right) \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-111}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-156}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+98}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* 1.0 (/ (fabs d) (sqrt (* l h)))))
        (t_1
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_2 (/ (* (- d) (sqrt (/ h l))) h)))
   (if (<= t_1 -5e-111)
     t_2
     (if (<= t_1 2e-156)
       t_0
       (if (<= t_1 1e+98)
         (sqrt (* (/ d l) (/ d h)))
         (if (<= t_1 INFINITY) t_0 t_2))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 * (fabs(d) / sqrt((l * h)));
	double t_1 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = (-d * sqrt((h / l))) / h;
	double tmp;
	if (t_1 <= -5e-111) {
		tmp = t_2;
	} else if (t_1 <= 2e-156) {
		tmp = t_0;
	} else if (t_1 <= 1e+98) {
		tmp = sqrt(((d / l) * (d / h)));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 * (Math.abs(d) / Math.sqrt((l * h)));
	double t_1 = (Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = (-d * Math.sqrt((h / l))) / h;
	double tmp;
	if (t_1 <= -5e-111) {
		tmp = t_2;
	} else if (t_1 <= 2e-156) {
		tmp = t_0;
	} else if (t_1 <= 1e+98) {
		tmp = Math.sqrt(((d / l) * (d / h)));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = 1.0 * (math.fabs(d) / math.sqrt((l * h)))
	t_1 = (math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_2 = (-d * math.sqrt((h / l))) / h
	tmp = 0
	if t_1 <= -5e-111:
		tmp = t_2
	elif t_1 <= 2e-156:
		tmp = t_0
	elif t_1 <= 1e+98:
		tmp = math.sqrt(((d / l) * (d / h)))
	elif t_1 <= math.inf:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(1.0 * Float64(abs(d) / sqrt(Float64(l * h))))
	t_1 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_2 = Float64(Float64(Float64(-d) * sqrt(Float64(h / l))) / h)
	tmp = 0.0
	if (t_1 <= -5e-111)
		tmp = t_2;
	elseif (t_1 <= 2e-156)
		tmp = t_0;
	elseif (t_1 <= 1e+98)
		tmp = sqrt(Float64(Float64(d / l) * Float64(d / h)));
	elseif (t_1 <= Inf)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = 1.0 * (abs(d) / sqrt((l * h)));
	t_1 = (((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_2 = (-d * sqrt((h / l))) / h;
	tmp = 0.0;
	if (t_1 <= -5e-111)
		tmp = t_2;
	elseif (t_1 <= 2e-156)
		tmp = t_0;
	elseif (t_1 <= 1e+98)
		tmp = sqrt(((d / l) * (d / h)));
	elseif (t_1 <= Inf)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(1.0 * N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-d) * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-111], t$95$2, If[LessEqual[t$95$1, 2e-156], t$95$0, If[LessEqual[t$95$1, 1e+98], N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$0, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5.0000000000000003e-111 or +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 58.5%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   5. Applied rewrites 40.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(-0.125 \cdot \left(M \cdot M\right)\right) \cdot \frac{D \cdot D}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   6. Taylor expanded in l around -inf

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

   8. Applied rewrites 27.1%

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

   if -5.0000000000000003e-111 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.00000000000000008e-156 or 9.99999999999999998e97 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 61.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 39.1

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

   4. Applied rewrites 39.1%

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

   6. Applied rewrites 87.4%

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

   7. Taylor expanded in d around inf

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

   9. Applied rewrites 86.8%

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

   if 2.00000000000000008e-156 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.99999999999999998e97

   1. Initial program 99.1%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 43.1

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 43.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 43.1%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

   9. Applied rewrites 98.0%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-111}:\\ \;\;\;\;\frac{\left(-d\right) \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2 \cdot 10^{-156}:\\ \;\;\;\;1 \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 10^{+98}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;1 \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-d\right) \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 49.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-83}:\\ \;\;\;\;\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-156} \lor \neg \left(t\_0 \leq 10^{+98}\right):\\ \;\;\;\;1 \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_0 -1e-83)
     (* (sqrt (pow (* l h) -1.0)) d)
     (if (or (<= t_0 2e-156) (not (<= t_0 1e+98)))
       (* 1.0 (/ (fabs d) (sqrt (* l h))))
       (sqrt (* (/ d l) (/ d h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -1e-83) {
		tmp = sqrt(pow((l * h), -1.0)) * d;
	} else if ((t_0 <= 2e-156) || !(t_0 <= 1e+98)) {
		tmp = 1.0 * (fabs(d) / sqrt((l * h)));
	} else {
		tmp = sqrt(((d / l) * (d / 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 = (((d / h) ** (2.0d0 ** (-1.0d0))) * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - (((2.0d0 ** (-1.0d0)) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_0 <= (-1d-83)) then
        tmp = sqrt(((l * h) ** (-1.0d0))) * d
    else if ((t_0 <= 2d-156) .or. (.not. (t_0 <= 1d+98))) then
        tmp = 1.0d0 * (abs(d) / sqrt((l * h)))
    else
        tmp = sqrt(((d / l) * (d / 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.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -1e-83) {
		tmp = Math.sqrt(Math.pow((l * h), -1.0)) * d;
	} else if ((t_0 <= 2e-156) || !(t_0 <= 1e+98)) {
		tmp = 1.0 * (Math.abs(d) / Math.sqrt((l * h)));
	} else {
		tmp = Math.sqrt(((d / l) * (d / h)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_0 <= -1e-83:
		tmp = math.sqrt(math.pow((l * h), -1.0)) * d
	elif (t_0 <= 2e-156) or not (t_0 <= 1e+98):
		tmp = 1.0 * (math.fabs(d) / math.sqrt((l * h)))
	else:
		tmp = math.sqrt(((d / l) * (d / h)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= -1e-83)
		tmp = Float64(sqrt((Float64(l * h) ^ -1.0)) * d);
	elseif ((t_0 <= 2e-156) || !(t_0 <= 1e+98))
		tmp = Float64(1.0 * Float64(abs(d) / sqrt(Float64(l * h))));
	else
		tmp = sqrt(Float64(Float64(d / l) * Float64(d / h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_0 <= -1e-83)
		tmp = sqrt(((l * h) ^ -1.0)) * d;
	elseif ((t_0 <= 2e-156) || ~((t_0 <= 1e+98)))
		tmp = 1.0 * (abs(d) / sqrt((l * h)));
	else
		tmp = sqrt(((d / l) * (d / h)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-83], N[(N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision], If[Or[LessEqual[t$95$0, 2e-156], N[Not[LessEqual[t$95$0, 1e+98]], $MachinePrecision]], N[(1.0 * N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1e-83

