Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.6% → 71.1%

Time: 18.6s

Alternatives: 17

Speedup: 0.7×

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: 66.6% 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: 71.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-0.125 \cdot {\left(D \cdot M\right)}^{2}, {\left(\frac{h}{\ell}\right)}^{1.5}, \sqrt{\frac{h}{\ell}} \cdot \left(d \cdot d\right)\right)}{d}}{h}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_0 INFINITY)
     t_0
     (/
      (/
       (fma
        (* -0.125 (pow (* D M) 2.0))
        (pow (/ h l) 1.5)
        (* (sqrt (/ h l)) (* d d)))
       d)
      h))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = (fma((-0.125 * pow((D * M), 2.0)), pow((h / l), 1.5), (sqrt((h / l)) * (d * d))) / d) / h;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = Float64(Float64(fma(Float64(-0.125 * (Float64(D * M) ^ 2.0)), (Float64(h / l) ^ 1.5), Float64(sqrt(Float64(h / l)) * Float64(d * d))) / 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[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], t$95$0, N[(N[(N[(N[(-0.125 * N[Power[N[(D * M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision] + N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / h), $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)))) < +inf.0

   1. Initial program 84.9%

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

   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

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

   5. Applied rewrites 25.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   6. Taylor expanded in d around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 30.5%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(-0.125 \cdot {\left(D \cdot M\right)}^{2}, {\left(\frac{h}{\ell}\right)}^{1.5}, \sqrt{\frac{h}{\ell}} \cdot \left(d \cdot d\right)\right)}{d}}{h} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 64.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{\frac{h}{\ell}} \cdot d\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-213}:\\ \;\;\;\;\left(1 - \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right) \cdot 0.5\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{t\_1}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, t\_1\right)}{h}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (* (sqrt (/ h l)) d)))
   (if (<= t_0 -5e-213)
     (*
      (- 1.0 (* (* (/ h l) (pow (* (/ D d) (* M 0.5)) 2.0)) 0.5))
      (sqrt (* (/ d l) (/ d h))))
     (if (<= t_0 INFINITY)
       (/ t_1 h)
       (/
        (fma (* -0.125 (/ (* (* D M) (* D M)) d)) (sqrt (pow (/ h l) 3.0)) t_1)
        h)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = sqrt((h / l)) * d;
	double tmp;
	if (t_0 <= -5e-213) {
		tmp = (1.0 - (((h / l) * pow(((D / d) * (M * 0.5)), 2.0)) * 0.5)) * sqrt(((d / l) * (d / h)));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_1 / h;
	} else {
		tmp = fma((-0.125 * (((D * M) * (D * M)) / d)), sqrt(pow((h / l), 3.0)), t_1) / h;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(sqrt(Float64(h / l)) * d)
	tmp = 0.0
	if (t_0 <= -5e-213)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(h / l) * (Float64(Float64(D / d) * Float64(M * 0.5)) ^ 2.0)) * 0.5)) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	elseif (t_0 <= Inf)
		tmp = Float64(t_1 / h);
	else
		tmp = Float64(fma(Float64(-0.125 * Float64(Float64(Float64(D * M) * Float64(D * M)) / d)), sqrt((Float64(h / l) ^ 3.0)), t_1) / 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[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-213], N[(N[(1.0 - N[(N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(t$95$1 / h), $MachinePrecision], N[(N[(N[(-0.125 * N[(N[(N[(D * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Power[N[(h / l), $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision] / h), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -4.99999999999999977e-213

   1. Initial program 88.4%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-pow.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-pow.f64 N/A

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

      14. lift-/.f64 88.5

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

   4. Applied rewrites 88.5%

      \[\leadsto \left(\color{blue}{\left({\left(\frac{d}{h}\right)}^{0.25} \cdot {\left(\frac{d}{h}\right)}^{0.25}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. metadata-eval N/A

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

      18. pow1/2 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lift-/.f64 88.5

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

   6. Applied rewrites 88.5%

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

   8. Applied rewrites 76.2%

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

   9. Taylor expanded in M around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 76.2

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

   11. Applied rewrites 76.2%

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

   if -4.99999999999999977e-213 < (*.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 81.5%

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

   3. 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

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

   5. Applied rewrites 64.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   6. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-*.f64 81.1

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

   8. Applied rewrites 81.1%

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

   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

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

   5. Applied rewrites 25.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h} \]

      6. lift-*.f64 25.2

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

   7. Applied rewrites 25.2%

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

Alternative 3: 64.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{\frac{h}{\ell}} \cdot d\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-213}:\\ \;\;\;\;\left(1 - \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right) \cdot 0.5\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{t\_1}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right), \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, t\_1\right)}{h}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (* (sqrt (/ h l)) d)))
   (if (<= t_0 -5e-213)
     (*
      (- 1.0 (* (* (/ h l) (pow (* (/ D d) (* M 0.5)) 2.0)) 0.5))
      (sqrt (* (/ d l) (/ d h))))
     (if (<= t_0 INFINITY)
       (/ t_1 h)
       (/
        (fma (* -0.125 (* (* D D) (/ (* M M) d))) (sqrt (pow (/ h l) 3.0)) t_1)
        h)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = sqrt((h / l)) * d;
	double tmp;
	if (t_0 <= -5e-213) {
		tmp = (1.0 - (((h / l) * pow(((D / d) * (M * 0.5)), 2.0)) * 0.5)) * sqrt(((d / l) * (d / h)));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_1 / h;
	} else {
		tmp = fma((-0.125 * ((D * D) * ((M * M) / d))), sqrt(pow((h / l), 3.0)), t_1) / h;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(sqrt(Float64(h / l)) * d)
	tmp = 0.0
	if (t_0 <= -5e-213)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(h / l) * (Float64(Float64(D / d) * Float64(M * 0.5)) ^ 2.0)) * 0.5)) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	elseif (t_0 <= Inf)
		tmp = Float64(t_1 / h);
	else
		tmp = Float64(fma(Float64(-0.125 * Float64(Float64(D * D) * Float64(Float64(M * M) / d))), sqrt((Float64(h / l) ^ 3.0)), t_1) / 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[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-213], N[(N[(1.0 - N[(N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(t$95$1 / h), $MachinePrecision], N[(N[(N[(-0.125 * N[(N[(D * D), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Power[N[(h / l), $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision] / h), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -4.99999999999999977e-213

   1. Initial program 88.4%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-pow.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-pow.f64 N/A

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

      14. lift-/.f64 88.5

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

   4. Applied rewrites 88.5%

      \[\leadsto \left(\color{blue}{\left({\left(\frac{d}{h}\right)}^{0.25} \cdot {\left(\frac{d}{h}\right)}^{0.25}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. metadata-eval N/A

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

      18. pow1/2 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lift-/.f64 88.5

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

   6. Applied rewrites 88.5%

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

   8. Applied rewrites 76.2%

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

   9. Taylor expanded in M around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 76.2

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

   11. Applied rewrites 76.2%

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

   if -4.99999999999999977e-213 < (*.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 81.5%

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

   3. 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

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

   5. Applied rewrites 64.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   6. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-*.f64 81.1

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

   8. Applied rewrites 81.1%

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

   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

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

   5. Applied rewrites 25.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow-prod-down N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right), \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h} \]

      11. lower-*.f64 24.7

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

   7. Applied rewrites 24.7%

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

Alternative 4: 63.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-213}:\\ \;\;\;\;\left(1 - \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right) \cdot 0.5\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot -1}{d}\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_0 -5e-213)
     (*
      (- 1.0 (* (* (/ h l) (pow (* (/ D d) (* M 0.5)) 2.0)) 0.5))
      (sqrt (* (/ d l) (/ d h))))
     (if (<= t_0 INFINITY)
       (/ (* (sqrt (/ h l)) d) h)
       (*
        (* -0.125 (/ (* (pow (* D M) 2.0) -1.0) d))
        (sqrt (/ h (* (* l l) l))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -5e-213) {
		tmp = (1.0 - (((h / l) * pow(((D / d) * (M * 0.5)), 2.0)) * 0.5)) * sqrt(((d / l) * (d / h)));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (sqrt((h / l)) * d) / h;
	} else {
		tmp = (-0.125 * ((pow((D * M), 2.0) * -1.0) / d)) * sqrt((h / ((l * l) * l)));
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -5e-213) {
		tmp = (1.0 - (((h / l) * Math.pow(((D / d) * (M * 0.5)), 2.0)) * 0.5)) * Math.sqrt(((d / l) * (d / h)));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (Math.sqrt((h / l)) * d) / h;
	} else {
		tmp = (-0.125 * ((Math.pow((D * M), 2.0) * -1.0) / d)) * Math.sqrt((h / ((l * l) * l)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_0 <= -5e-213:
		tmp = (1.0 - (((h / l) * math.pow(((D / d) * (M * 0.5)), 2.0)) * 0.5)) * math.sqrt(((d / l) * (d / h)))
	elif t_0 <= math.inf:
		tmp = (math.sqrt((h / l)) * d) / h
	else:
		tmp = (-0.125 * ((math.pow((D * M), 2.0) * -1.0) / d)) * math.sqrt((h / ((l * l) * l)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= -5e-213)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(h / l) * (Float64(Float64(D / d) * Float64(M * 0.5)) ^ 2.0)) * 0.5)) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * d) / h);
	else
		tmp = Float64(Float64(-0.125 * Float64(Float64((Float64(D * M) ^ 2.0) * -1.0) / d)) * sqrt(Float64(h / Float64(Float64(l * l) * l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_0 <= -5e-213)
		tmp = (1.0 - (((h / l) * (((D / d) * (M * 0.5)) ^ 2.0)) * 0.5)) * sqrt(((d / l) * (d / h)));
	elseif (t_0 <= Inf)
		tmp = (sqrt((h / l)) * d) / h;
	else
		tmp = (-0.125 * ((((D * M) ^ 2.0) * -1.0) / d)) * sqrt((h / ((l * l) * l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-213], N[(N[(1.0 - N[(N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] / h), $MachinePrecision], N[(N[(-0.125 * N[(N[(N[Power[N[(D * M), $MachinePrecision], 2.0], $MachinePrecision] * -1.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $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)))) < -4.99999999999999977e-213

   1. Initial program 88.4%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-pow.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-pow.f64 N/A

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

      14. lift-/.f64 88.5

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

   4. Applied rewrites 88.5%

      \[\leadsto \left(\color{blue}{\left({\left(\frac{d}{h}\right)}^{0.25} \cdot {\left(\frac{d}{h}\right)}^{0.25}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. metadata-eval N/A

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

      18. pow1/2 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lift-/.f64 88.5

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

   6. Applied rewrites 88.5%

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

   8. Applied rewrites 76.2%

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

   9. Taylor expanded in M around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 76.2

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

   11. Applied rewrites 76.2%

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

   if -4.99999999999999977e-213 < (*.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 81.5%

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

   3. 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

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

   5. Applied rewrites 64.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   6. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-*.f64 81.1

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

   8. Applied rewrites 81.1%

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

   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-/.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-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-pow.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-pow.f64 N/A

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

      14. lift-/.f64 0.0

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

   4. Applied rewrites 0.0%

      \[\leadsto \left(\color{blue}{\left({\left(\frac{d}{h}\right)}^{0.25} \cdot {\left(\frac{d}{h}\right)}^{0.25}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. metadata-eval N/A

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

      18. pow1/2 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lift-/.f64 0.0

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

   6. Applied rewrites 0.0%

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

   7. Taylor expanded in h around -inf

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

   9. Applied rewrites 19.9%

      \[\leadsto \color{blue}{\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot -1}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]
   10. Step-by-step derivation