   1. Initial program 90.8%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 12.9

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 12.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

   if -1e-83 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.00000000000000008e-156 or 9.99999999999999998e97 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 40.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 28.1

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

   4. Applied rewrites 28.1%

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

   6. Applied rewrites 65.2%

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

   7. Taylor expanded in d around inf

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

   9. Applied rewrites 60.1%

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

   if 2.00000000000000008e-156 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.99999999999999998e97

   1. Initial program 99.1%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 43.1

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 43.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 43.1%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

   9. Applied rewrites 98.0%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -1 \cdot 10^{-83}:\\ \;\;\;\;\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2 \cdot 10^{-156} \lor \neg \left(\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 10^{+98}\right):\\ \;\;\;\;1 \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 77.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{D}{d}}{-2} \cdot M\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq 0 \lor \neg \left(t\_1 \leq 10^{+291}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{h}{\ell} \cdot \left(t\_0 \cdot -0.5\right), t\_0, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ (/ D d) -2.0) M))
        (t_1
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (or (<= t_1 0.0) (not (<= t_1 1e+291)))
     (* (fma (* (/ h l) (* t_0 -0.5)) t_0 1.0) (/ (fabs d) (sqrt (* l h))))
     (* (sqrt (/ d l)) (sqrt (/ d 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 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if ((t_1 <= 0.0) || !(t_1 <= 1e+291)) {
		tmp = fma(((h / l) * (t_0 * -0.5)), t_0, 1.0) * (fabs(d) / sqrt((l * h)));
	} else {
		tmp = sqrt((d / l)) * sqrt((d / h));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(Float64(D / d) / -2.0) * M)
	t_1 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if ((t_1 <= 0.0) || !(t_1 <= 1e+291))
		tmp = Float64(fma(Float64(Float64(h / l) * Float64(t_0 * -0.5)), t_0, 1.0) * Float64(abs(d) / sqrt(Float64(l * h))));
	else
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / 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[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, 0.0], N[Not[LessEqual[t$95$1, 1e+291]], $MachinePrecision]], N[(N[(N[(N[(h / l), $MachinePrecision] * N[(t$95$0 * -0.5), $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 9.9999999999999996e290 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 53.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 35.9

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

   4. Applied rewrites 35.9%

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

   6. Applied rewrites 70.7%

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

      1. unpow1 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. pow-prod-down N/A

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

      5. pow-sqr N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/l/ N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. metadata-eval N/A

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

      13. unpow2 N/A

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

      14. associate-/r* N/A

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

      15. associate-/r* N/A

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

      16. frac-times N/A

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

      17. sqr-neg-rev N/A

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

      18. times-frac N/A

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

      19. lower-*.f64 N/A

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

   8. Applied rewrites 70.9%

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

   10. Applied rewrites 74.9%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.9999999999999996e290

   1. Initial program 99.1%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 39.9

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 39.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 40.0%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

   9. Applied rewrites 98.1%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0 \lor \neg \left(\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 10^{+291}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{h}{\ell} \cdot \left(\left(\frac{\frac{D}{d}}{-2} \cdot M\right) \cdot -0.5\right), \frac{\frac{D}{d}}{-2} \cdot M, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 75.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{\frac{D}{d}}{2} \cdot M\\ t_2 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-229}:\\ \;\;\;\;\mathsf{fma}\left(\left(--0.5\right) \cdot \left(\frac{M \cdot D}{-2 \cdot d} \cdot \frac{D \cdot \frac{M}{2}}{d}\right), \frac{h}{\ell}, 1\right) \cdot t\_2\\ \mathbf{elif}\;t\_0 \leq 10^{+291}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1 \cdot \left(t\_1 \cdot -0.5\right), \frac{h}{\ell}, 1\right) \cdot t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (* (/ (/ D d) 2.0) M))
        (t_2 (/ (fabs d) (sqrt (* l h)))))
   (if (<= t_0 5e-229)
     (*
      (fma
       (* (- -0.5) (* (/ (* M D) (* -2.0 d)) (/ (* D (/ M 2.0)) d)))
       (/ h l)
       1.0)
      t_2)
     (if (<= t_0 1e+291)
       (* (sqrt (/ d l)) (sqrt (/ d h)))
       (* (fma (* t_1 (* t_1 -0.5)) (/ h l) 1.0) t_2)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = ((D / d) / 2.0) * M;
	double t_2 = fabs(d) / sqrt((l * h));
	double tmp;
	if (t_0 <= 5e-229) {
		tmp = fma((-(-0.5) * (((M * D) / (-2.0 * d)) * ((D * (M / 2.0)) / d))), (h / l), 1.0) * t_2;
	} else if (t_0 <= 1e+291) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = fma((t_1 * (t_1 * -0.5)), (h / l), 1.0) * t_2;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(Float64(Float64(D / d) / 2.0) * M)
	t_2 = Float64(abs(d) / sqrt(Float64(l * h)))
	tmp = 0.0
	if (t_0 <= 5e-229)
		tmp = Float64(fma(Float64(Float64(-(-0.5)) * Float64(Float64(Float64(M * D) / Float64(-2.0 * d)) * Float64(Float64(D * Float64(M / 2.0)) / d))), Float64(h / l), 1.0) * t_2);
	elseif (t_0 <= 1e+291)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = Float64(fma(Float64(t_1 * Float64(t_1 * -0.5)), Float64(h / l), 1.0) * t_2);
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(D / d), $MachinePrecision] / 2.0), $MachinePrecision] * M), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-229], N[(N[(N[((--0.5) * N[(N[(N[(M * D), $MachinePrecision] / N[(-2.0 * d), $MachinePrecision]), $MachinePrecision] * N[(N[(D * N[(M / 2.0), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$0, 1e+291], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$1 * N[(t$95$1 * -0.5), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.00000000000000016e-229