       1. lift-pow.f64 N/A

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

       2. unpow3 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 19.9

          \[\leadsto \left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot -1}{d}\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

   11. Applied rewrites 19.9%

       \[\leadsto \left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot -1}{d}\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-213}:\\ \;\;\;\;\left(1 - \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right) \cdot 0.5\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot -1}{d}\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(D \cdot M\right)}^{2}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-68}:\\ \;\;\;\;\left(1 - \left(\frac{t\_0 \cdot h}{\left(d \cdot d\right) \cdot \ell} \cdot 0.25\right) \cdot 0.5\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(-0.125 \cdot \frac{t\_0 \cdot -1}{d}\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (pow (* D M) 2.0))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_1 -1e-68)
     (*
      (- 1.0 (* (* (/ (* t_0 h) (* (* d d) l)) 0.25) 0.5))
      (sqrt (* (/ d l) (/ d h))))
     (if (<= t_1 INFINITY)
       (/ (* (sqrt (/ h l)) d) h)
       (* (* -0.125 (/ (* t_0 -1.0) d)) (sqrt (/ h (* (* l l) l))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = pow((D * M), 2.0);
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= -1e-68) {
		tmp = (1.0 - ((((t_0 * h) / ((d * d) * l)) * 0.25) * 0.5)) * sqrt(((d / l) * (d / h)));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (sqrt((h / l)) * d) / h;
	} else {
		tmp = (-0.125 * ((t_0 * -1.0) / d)) * sqrt((h / ((l * l) * l)));
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.pow((D * M), 2.0);
	double t_1 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= -1e-68) {
		tmp = (1.0 - ((((t_0 * h) / ((d * d) * l)) * 0.25) * 0.5)) * Math.sqrt(((d / l) * (d / h)));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (Math.sqrt((h / l)) * d) / h;
	} else {
		tmp = (-0.125 * ((t_0 * -1.0) / d)) * Math.sqrt((h / ((l * l) * l)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.pow((D * M), 2.0)
	t_1 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_1 <= -1e-68:
		tmp = (1.0 - ((((t_0 * h) / ((d * d) * l)) * 0.25) * 0.5)) * math.sqrt(((d / l) * (d / h)))
	elif t_1 <= math.inf:
		tmp = (math.sqrt((h / l)) * d) / h
	else:
		tmp = (-0.125 * ((t_0 * -1.0) / d)) * math.sqrt((h / ((l * l) * l)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(D * M) ^ 2.0
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_1 <= -1e-68)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(Float64(t_0 * h) / Float64(Float64(d * d) * l)) * 0.25) * 0.5)) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * d) / h);
	else
		tmp = Float64(Float64(-0.125 * Float64(Float64(t_0 * -1.0) / d)) * sqrt(Float64(h / Float64(Float64(l * l) * l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (D * M) ^ 2.0;
	t_1 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_1 <= -1e-68)
		tmp = (1.0 - ((((t_0 * h) / ((d * d) * l)) * 0.25) * 0.5)) * sqrt(((d / l) * (d / h)));
	elseif (t_1 <= Inf)
		tmp = (sqrt((h / l)) * d) / h;
	else
		tmp = (-0.125 * ((t_0 * -1.0) / d)) * sqrt((h / ((l * l) * l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Power[N[(D * M), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-68], N[(N[(1.0 - N[(N[(N[(N[(t$95$0 * h), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] / h), $MachinePrecision], N[(N[(-0.125 * N[(N[(t$95$0 * -1.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $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)))) < -1.00000000000000007e-68

   1. Initial program 88.3%

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

      1. lift-/.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-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-pow.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-pow.f64 N/A

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

      14. lift-/.f64 88.3

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

   4. Applied rewrites 88.3%

      \[\leadsto \left(\color{blue}{\left({\left(\frac{d}{h}\right)}^{0.25} \cdot {\left(\frac{d}{h}\right)}^{0.25}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. metadata-eval N/A

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

      18. pow1/2 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lift-/.f64 88.3

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

   6. Applied rewrites 88.3%

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

   8. Applied rewrites 77.6%

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

   9. Taylor expanded in d around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. associate-*r* N/A

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

       5. lower-*.f64 N/A

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

       6. unpow-prod-down N/A

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

       7. lift-pow.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 58.6

           \[\leadsto \left(1 - \left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{\left(d \cdot d\right) \cdot \ell} \cdot 0.25\right) \cdot 0.5\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   11. Applied rewrites 58.6%

       \[\leadsto \left(1 - \color{blue}{\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{\left(d \cdot d\right) \cdot \ell} \cdot 0.25\right)} \cdot 0.5\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   if -1.00000000000000007e-68 < (*.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 81.8%

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

   3. Taylor expanded in 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

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

   5. Applied rewrites 64.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   6. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-*.f64 79.7

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

   8. Applied rewrites 79.7%

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

   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-/.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-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-pow.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-pow.f64 N/A

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

      14. lift-/.f64 0.0

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

   4. Applied rewrites 0.0%

      \[\leadsto \left(\color{blue}{\left({\left(\frac{d}{h}\right)}^{0.25} \cdot {\left(\frac{d}{h}\right)}^{0.25}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. metadata-eval N/A

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

      18. pow1/2 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lift-/.f64 0.0

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

   6. Applied rewrites 0.0%

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

   7. Taylor expanded in h around -inf

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

   9. Applied rewrites 19.9%

      \[\leadsto \color{blue}{\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot -1}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]
   10. Step-by-step derivation

       1. lift-pow.f64 N/A

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

       2. unpow3 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 19.9

          \[\leadsto \left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot -1}{d}\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

   11. Applied rewrites 19.9%

       \[\leadsto \left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot -1}{d}\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -1 \cdot 10^{-68}:\\ \;\;\;\;\left(1 - \left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{\left(d \cdot d\right) \cdot \ell} \cdot 0.25\right) \cdot 0.5\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot -1}{d}\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\left(D \cdot M\right)}^{2}}{d}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-213}:\\ \;\;\;\;\left(-0.125 \cdot t\_0\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{\ell \cdot h} \cdot t\_0\right) \cdot -0.125}{\ell \cdot \ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ (pow (* D M) 2.0) d))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_1 -5e-213)
     (* (* -0.125 t_0) (sqrt (/ h (* (* l l) l))))
     (if (<= t_1 INFINITY)
       (/ (* (sqrt (/ h l)) d) h)
       (/ (* (* (sqrt (* l h)) t_0) -0.125) (* l l))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = pow((D * M), 2.0) / d;
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= -5e-213) {
		tmp = (-0.125 * t_0) * sqrt((h / ((l * l) * l)));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (sqrt((h / l)) * d) / h;
	} else {
		tmp = ((sqrt((l * h)) * t_0) * -0.125) / (l * l);
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.pow((D * M), 2.0) / d;
	double t_1 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= -5e-213) {
		tmp = (-0.125 * t_0) * Math.sqrt((h / ((l * l) * l)));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (Math.sqrt((h / l)) * d) / h;
	} else {
		tmp = ((Math.sqrt((l * h)) * t_0) * -0.125) / (l * l);
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.pow((D * M), 2.0) / d
	t_1 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_1 <= -5e-213:
		tmp = (-0.125 * t_0) * math.sqrt((h / ((l * l) * l)))
	elif t_1 <= math.inf:
		tmp = (math.sqrt((h / l)) * d) / h
	else:
		tmp = ((math.sqrt((l * h)) * t_0) * -0.125) / (l * l)
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64((Float64(D * M) ^ 2.0) / d)
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_1 <= -5e-213)
		tmp = Float64(Float64(-0.125 * t_0) * sqrt(Float64(h / Float64(Float64(l * l) * l))));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * d) / h);
	else
		tmp = Float64(Float64(Float64(sqrt(Float64(l * h)) * t_0) * -0.125) / Float64(l * l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = ((D * M) ^ 2.0) / d;
	t_1 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_1 <= -5e-213)
		tmp = (-0.125 * t_0) * sqrt((h / ((l * l) * l)));
	elseif (t_1 <= Inf)
		tmp = (sqrt((h / l)) * d) / h;
	else
		tmp = ((sqrt((l * h)) * t_0) * -0.125) / (l * l);
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Power[N[(D * M), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-213], N[(N[(-0.125 * t$95$0), $MachinePrecision] * N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] * -0.125), $MachinePrecision] / N[(l * l), $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)))) < -4.99999999999999977e-213

   1. Initial program 88.4%

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

   3. Taylor expanded in d around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow-prod-down N/A

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

      6. *-commutative N/A

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

      7. lower-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-pow.f64 38.9

          \[\leadsto \left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

   5. Applied rewrites 38.9%

      \[\leadsto \color{blue}{\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-*.f64 38.9

         \[\leadsto \left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

   7. Applied rewrites 38.9%

      \[\leadsto \left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

   if -4.99999999999999977e-213 < (*.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 81.5%

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

   3. 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

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

   5. Applied rewrites 64.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   6. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-*.f64 81.1

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

   8. Applied rewrites 81.1%

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

   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 l around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 12.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, \sqrt{\ell \cdot h}, \sqrt{\frac{{\ell}^{3}}{h}} \cdot d\right)}{\ell \cdot \ell}} \]

   6. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. unpow-prod-down N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 17.3

          \[\leadsto \frac{\left(\sqrt{\ell \cdot h} \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}\right) \cdot -0.125}{\ell \cdot \ell} \]

   8. Applied rewrites 17.3%

      \[\leadsto \frac{\left(\sqrt{\ell \cdot h} \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}\right) \cdot -0.125}{\color{blue}{\ell} \cdot \ell} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 50.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-213}:\\ \;\;\;\;\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\frac{{\ell}^{3}}{h}} \cdot d}{\ell}}{\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_0 -5e-213)
     (* (* -0.125 (/ (pow (* D M) 2.0) d)) (sqrt (/ h (* (* l l) l))))
     (if (<= t_0 INFINITY)
       (/ (* (sqrt (/ h l)) d) h)
       (/ (/ (* (sqrt (/ (pow l 3.0) h)) d) l) l)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -5e-213) {
		tmp = (-0.125 * (pow((D * M), 2.0) / d)) * sqrt((h / ((l * l) * l)));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (sqrt((h / l)) * d) / h;
	} else {
		tmp = ((sqrt((pow(l, 3.0) / h)) * d) / l) / l;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -5e-213) {
		tmp = (-0.125 * (Math.pow((D * M), 2.0) / d)) * Math.sqrt((h / ((l * l) * l)));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (Math.sqrt((h / l)) * d) / h;
	} else {
		tmp = ((Math.sqrt((Math.pow(l, 3.0) / h)) * d) / l) / l;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_0 <= -5e-213:
		tmp = (-0.125 * (math.pow((D * M), 2.0) / d)) * math.sqrt((h / ((l * l) * l)))
	elif t_0 <= math.inf:
		tmp = (math.sqrt((h / l)) * d) / h
	else:
		tmp = ((math.sqrt((math.pow(l, 3.0) / h)) * d) / l) / l
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= -5e-213)
		tmp = Float64(Float64(-0.125 * Float64((Float64(D * M) ^ 2.0) / d)) * sqrt(Float64(h / Float64(Float64(l * l) * l))));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * d) / h);
	else
		tmp = Float64(Float64(Float64(sqrt(Float64((l ^ 3.0) / h)) * d) / l) / l);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_0 <= -5e-213)
		tmp = (-0.125 * (((D * M) ^ 2.0) / d)) * sqrt((h / ((l * l) * l)));
	elseif (t_0 <= Inf)
		tmp = (sqrt((h / l)) * d) / h;
	else
		tmp = ((sqrt(((l ^ 3.0) / h)) * d) / l) / l;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-213], N[(N[(-0.125 * N[(N[Power[N[(D * M), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(N[Power[l, 3.0], $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] / l), $MachinePrecision] / l), $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)))) < -4.99999999999999977e-213