   1. Initial program 81.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 46.3

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

   4. Applied rewrites 46.3%

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

   6. Applied rewrites 84.9%

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

      1. unpow1 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. pow-prod-down N/A

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

      5. pow-sqr N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/l/ N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. metadata-eval N/A

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

      13. unpow2 N/A

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

      14. associate-/r* N/A

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

      15. associate-/r* N/A

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

      16. frac-times N/A

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

      17. sqr-neg-rev N/A

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

      18. times-frac N/A

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

      19. lower-*.f64 N/A

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

   8. Applied rewrites 86.2%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-prod N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. sqr-neg-rev N/A

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

      6. sqrt-prod N/A

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

      7. rem-square-sqrt 15.6

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r/ N/A

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

      12. lift-/.f64 N/A

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

      13. times-frac N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. count-2-rev N/A

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

      17. rem-square-sqrt N/A

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

      18. sqrt-prod N/A

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

      19. sqr-neg-rev N/A

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

      20. lift-neg.f64 N/A

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

      21. lift-neg.f64 N/A

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

      22. sqrt-prod N/A

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

      23. rem-square-sqrt N/A

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

      24. lift-neg.f64 N/A

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

      25. rem-square-sqrt N/A

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

      26. sqrt-prod N/A

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

      27. sqr-neg-rev N/A

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

      28. lift-neg.f64 N/A

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

      29. lift-neg.f64 N/A

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

      30. sqrt-prod N/A

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

      31. rem-square-sqrt N/A

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

   10. Applied rewrites 86.2%

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

   if 5.00000000000000016e-229 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.9999999999999996e290

   1. Initial program 99.2%

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 39.5

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 39.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 39.6%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

   9. Applied rewrites 98.3%

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

   if 9.9999999999999996e290 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 21.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 23.5

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

   4. Applied rewrites 23.5%

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

   6. Applied rewrites 54.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 54.0

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 54.0

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

   8. Applied rewrites 54.0%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{-229}:\\ \;\;\;\;\mathsf{fma}\left(\left(--0.5\right) \cdot \left(\frac{M \cdot D}{-2 \cdot d} \cdot \frac{D \cdot \frac{M}{2}}{d}\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 10^{+291}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot -0.5\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 74.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 10^{+291}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \frac{\left(\frac{D}{d} \cdot M\right) \cdot \left(D \cdot M\right)}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (or (<= t_0 0.0) (not (<= t_0 1e+291)))
     (*
      (fma
       (* -0.5 (/ (* (* (/ D d) M) (* D M)) (* 2.0 (* 2.0 d))))
       (/ h l)
       1.0)
      (/ (fabs d) (sqrt (* l h))))
     (* (sqrt (/ d l)) (sqrt (/ d h))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 1e+291)) {
		tmp = fma((-0.5 * ((((D / d) * M) * (D * M)) / (2.0 * (2.0 * d)))), (h / l), 1.0) * (fabs(d) / sqrt((l * h)));
	} else {
		tmp = sqrt((d / l)) * sqrt((d / h));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 1e+291))
		tmp = Float64(fma(Float64(-0.5 * Float64(Float64(Float64(Float64(D / d) * M) * Float64(D * M)) / Float64(2.0 * Float64(2.0 * d)))), Float64(h / l), 1.0) * Float64(abs(d) / sqrt(Float64(l * h))));
	else
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 1e+291]], $MachinePrecision]], N[(N[(N[(-0.5 * N[(N[(N[(N[(D / d), $MachinePrecision] * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 9.9999999999999996e290 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 53.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-sqrt.f64 35.9

         \[\leadsto \left(\frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 35.9%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Applied rewrites 70.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5 \cdot {\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \]
   7. Step-by-step derivation

      1. unpow1 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{{\left({\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2}\right)}^{1}}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot {\color{blue}{\left({\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2}\right)}}^{1}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot {\color{blue}{\left(\left(M \cdot \frac{\frac{D}{d}}{2}\right) \cdot \left(M \cdot \frac{\frac{D}{d}}{2}\right)\right)}}^{1}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      4. pow-prod-down N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\left({\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{1} \cdot {\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{1}\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      5. pow-sqr N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{\left(2 \cdot 1\right)}}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot {\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{\left(2 \cdot 1\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot {\left(M \cdot \color{blue}{\frac{\frac{D}{d}}{2}}\right)}^{\left(2 \cdot 1\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      8. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot {\left(M \cdot \frac{\color{blue}{\frac{D}{d}}}{2}\right)}^{\left(2 \cdot 1\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      9. associate-/l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot {\left(M \cdot \color{blue}{\frac{D}{d \cdot 2}}\right)}^{\left(2 \cdot 1\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot {\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{\left(2 \cdot 1\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{\left(2 \cdot 1\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\color{blue}{2}}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      14. times-frac N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \frac{M \cdot D}{2 \cdot d}\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      15. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{M}{2} \cdot \color{blue}{\frac{D}{d}}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      16. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\color{blue}{\frac{M \cdot \frac{D}{d}}{2}} \cdot \frac{M \cdot D}{2 \cdot d}\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      17. frac-times N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\frac{\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      18. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\frac{\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      19. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot D\right)}}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      20. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(\frac{D}{d} \cdot M\right)} \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      21. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(\frac{D}{d} \cdot M\right)} \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      22. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{D}{d} \cdot M\right) \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      23. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{D}{d} \cdot M\right) \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

   8. Applied rewrites 69.4%

      \[\leadsto \mathsf{fma}\left(-0.5 \cdot \color{blue}{\frac{\left(\frac{D}{d} \cdot M\right) \cdot \left(D \cdot M\right)}{2 \cdot \left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.9999999999999996e290

   1. Initial program 99.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 39.9

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 39.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 40.0%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

   9. Applied rewrites 98.1%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0 \lor \neg \left(\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 10^{+291}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \frac{\left(\frac{D}{d} \cdot M\right) \cdot \left(D \cdot M\right)}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 75.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-229}:\\ \;\;\;\;\mathsf{fma}\left(\left(--0.5\right) \cdot \left(\frac{M \cdot D}{-2 \cdot d} \cdot \frac{D \cdot \frac{M}{2}}{d}\right), \frac{h}{\ell}, 1\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+291}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \frac{\left(\frac{D}{d} \cdot M\right) \cdot \left(D \cdot M\right)}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (/ (fabs d) (sqrt (* l h)))))
   (if (<= t_0 5e-229)
     (*
      (fma
       (* (- -0.5) (* (/ (* M D) (* -2.0 d)) (/ (* D (/ M 2.0)) d)))
       (/ h l)
       1.0)
      t_1)
     (if (<= t_0 1e+291)
       (* (sqrt (/ d l)) (sqrt (/ d h)))
       (*
        (fma
         (* -0.5 (/ (* (* (/ D d) M) (* D M)) (* 2.0 (* 2.0 d))))
         (/ h l)
         1.0)
        t_1)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = fabs(d) / sqrt((l * h));
	double tmp;
	if (t_0 <= 5e-229) {
		tmp = fma((-(-0.5) * (((M * D) / (-2.0 * d)) * ((D * (M / 2.0)) / d))), (h / l), 1.0) * t_1;
	} else if (t_0 <= 1e+291) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = fma((-0.5 * ((((D / d) * M) * (D * M)) / (2.0 * (2.0 * d)))), (h / l), 1.0) * t_1;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(abs(d) / sqrt(Float64(l * h)))
	tmp = 0.0
	if (t_0 <= 5e-229)
		tmp = Float64(fma(Float64(Float64(-(-0.5)) * Float64(Float64(Float64(M * D) / Float64(-2.0 * d)) * Float64(Float64(D * Float64(M / 2.0)) / d))), Float64(h / l), 1.0) * t_1);
	elseif (t_0 <= 1e+291)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = Float64(fma(Float64(-0.5 * Float64(Float64(Float64(Float64(D / d) * M) * Float64(D * M)) / Float64(2.0 * Float64(2.0 * d)))), Float64(h / l), 1.0) * t_1);
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-229], N[(N[(N[((--0.5) * N[(N[(N[(M * D), $MachinePrecision] / N[(-2.0 * d), $MachinePrecision]), $MachinePrecision] * N[(N[(D * N[(M / 2.0), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 1e+291], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.5 * N[(N[(N[(N[(D / d), $MachinePrecision] * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.00000000000000016e-229

   1. Initial program 81.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-sqrt.f64 46.3

         \[\leadsto \left(\frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 46.3%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Applied rewrites 84.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5 \cdot {\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \]
   7. Step-by-step derivation

      1. unpow1 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{{\left({\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2}\right)}^{1}}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot {\color{blue}{\left({\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2}\right)}}^{1}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot {\color{blue}{\left(\left(M \cdot \frac{\frac{D}{d}}{2}\right) \cdot \left(M \cdot \frac{\frac{D}{d}}{2}\right)\right)}}^{1}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      4. pow-prod-down N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\left({\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{1} \cdot {\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{1}\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      5. pow-sqr N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{\left(2 \cdot 1\right)}}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot {\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{\left(2 \cdot 1\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot {\left(M \cdot \color{blue}{\frac{\frac{D}{d}}{2}}\right)}^{\left(2 \cdot 1\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      8. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot {\left(M \cdot \frac{\color{blue}{\frac{D}{d}}}{2}\right)}^{\left(2 \cdot 1\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      9. associate-/l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot {\left(M \cdot \color{blue}{\frac{D}{d \cdot 2}}\right)}^{\left(2 \cdot 1\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot {\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{\left(2 \cdot 1\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{\left(2 \cdot 1\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\color{blue}{2}}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\color{blue}{\frac{\frac{M \cdot D}{2}}{d}} \cdot \frac{M \cdot D}{2 \cdot d}\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      15. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{\frac{M \cdot D}{2}}{d} \cdot \color{blue}{\frac{\frac{M \cdot D}{2}}{d}}\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      16. frac-times N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\frac{\frac{M \cdot D}{2} \cdot \frac{M \cdot D}{2}}{d \cdot d}}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      17. sqr-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\frac{M \cdot D}{2} \cdot \frac{M \cdot D}{2}}{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \left(\mathsf{neg}\left(d\right)\right)}}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      18. times-frac N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\left(\frac{\frac{M \cdot D}{2}}{\mathsf{neg}\left(d\right)} \cdot \frac{\frac{M \cdot D}{2}}{\mathsf{neg}\left(d\right)}\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      19. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\left(\frac{\frac{M \cdot D}{2}}{\mathsf{neg}\left(d\right)} \cdot \frac{\frac{M \cdot D}{2}}{\mathsf{neg}\left(d\right)}\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