   1. Initial program 88.4%

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

   3. Taylor expanded in d around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow-prod-down N/A

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

      6. *-commutative N/A

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

      7. lower-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-pow.f64 38.9

          \[\leadsto \left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

   5. Applied rewrites 38.9%

      \[\leadsto \color{blue}{\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-*.f64 38.9

         \[\leadsto \left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

   7. Applied rewrites 38.9%

      \[\leadsto \left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

   if -4.99999999999999977e-213 < (*.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 81.5%

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

   3. 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

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

   5. Applied rewrites 64.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   6. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-*.f64 81.1

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

   8. Applied rewrites 81.1%

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

   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 l around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 12.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, \sqrt{\ell \cdot h}, \sqrt{\frac{{\ell}^{3}}{h}} \cdot d\right)}{\ell \cdot \ell}} \]

   6. Taylor expanded in d around inf

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

      1. *-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{{\ell}^{3}}{h}} \cdot d}{\ell \cdot \ell} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{{\ell}^{3}}{h}} \cdot d}{\ell \cdot \ell} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{{\ell}^{3}}{h}} \cdot d}{\ell \cdot \ell} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{{\ell}^{3}}{h}} \cdot d}{\ell \cdot \ell} \]

      5. lift-*.f64 5.5

         \[\leadsto \frac{\sqrt{\frac{{\ell}^{3}}{h}} \cdot d}{\ell \cdot \ell} \]

   8. Applied rewrites 5.5%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{{\ell}^{3}}{h}} \cdot d}{\ell \cdot \color{blue}{\ell}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{{\ell}^{3}}{h}} \cdot d}{\color{blue}{\ell \cdot \ell}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{\sqrt{\frac{{\ell}^{3}}{h}} \cdot d}{\ell}}{\color{blue}{\ell}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{\frac{{\ell}^{3}}{h}} \cdot d}{\ell}}{\color{blue}{\ell}} \]

      5. lower-/.f64 16.8

         \[\leadsto \frac{\frac{\sqrt{\frac{{\ell}^{3}}{h}} \cdot d}{\ell}}{\ell} \]

   10. Applied rewrites 16.8%

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

Alternative 8: 48.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-98}:\\ \;\;\;\;\left(\left(\frac{M \cdot M}{d} \cdot -0.125\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \left(D \cdot D\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\frac{{\ell}^{3}}{h}} \cdot d}{\ell}}{\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_0 -4e-98)
     (* (* (* (/ (* M M) d) -0.125) (sqrt (/ h (pow l 3.0)))) (* D D))
     (if (<= t_0 INFINITY)
       (/ (* (sqrt (/ h l)) d) h)
       (/ (/ (* (sqrt (/ (pow l 3.0) h)) d) l) l)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -4e-98) {
		tmp = ((((M * M) / d) * -0.125) * sqrt((h / pow(l, 3.0)))) * (D * D);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (sqrt((h / l)) * d) / h;
	} else {
		tmp = ((sqrt((pow(l, 3.0) / h)) * d) / l) / l;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -4e-98) {
		tmp = ((((M * M) / d) * -0.125) * Math.sqrt((h / Math.pow(l, 3.0)))) * (D * D);
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (Math.sqrt((h / l)) * d) / h;
	} else {
		tmp = ((Math.sqrt((Math.pow(l, 3.0) / h)) * d) / l) / l;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_0 <= -4e-98:
		tmp = ((((M * M) / d) * -0.125) * math.sqrt((h / math.pow(l, 3.0)))) * (D * D)
	elif t_0 <= math.inf:
		tmp = (math.sqrt((h / l)) * d) / h
	else:
		tmp = ((math.sqrt((math.pow(l, 3.0) / h)) * d) / l) / l
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= -4e-98)
		tmp = Float64(Float64(Float64(Float64(Float64(M * M) / d) * -0.125) * sqrt(Float64(h / (l ^ 3.0)))) * Float64(D * D));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * d) / h);
	else
		tmp = Float64(Float64(Float64(sqrt(Float64((l ^ 3.0) / h)) * d) / l) / l);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_0 <= -4e-98)
		tmp = ((((M * M) / d) * -0.125) * sqrt((h / (l ^ 3.0)))) * (D * D);
	elseif (t_0 <= Inf)
		tmp = (sqrt((h / l)) * d) / h;
	else
		tmp = ((sqrt(((l ^ 3.0) / h)) * d) / l) / l;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-98], N[(N[(N[(N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision] * N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(N[Power[l, 3.0], $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] / l), $MachinePrecision] / l), $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)))) < -3.99999999999999976e-98

   1. Initial program 88.3%

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

   3. Taylor expanded in D around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 23.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.125 \cdot \frac{M \cdot M}{d}, \sqrt{\frac{h}{{\ell}^{3}}}, \frac{d}{D \cdot D} \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\right) \cdot \left(D \cdot D\right)} \]

   6. Taylor expanded in d around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-sqrt.f64 30.3

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

   8. Applied rewrites 30.3%

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

   if -3.99999999999999976e-98 < (*.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 81.6%

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

   3. Taylor expanded in 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

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

   5. Applied rewrites 64.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   6. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-*.f64 80.4

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

   8. Applied rewrites 80.4%

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

   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 l around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 12.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, \sqrt{\ell \cdot h}, \sqrt{\frac{{\ell}^{3}}{h}} \cdot d\right)}{\ell \cdot \ell}} \]

   6. Taylor expanded in d around inf

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

      1. *-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{{\ell}^{3}}{h}} \cdot d}{\ell \cdot \ell} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{{\ell}^{3}}{h}} \cdot d}{\ell \cdot \ell} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{{\ell}^{3}}{h}} \cdot d}{\ell \cdot \ell} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{{\ell}^{3}}{h}} \cdot d}{\ell \cdot \ell} \]