   8. Applied rewrites 86.2%

      \[\leadsto \mathsf{fma}\left(-0.5 \cdot \color{blue}{\left(\frac{D \cdot \frac{M}{2}}{-d} \cdot \frac{D \cdot \frac{M}{2}}{-d}\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]
   9. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{2}}{\color{blue}{\sqrt{-d} \cdot \sqrt{-d}}} \cdot \frac{D \cdot \frac{M}{2}}{-d}\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      2. sqrt-prod N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{2}}{\color{blue}{\sqrt{\left(-d\right) \cdot \left(-d\right)}}} \cdot \frac{D \cdot \frac{M}{2}}{-d}\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      3. lift-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{2}}{\sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \left(-d\right)}} \cdot \frac{D \cdot \frac{M}{2}}{-d}\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      4. lift-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{2}}{\sqrt{\left(\mathsf{neg}\left(d\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)}}} \cdot \frac{D \cdot \frac{M}{2}}{-d}\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      5. sqr-neg-rev N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{2}}{\sqrt{\color{blue}{d \cdot d}}} \cdot \frac{D \cdot \frac{M}{2}}{-d}\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      6. sqrt-prod N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{2}}{\color{blue}{\sqrt{d} \cdot \sqrt{d}}} \cdot \frac{D \cdot \frac{M}{2}}{-d}\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      7. rem-square-sqrt 15.6

         \[\leadsto \mathsf{fma}\left(-0.5 \cdot \left(\frac{D \cdot \frac{M}{2}}{\color{blue}{d}} \cdot \frac{D \cdot \frac{M}{2}}{-d}\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\color{blue}{\frac{D \cdot \frac{M}{2}}{d}} \cdot \frac{D \cdot \frac{M}{2}}{-d}\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{\color{blue}{D \cdot \frac{M}{2}}}{d} \cdot \frac{D \cdot \frac{M}{2}}{-d}\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{\color{blue}{\frac{M}{2} \cdot D}}{d} \cdot \frac{D \cdot \frac{M}{2}}{-d}\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \frac{D \cdot \frac{M}{2}}{-d}\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      12. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\color{blue}{\frac{M}{2}} \cdot \frac{D}{d}\right) \cdot \frac{D \cdot \frac{M}{2}}{-d}\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      13. times-frac N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \frac{D \cdot \frac{M}{2}}{-d}\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      14. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \frac{D \cdot \frac{M}{2}}{-d}\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \frac{D \cdot \frac{M}{2}}{-d}\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      16. count-2-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{M \cdot D}{\color{blue}{d + d}} \cdot \frac{D \cdot \frac{M}{2}}{-d}\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      17. rem-square-sqrt N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{M \cdot D}{\color{blue}{\sqrt{d} \cdot \sqrt{d}} + d} \cdot \frac{D \cdot \frac{M}{2}}{-d}\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      18. sqrt-prod N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{M \cdot D}{\color{blue}{\sqrt{d \cdot d}} + d} \cdot \frac{D \cdot \frac{M}{2}}{-d}\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      19. sqr-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{M \cdot D}{\sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \left(\mathsf{neg}\left(d\right)\right)}} + d} \cdot \frac{D \cdot \frac{M}{2}}{-d}\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      20. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{M \cdot D}{\sqrt{\color{blue}{\left(-d\right)} \cdot \left(\mathsf{neg}\left(d\right)\right)} + d} \cdot \frac{D \cdot \frac{M}{2}}{-d}\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      21. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{M \cdot D}{\sqrt{\left(-d\right) \cdot \color{blue}{\left(-d\right)}} + d} \cdot \frac{D \cdot \frac{M}{2}}{-d}\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      22. sqrt-prod N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{M \cdot D}{\color{blue}{\sqrt{-d} \cdot \sqrt{-d}} + d} \cdot \frac{D \cdot \frac{M}{2}}{-d}\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      23. rem-square-sqrt N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{M \cdot D}{\color{blue}{\left(-d\right)} + d} \cdot \frac{D \cdot \frac{M}{2}}{-d}\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      24. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{M \cdot D}{\color{blue}{\left(\mathsf{neg}\left(d\right)\right)} + d} \cdot \frac{D \cdot \frac{M}{2}}{-d}\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      25. rem-square-sqrt N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{M \cdot D}{\left(\mathsf{neg}\left(d\right)\right) + \color{blue}{\sqrt{d} \cdot \sqrt{d}}} \cdot \frac{D \cdot \frac{M}{2}}{-d}\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      26. sqrt-prod N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{M \cdot D}{\left(\mathsf{neg}\left(d\right)\right) + \color{blue}{\sqrt{d \cdot d}}} \cdot \frac{D \cdot \frac{M}{2}}{-d}\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      27. sqr-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{M \cdot D}{\left(\mathsf{neg}\left(d\right)\right) + \sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \left(\mathsf{neg}\left(d\right)\right)}}} \cdot \frac{D \cdot \frac{M}{2}}{-d}\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      28. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{M \cdot D}{\left(\mathsf{neg}\left(d\right)\right) + \sqrt{\color{blue}{\left(-d\right)} \cdot \left(\mathsf{neg}\left(d\right)\right)}} \cdot \frac{D \cdot \frac{M}{2}}{-d}\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      29. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{M \cdot D}{\left(\mathsf{neg}\left(d\right)\right) + \sqrt{\left(-d\right) \cdot \color{blue}{\left(-d\right)}}} \cdot \frac{D \cdot \frac{M}{2}}{-d}\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      30. sqrt-prod N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{M \cdot D}{\left(\mathsf{neg}\left(d\right)\right) + \color{blue}{\sqrt{-d} \cdot \sqrt{-d}}} \cdot \frac{D \cdot \frac{M}{2}}{-d}\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      31. rem-square-sqrt N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{M \cdot D}{\left(\mathsf{neg}\left(d\right)\right) + \color{blue}{\left(-d\right)}} \cdot \frac{D \cdot \frac{M}{2}}{-d}\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