      5. lift-*.f64 5.5

         \[\leadsto \frac{\sqrt{\frac{{\ell}^{3}}{h}} \cdot d}{\ell \cdot \ell} \]

   8. Applied rewrites 5.5%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{{\ell}^{3}}{h}} \cdot d}{\ell \cdot \color{blue}{\ell}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{{\ell}^{3}}{h}} \cdot d}{\color{blue}{\ell \cdot \ell}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{\sqrt{\frac{{\ell}^{3}}{h}} \cdot d}{\ell}}{\color{blue}{\ell}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{\frac{{\ell}^{3}}{h}} \cdot d}{\ell}}{\color{blue}{\ell}} \]

      5. lower-/.f64 16.8

         \[\leadsto \frac{\frac{\sqrt{\frac{{\ell}^{3}}{h}} \cdot d}{\ell}}{\ell} \]

   10. Applied rewrites 16.8%

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

Alternative 9: 45.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-213}:\\ \;\;\;\;\frac{\left(-d\right) \cdot t\_0}{h}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{t\_0 \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\frac{{\ell}^{3}}{h}} \cdot d}{\ell}}{\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ h l)))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_1 -5e-213)
     (/ (* (- d) t_0) h)
     (if (<= t_1 INFINITY)
       (/ (* t_0 d) h)
       (/ (/ (* (sqrt (/ (pow l 3.0) h)) d) l) l)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h / l));
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= -5e-213) {
		tmp = (-d * t_0) / h;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = ((sqrt((pow(l, 3.0) / h)) * d) / l) / l;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((h / l));
	double t_1 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= -5e-213) {
		tmp = (-d * t_0) / h;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = ((Math.sqrt((Math.pow(l, 3.0) / h)) * d) / l) / l;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((h / l))
	t_1 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_1 <= -5e-213:
		tmp = (-d * t_0) / h
	elif t_1 <= math.inf:
		tmp = (t_0 * d) / h
	else:
		tmp = ((math.sqrt((math.pow(l, 3.0) / h)) * d) / l) / l
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h / l))
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_1 <= -5e-213)
		tmp = Float64(Float64(Float64(-d) * t_0) / h);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(t_0 * d) / h);
	else
		tmp = Float64(Float64(Float64(sqrt(Float64((l ^ 3.0) / h)) * d) / l) / l);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((h / l));
	t_1 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_1 <= -5e-213)
		tmp = (-d * t_0) / h;
	elseif (t_1 <= Inf)
		tmp = (t_0 * d) / h;
	else
		tmp = ((sqrt(((l ^ 3.0) / h)) * d) / l) / l;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-213], N[(N[((-d) * t$95$0), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(t$95$0 * d), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(N[Power[l, 3.0], $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] / l), $MachinePrecision] / l), $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)))) < -4.99999999999999977e-213

   1. Initial program 88.4%

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

   3. 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

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

   5. Applied rewrites 52.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   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

      1. sqrt-pow2 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-sqrt.f64 28.3

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

   8. Applied rewrites 28.3%

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

   if -4.99999999999999977e-213 < (*.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 81.5%

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

   3. 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

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

   5. Applied rewrites 64.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   6. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-*.f64 81.1

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

   8. Applied rewrites 81.1%

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

   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 l around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 12.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, \sqrt{\ell \cdot h}, \sqrt{\frac{{\ell}^{3}}{h}} \cdot d\right)}{\ell \cdot \ell}} \]

   6. Taylor expanded in d around inf

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

      1. *-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{{\ell}^{3}}{h}} \cdot d}{\ell \cdot \ell} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{{\ell}^{3}}{h}} \cdot d}{\ell \cdot \ell} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{{\ell}^{3}}{h}} \cdot d}{\ell \cdot \ell} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{{\ell}^{3}}{h}} \cdot d}{\ell \cdot \ell} \]

      5. lift-*.f64 5.5

         \[\leadsto \frac{\sqrt{\frac{{\ell}^{3}}{h}} \cdot d}{\ell \cdot \ell} \]

   8. Applied rewrites 5.5%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{{\ell}^{3}}{h}} \cdot d}{\ell \cdot \color{blue}{\ell}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{{\ell}^{3}}{h}} \cdot d}{\color{blue}{\ell \cdot \ell}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{\sqrt{\frac{{\ell}^{3}}{h}} \cdot d}{\ell}}{\color{blue}{\ell}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{\frac{{\ell}^{3}}{h}} \cdot d}{\ell}}{\color{blue}{\ell}} \]

      5. lower-/.f64 16.8

         \[\leadsto \frac{\frac{\sqrt{\frac{{\ell}^{3}}{h}} \cdot d}{\ell}}{\ell} \]

   10. Applied rewrites 16.8%

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

Alternative 10: 46.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-213} \lor \neg \left(t\_1 \leq \infty\right):\\ \;\;\;\;\frac{\left(-d\right) \cdot t\_0}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot d}{h}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ h l)))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (or (<= t_1 -5e-213) (not (<= t_1 INFINITY)))
     (/ (* (- d) t_0) h)
     (/ (* t_0 d) h))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h / l));
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if ((t_1 <= -5e-213) || !(t_1 <= ((double) INFINITY))) {
		tmp = (-d * t_0) / h;
	} else {
		tmp = (t_0 * d) / h;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((h / l));
	double t_1 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if ((t_1 <= -5e-213) || !(t_1 <= Double.POSITIVE_INFINITY)) {
		tmp = (-d * t_0) / h;
	} else {
		tmp = (t_0 * d) / h;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((h / l))
	t_1 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if (t_1 <= -5e-213) or not (t_1 <= math.inf):
		tmp = (-d * t_0) / h
	else:
		tmp = (t_0 * d) / h
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h / l))
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if ((t_1 <= -5e-213) || !(t_1 <= Inf))
		tmp = Float64(Float64(Float64(-d) * t_0) / h);
	else
		tmp = Float64(Float64(t_0 * d) / h);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((h / l));
	t_1 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if ((t_1 <= -5e-213) || ~((t_1 <= Inf)))
		tmp = (-d * t_0) / h;
	else
		tmp = (t_0 * d) / h;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5e-213], N[Not[LessEqual[t$95$1, Infinity]], $MachinePrecision]], N[(N[((-d) * t$95$0), $MachinePrecision] / h), $MachinePrecision], N[(N[(t$95$0 * d), $MachinePrecision] / h), $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)))) < -4.99999999999999977e-213 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 63.9%

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

   3. Taylor expanded in 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

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

   5. Applied rewrites 45.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   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

      1. sqrt-pow2 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-sqrt.f64 25.1

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

   8. Applied rewrites 25.1%

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

   if -4.99999999999999977e-213 < (*.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 81.5%

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

   3. 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

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

   5. Applied rewrites 64.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   6. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-*.f64 81.1

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

   8. Applied rewrites 81.1%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-213} \lor \neg \left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty\right):\\ \;\;\;\;\frac{\left(-d\right) \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 71.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{d}{h}\right)}^{0.25}\\ t_1 := {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\\ \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot t\_1\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\left(t\_0 \cdot \left(t\_0 \cdot \sqrt{\frac{d}{\ell}}\right)\right) \cdot \left(1 - \left(0.5 \cdot t\_1\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-0.125 \cdot {\left(D \cdot M\right)}^{2}, {\left(\frac{h}{\ell}\right)}^{1.5}, \sqrt{\frac{h}{\ell}} \cdot \left(d \cdot d\right)\right)}{d}}{h}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (pow (/ d h) 0.25)) (t_1 (pow (/ (* M D) (* 2.0 d)) 2.0)))
   (if (<=
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (- 1.0 (* (* (/ 1.0 2.0) t_1) (/ h l))))
        INFINITY)
     (* (* t_0 (* t_0 (sqrt (/ d l)))) (- 1.0 (* (* 0.5 t_1) (/ h l))))
     (/
      (/
       (fma
        (* -0.125 (pow (* D M) 2.0))
        (pow (/ h l) 1.5)
        (* (sqrt (/ h l)) (* d d)))
       d)
      h))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = pow((d / h), 0.25);
	double t_1 = pow(((M * D) / (2.0 * d)), 2.0);
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * t_1) * (h / l)))) <= ((double) INFINITY)) {
		tmp = (t_0 * (t_0 * sqrt((d / l)))) * (1.0 - ((0.5 * t_1) * (h / l)));
	} else {
		tmp = (fma((-0.125 * pow((D * M), 2.0)), pow((h / l), 1.5), (sqrt((h / l)) * (d * d))) / d) / h;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(d / h) ^ 0.25
	t_1 = Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * t_1) * Float64(h / l)))) <= Inf)
		tmp = Float64(Float64(t_0 * Float64(t_0 * sqrt(Float64(d / l)))) * Float64(1.0 - Float64(Float64(0.5 * t_1) * Float64(h / l))));
	else
		tmp = Float64(Float64(fma(Float64(-0.125 * (Float64(D * M) ^ 2.0)), (Float64(h / l) ^ 1.5), Float64(sqrt(Float64(h / l)) * Float64(d * d))) / d) / h);
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Power[N[(d / h), $MachinePrecision], 0.25], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(t$95$0 * N[(t$95$0 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(0.5 * t$95$1), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.125 * N[Power[N[(D * M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision] + N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / h), $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)))) < +inf.0

   1. Initial program 84.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-pow.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-pow.f64 N/A

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

      14. lift-/.f64 84.9

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

   4. Applied rewrites 84.9%

      \[\leadsto \left(\color{blue}{\left({\left(\frac{d}{h}\right)}^{0.25} \cdot {\left(\frac{d}{h}\right)}^{0.25}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. metadata-eval N/A

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

      18. pow1/2 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lift-/.f64 84.9

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

   6. Applied rewrites 84.9%

      \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{0.25} \cdot \left({\left(\frac{d}{h}\right)}^{0.25} \cdot \sqrt{\frac{d}{\ell}}\right)\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. metadata-eval 84.9

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{0.25} \cdot \left({\left(\frac{d}{h}\right)}^{0.25} \cdot \sqrt{\frac{d}{\ell}}\right)\right) \cdot \left(1 - \left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   8. Applied rewrites 84.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{0.25} \cdot \left({\left(\frac{d}{h}\right)}^{0.25} \cdot \sqrt{\frac{d}{\ell}}\right)\right) \cdot \left(1 - \left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   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

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

   5. Applied rewrites 25.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   6. Taylor expanded in d around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 30.5%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(-0.125 \cdot {\left(D \cdot M\right)}^{2}, {\left(\frac{h}{\ell}\right)}^{1.5}, \sqrt{\frac{h}{\ell}} \cdot \left(d \cdot d\right)\right)}{d}}{h} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 70.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-0.125 \cdot {\left(D \cdot M\right)}^{2}, {\left(\frac{h}{\ell}\right)}^{1.5}, \sqrt{\frac{h}{\ell}} \cdot \left(d \cdot d\right)\right)}{d}}{h}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
       (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))
      INFINITY)
   (*
    (sqrt (/ d h))
    (*
     (sqrt (/ d l))
     (- 1.0 (* (/ h l) (* (pow (* (/ M 2.0) (/ D d)) 2.0) 0.5)))))
   (/
    (/
     (fma
      (* -0.125 (pow (* D M) 2.0))
      (pow (/ h l) 1.5)
      (* (sqrt (/ h l)) (* d d)))
     d)
    h)))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= ((double) INFINITY)) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - ((h / l) * (pow(((M / 2.0) * (D / d)), 2.0) * 0.5))));
	} else {
		tmp = (fma((-0.125 * pow((D * M), 2.0)), pow((h / l), 1.5), (sqrt((h / l)) * (d * d))) / d) / h;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= Inf)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * 0.5)))));
	else
		tmp = Float64(Float64(fma(Float64(-0.125 * (Float64(D * M) ^ 2.0)), (Float64(h / l) ^ 1.5), Float64(sqrt(Float64(h / l)) * Float64(d * d))) / d) / h);
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.125 * N[Power[N[(D * M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision] + N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / h), $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)))) < +inf.0