   10. Applied rewrites 86.2%

       \[\leadsto \mathsf{fma}\left(-0.5 \cdot \left(\color{blue}{\frac{M \cdot D}{-2 \cdot d}} \cdot \frac{D \cdot \frac{M}{2}}{-d}\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

   if 5.00000000000000016e-229 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.9999999999999996e290

   1. Initial program 99.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 39.5

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 39.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 39.6%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

   9. Applied rewrites 98.3%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}} \]

   if 9.9999999999999996e290 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 21.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-sqrt.f64 23.5

         \[\leadsto \left(\frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 23.5%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Applied rewrites 54.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5 \cdot {\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \]
   7. Step-by-step derivation

      1. unpow1 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{{\left({\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2}\right)}^{1}}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot {\color{blue}{\left({\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2}\right)}}^{1}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot {\color{blue}{\left(\left(M \cdot \frac{\frac{D}{d}}{2}\right) \cdot \left(M \cdot \frac{\frac{D}{d}}{2}\right)\right)}}^{1}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      4. pow-prod-down N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\left({\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{1} \cdot {\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{1}\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      5. pow-sqr N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{\left(2 \cdot 1\right)}}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot {\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{\left(2 \cdot 1\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot {\left(M \cdot \color{blue}{\frac{\frac{D}{d}}{2}}\right)}^{\left(2 \cdot 1\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      8. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot {\left(M \cdot \frac{\color{blue}{\frac{D}{d}}}{2}\right)}^{\left(2 \cdot 1\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      9. associate-/l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot {\left(M \cdot \color{blue}{\frac{D}{d \cdot 2}}\right)}^{\left(2 \cdot 1\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot {\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{\left(2 \cdot 1\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{\left(2 \cdot 1\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\color{blue}{2}}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      14. times-frac N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \frac{M \cdot D}{2 \cdot d}\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      15. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{M}{2} \cdot \color{blue}{\frac{D}{d}}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      16. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\color{blue}{\frac{M \cdot \frac{D}{d}}{2}} \cdot \frac{M \cdot D}{2 \cdot d}\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      17. frac-times N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\frac{\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      18. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\frac{\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      19. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot D\right)}}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      20. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(\frac{D}{d} \cdot M\right)} \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      21. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(\frac{D}{d} \cdot M\right)} \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      22. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{D}{d} \cdot M\right) \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      23. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{D}{d} \cdot M\right) \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

   8. Applied rewrites 52.8%

      \[\leadsto \mathsf{fma}\left(-0.5 \cdot \color{blue}{\frac{\left(\frac{D}{d} \cdot M\right) \cdot \left(D \cdot M\right)}{2 \cdot \left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{-229}:\\ \;\;\;\;\mathsf{fma}\left(\left(--0.5\right) \cdot \left(\frac{M \cdot D}{-2 \cdot d} \cdot \frac{D \cdot \frac{M}{2}}{d}\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 10^{+291}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \frac{\left(\frac{D}{d} \cdot M\right) \cdot \left(D \cdot M\right)}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 45.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -1 \cdot 10^{-83}:\\ \;\;\;\;\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
       (- 1.0 (* (* (pow 2.0 -1.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))
      -1e-83)
   (* (sqrt (pow (* l h) -1.0)) d)
   (* 1.0 (/ (fabs d) (sqrt (* l h))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -1e-83) {
		tmp = sqrt(pow((l * h), -1.0)) * d;
	} else {
		tmp = 1.0 * (fabs(d) / sqrt((l * h)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((((d / h) ** (2.0d0 ** (-1.0d0))) * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - (((2.0d0 ** (-1.0d0)) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))) <= (-1d-83)) then
        tmp = sqrt(((l * h) ** (-1.0d0))) * d
    else
        tmp = 1.0d0 * (abs(d) / sqrt((l * h)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -1e-83) {
		tmp = Math.sqrt(Math.pow((l * h), -1.0)) * d;
	} else {
		tmp = 1.0 * (Math.abs(d) / Math.sqrt((l * h)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if ((math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -1e-83:
		tmp = math.sqrt(math.pow((l * h), -1.0)) * d
	else:
		tmp = 1.0 * (math.fabs(d) / math.sqrt((l * h)))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= -1e-83)
		tmp = Float64(sqrt((Float64(l * h) ^ -1.0)) * d);
	else
		tmp = Float64(1.0 * Float64(abs(d) / sqrt(Float64(l * h))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (((((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)))) <= -1e-83)
		tmp = sqrt(((l * h) ^ -1.0)) * d;
	else
		tmp = 1.0 * (abs(d) / sqrt((l * h)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $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], -1e-83], N[(N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision], N[(1.0 * N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1e-83

   1. Initial program 90.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 12.9

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 12.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

   if -1e-83 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 53.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-sqrt.f64 34.0

         \[\leadsto \left(\frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 34.0%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Applied rewrites 65.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5 \cdot {\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \]

   7. Taylor expanded in d around inf

      \[\leadsto \color{blue}{1} \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]
   8. Step-by-step derivation

   9. Applied rewrites 61.2%

      \[\leadsto \color{blue}{1} \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -1 \cdot 10^{-83}:\\ \;\;\;\;\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 77.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{D}{d}}{2}\\ \mathbf{if}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left(t\_0 \cdot \left(M \cdot \frac{h}{\ell}\right)\right) \cdot \left(\left(0.5 \cdot \frac{D}{2}\right) \cdot \frac{M}{d}\right)\right)\\ \mathbf{elif}\;d \leq 1.12 \cdot 10^{-169}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{h}{\ell}, {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot -0.5, 1\right) \cdot d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5 \cdot {\left(M \cdot t\_0\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ (/ D d) 2.0)))
   (if (<= d -2e-310)
     (*
      (* (/ (sqrt (- d)) (sqrt (- h))) (pow (/ d l) (pow 2.0 -1.0)))
      (- 1.0 (* (* t_0 (* M (/ h l))) (* (* 0.5 (/ D 2.0)) (/ M d)))))
     (if (<= d 1.12e-169)
       (/
        (* (fma (/ h l) (* (pow (* 0.5 (/ (* M D) d)) 2.0) -0.5) 1.0) d)
        (sqrt (* l h)))
       (/
        (*
         (fma (* -0.5 (pow (* M t_0) 2.0)) (/ h l) 1.0)
         (/ (fabs d) (sqrt l)))
        (sqrt h))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (D / d) / 2.0;
	double tmp;
	if (d <= -2e-310) {
		tmp = ((sqrt(-d) / sqrt(-h)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((t_0 * (M * (h / l))) * ((0.5 * (D / 2.0)) * (M / d))));
	} else if (d <= 1.12e-169) {
		tmp = (fma((h / l), (pow((0.5 * ((M * D) / d)), 2.0) * -0.5), 1.0) * d) / sqrt((l * h));
	} else {
		tmp = (fma((-0.5 * pow((M * t_0), 2.0)), (h / l), 1.0) * (fabs(d) / sqrt(l))) / sqrt(h);
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(D / d) / 2.0)
	tmp = 0.0
	if (d <= -2e-310)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64(t_0 * Float64(M * Float64(h / l))) * Float64(Float64(0.5 * Float64(D / 2.0)) * Float64(M / d)))));
	elseif (d <= 1.12e-169)
		tmp = Float64(Float64(fma(Float64(h / l), Float64((Float64(0.5 * Float64(Float64(M * D) / d)) ^ 2.0) * -0.5), 1.0) * d) / sqrt(Float64(l * h)));
	else
		tmp = Float64(Float64(fma(Float64(-0.5 * (Float64(M * t_0) ^ 2.0)), Float64(h / l), 1.0) * Float64(abs(d) / sqrt(l))) / sqrt(h));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(D / d), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[d, -2e-310], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(t$95$0 * N[(M * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[(D / 2.0), $MachinePrecision]), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.12e-169], N[(N[(N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(0.5 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * d), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.5 * N[Power[N[(M * t$95$0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.999999999999994e-310

   1. Initial program 62.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right)}\right) \]

      8. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \]

   4. Applied rewrites 60.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot \left(M \cdot \frac{h}{\ell}\right)\right) \cdot \left(\left(0.5 \cdot \frac{D}{2}\right) \cdot \frac{M}{d}\right)}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{\frac{D}{d}}{2} \cdot \left(M \cdot \frac{h}{\ell}\right)\right) \cdot \left(\left(\frac{1}{2} \cdot \frac{D}{2}\right) \cdot \frac{M}{d}\right)\right) \]

      2. metadata-eval 60.2

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{\frac{D}{d}}{2} \cdot \left(M \cdot \frac{h}{\ell}\right)\right) \cdot \left(\left(0.5 \cdot \frac{D}{2}\right) \cdot \frac{M}{d}\right)\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{\frac{D}{d}}{2} \cdot \left(M \cdot \frac{h}{\ell}\right)\right) \cdot \left(\left(\frac{1}{2} \cdot \frac{D}{2}\right) \cdot \frac{M}{d}\right)\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{\frac{D}{d}}{2} \cdot \left(M \cdot \frac{h}{\ell}\right)\right) \cdot \left(\left(\frac{1}{2} \cdot \frac{D}{2}\right) \cdot \frac{M}{d}\right)\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{\frac{D}{d}}{2} \cdot \left(M \cdot \frac{h}{\ell}\right)\right) \cdot \left(\left(\frac{1}{2} \cdot \frac{D}{2}\right) \cdot \frac{M}{d}\right)\right) \]

      6. frac-2neg N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{\frac{D}{d}}{2} \cdot \left(M \cdot \frac{h}{\ell}\right)\right) \cdot \left(\left(\frac{1}{2} \cdot \frac{D}{2}\right) \cdot \frac{M}{d}\right)\right) \]

      7. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{\frac{D}{d}}{2} \cdot \left(M \cdot \frac{h}{\ell}\right)\right) \cdot \left(\left(\frac{1}{2} \cdot \frac{D}{2}\right) \cdot \frac{M}{d}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{\frac{D}{d}}{2} \cdot \left(M \cdot \frac{h}{\ell}\right)\right) \cdot \left(\left(\frac{1}{2} \cdot \frac{D}{2}\right) \cdot \frac{M}{d}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{\frac{D}{d}}{2} \cdot \left(M \cdot \frac{h}{\ell}\right)\right) \cdot \left(\left(\frac{1}{2} \cdot \frac{D}{2}\right) \cdot \frac{M}{d}\right)\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto \left(\frac{\sqrt{\color{blue}{-d}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{\frac{D}{d}}{2} \cdot \left(M \cdot \frac{h}{\ell}\right)\right) \cdot \left(\left(\frac{1}{2} \cdot \frac{D}{2}\right) \cdot \frac{M}{d}\right)\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{\sqrt{-d}}{\color{blue}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{\frac{D}{d}}{2} \cdot \left(M \cdot \frac{h}{\ell}\right)\right) \cdot \left(\left(\frac{1}{2} \cdot \frac{D}{2}\right) \cdot \frac{M}{d}\right)\right) \]

      12. lower-neg.f64 73.5

          \[\leadsto \left(\frac{\sqrt{-d}}{\sqrt{\color{blue}{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{\frac{D}{d}}{2} \cdot \left(M \cdot \frac{h}{\ell}\right)\right) \cdot \left(\left(0.5 \cdot \frac{D}{2}\right) \cdot \frac{M}{d}\right)\right) \]

   6. Applied rewrites 73.5%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{\frac{D}{d}}{2} \cdot \left(M \cdot \frac{h}{\ell}\right)\right) \cdot \left(\left(0.5 \cdot \frac{D}{2}\right) \cdot \frac{M}{d}\right)\right) \]

   if -1.999999999999994e-310 < d < 1.11999999999999998e-169

   1. Initial program 42.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-sqrt.f64 49.4

         \[\leadsto \left(\frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 49.4%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Applied rewrites 62.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5 \cdot {\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{2} \cdot {\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot {\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{-1}{2} \cdot {\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \left|d\right|}{\sqrt{\ell \cdot h}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{-1}{2} \cdot {\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \left|d\right|}{\sqrt{\ell \cdot h}}} \]

   8. Applied rewrites 73.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{h}{\ell}, {\left(\frac{\frac{D}{d}}{2} \cdot M\right)}^{2} \cdot -0.5, 1\right) \cdot d}{\sqrt{\ell \cdot h}}} \]

   9. Taylor expanded in d around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{h}{\ell}, {\color{blue}{\left(\frac{1}{2} \cdot \frac{D \cdot M}{d}\right)}}^{2} \cdot \frac{-1}{2}, 1\right) \cdot d}{\sqrt{\ell \cdot h}} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{h}{\ell}, {\color{blue}{\left(\frac{1}{2} \cdot \frac{D \cdot M}{d}\right)}}^{2} \cdot \frac{-1}{2}, 1\right) \cdot d}{\sqrt{\ell \cdot h}} \]

       2. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{h}{\ell}, {\left(\frac{1}{2} \cdot \color{blue}{\frac{D \cdot M}{d}}\right)}^{2} \cdot \frac{-1}{2}, 1\right) \cdot d}{\sqrt{\ell \cdot h}} \]

       3. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{h}{\ell}, {\left(\frac{1}{2} \cdot \frac{\color{blue}{M \cdot D}}{d}\right)}^{2} \cdot \frac{-1}{2}, 1\right) \cdot d}{\sqrt{\ell \cdot h}} \]

       4. lower-*.f64 76.4

          \[\leadsto \frac{\mathsf{fma}\left(\frac{h}{\ell}, {\left(0.5 \cdot \frac{\color{blue}{M \cdot D}}{d}\right)}^{2} \cdot -0.5, 1\right) \cdot d}{\sqrt{\ell \cdot h}} \]

   11. Applied rewrites 76.4%

       \[\leadsto \frac{\mathsf{fma}\left(\frac{h}{\ell}, {\color{blue}{\left(0.5 \cdot \frac{M \cdot D}{d}\right)}}^{2} \cdot -0.5, 1\right) \cdot d}{\sqrt{\ell \cdot h}} \]

   if 1.11999999999999998e-169 < d

   1. Initial program 76.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-sqrt.f64 84.4

         \[\leadsto \left(\frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 84.4%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Applied rewrites 90.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.5 \cdot {\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell}}}{\sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left(\frac{\frac{D}{d}}{2} \cdot \left(M \cdot \frac{h}{\ell}\right)\right) \cdot \left(\left(0.5 \cdot \frac{D}{2}\right) \cdot \frac{M}{d}\right)\right)\\ \mathbf{elif}\;d \leq 1.12 \cdot 10^{-169}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{h}{\ell}, {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot -0.5, 1\right) \cdot d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5 \cdot {\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 42.6% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;\ell \leq -9.8 \cdot 10^{-266}:\\ \;\;\;\;\frac{-d}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (* l h)))) (if (<= l -9.8e-266) (/ (- d) t_0) (/ d t_0))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((l * h));
	double tmp;
	if (l <= -9.8e-266) {
		tmp = -d / t_0;
	} else {
		tmp = d / t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((l * h))
    if (l <= (-9.8d-266)) then
        tmp = -d / t_0
    else
        tmp = d / t_0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((l * h));
	double tmp;
	if (l <= -9.8e-266) {
		tmp = -d / t_0;
	} else {
		tmp = d / t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((l * h))
	tmp = 0
	if l <= -9.8e-266:
		tmp = -d / t_0
	else:
		tmp = d / t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(l * h))
	tmp = 0.0
	if (l <= -9.8e-266)
		tmp = Float64(Float64(-d) / t_0);
	else
		tmp = Float64(d / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((l * h));
	tmp = 0.0;
	if (l <= -9.8e-266)
		tmp = -d / t_0;
	else
		tmp = d / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -9.8e-266], N[((-d) / t$95$0), $MachinePrecision], N[(d / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -9.8000000000000005e-266

   1. Initial program 61.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 7.6

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 7.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 6.8%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

   9. Applied rewrites 6.8%

      \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 40.4%

       \[\leadsto \frac{d}{\color{blue}{-\sqrt{\ell \cdot h}}} \]

   if -9.8000000000000005e-266 < l

   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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 45.0

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 45.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 45.0%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

   9. Applied rewrites 45.1%

      \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -9.8 \cdot 10^{-266}:\\ \;\;\;\;\frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 26.3% 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 65.7%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Add Preprocessing

3. Taylor expanded in d around inf

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

   4. lower-/.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

   5. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   6. lower-*.f64 27.8

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

5. Applied rewrites 27.8%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
6. Step-by-step derivation

7. Applied rewrites 27.4%

   \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
8. Step-by-step derivation

9. Applied rewrites 27.5%

   \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024358 
(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)))))