   1. Initial program 84.9%

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

   4. Applied rewrites 84.2%

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

   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

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

   5. Applied rewrites 25.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   6. Taylor expanded in d around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 30.5%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(-0.125 \cdot {\left(D \cdot M\right)}^{2}, {\left(\frac{h}{\ell}\right)}^{1.5}, \sqrt{\frac{h}{\ell}} \cdot \left(d \cdot d\right)\right)}{d}}{h} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 69.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
       (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))
      INFINITY)
   (*
    (sqrt (/ d h))
    (*
     (sqrt (/ d l))
     (- 1.0 (* (/ h l) (* (pow (* (/ M 2.0) (/ D d)) 2.0) 0.5)))))
   (/
    (fma
     (* -0.125 (/ (* (* D M) (* D M)) d))
     (sqrt (pow (/ h l) 3.0))
     (* (sqrt (/ h l)) d))
    h)))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= ((double) INFINITY)) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - ((h / l) * (pow(((M / 2.0) * (D / d)), 2.0) * 0.5))));
	} else {
		tmp = fma((-0.125 * (((D * M) * (D * M)) / d)), sqrt(pow((h / l), 3.0)), (sqrt((h / l)) * d)) / h;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= Inf)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * 0.5)))));
	else
		tmp = Float64(fma(Float64(-0.125 * Float64(Float64(Float64(D * M) * Float64(D * M)) / d)), sqrt((Float64(h / l) ^ 3.0)), Float64(sqrt(Float64(h / l)) * d)) / h);
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.125 * N[(N[(N[(D * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Power[N[(h / l), $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / h), $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)))) < +inf.0

   1. Initial program 84.9%

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

   4. Applied rewrites 84.2%

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

   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

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

   5. Applied rewrites 25.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h} \]

      6. lift-*.f64 25.2

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

   7. Applied rewrites 25.2%

      \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 69.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right)\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
       (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))
      INFINITY)
   (*
    (- 1.0 (* (/ h l) (* (pow (* (* 0.5 M) (/ D d)) 2.0) 0.5)))
    (* (sqrt (/ d l)) (sqrt (/ d h))))
   (/
    (fma
     (* -0.125 (/ (* (* D M) (* D M)) d))
     (sqrt (pow (/ h l) 3.0))
     (* (sqrt (/ h l)) d))
    h)))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= ((double) INFINITY)) {
		tmp = (1.0 - ((h / l) * (pow(((0.5 * M) * (D / d)), 2.0) * 0.5))) * (sqrt((d / l)) * sqrt((d / h)));
	} else {
		tmp = fma((-0.125 * (((D * M) * (D * M)) / d)), sqrt(pow((h / l), 3.0)), (sqrt((h / l)) * d)) / h;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= Inf)
		tmp = Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(0.5 * M) * Float64(D / d)) ^ 2.0) * 0.5))) * Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))));
	else
		tmp = Float64(fma(Float64(-0.125 * Float64(Float64(Float64(D * M) * Float64(D * M)) / d)), sqrt((Float64(h / l) ^ 3.0)), Float64(sqrt(Float64(h / l)) * d)) / h);
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(0.5 * M), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.125 * N[(N[(N[(D * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Power[N[(h / l), $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / h), $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)))) < +inf.0

   1. Initial program 84.9%

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

   4. Applied rewrites 84.2%

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

   5. Taylor expanded in M around 0

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval 84.2

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

   7. Applied rewrites 84.2%

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

   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

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

   5. Applied rewrites 25.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h} \]

      6. lift-*.f64 25.2

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

   7. Applied rewrites 25.2%

      \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 40.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-213}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot d\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
       (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))
      -5e-213)
   (* (sqrt (/ 1.0 (* l h))) d)
   (/ (* (sqrt (/ h l)) d) h)))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e-213) {
		tmp = sqrt((1.0 / (l * h))) * d;
	} else {
		tmp = (sqrt((h / 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) :: tmp
    if (((((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))) <= (-5d-213)) then
        tmp = sqrt((1.0d0 / (l * h))) * d
    else
        tmp = (sqrt((h / 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 tmp;
	if (((Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e-213) {
		tmp = Math.sqrt((1.0 / (l * h))) * d;
	} else {
		tmp = (Math.sqrt((h / l)) * d) / h;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if ((math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e-213:
		tmp = math.sqrt((1.0 / (l * h))) * d
	else:
		tmp = (math.sqrt((h / l)) * d) / h
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= -5e-213)
		tmp = Float64(sqrt(Float64(1.0 / Float64(l * h))) * d);
	else
		tmp = Float64(Float64(sqrt(Float64(h / l)) * d) / h);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (((((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)))) <= -5e-213)
		tmp = sqrt((1.0 / (l * h))) * d;
	else
		tmp = (sqrt((h / l)) * d) / h;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-213], N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] / h), $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)))) < -4.99999999999999977e-213

   1. Initial program 88.4%

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

   3. 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

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 11.3

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

   5. Applied rewrites 11.3%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow-1 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 11.3

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

   7. Applied rewrites 11.3%

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

   if -4.99999999999999977e-213 < (*.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 59.0%

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

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

   5. Applied rewrites 53.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   6. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-*.f64 59.9

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

   8. Applied rewrites 59.9%

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

Alternative 16: 51.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(D \cdot M\right)}^{2}\\ t_1 := \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\\ \mathbf{if}\;\ell \leq -7.4 \cdot 10^{-38}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq -1.25 \cdot 10^{-303}:\\ \;\;\;\;\left(-0.125 \cdot \frac{t\_0 \cdot -1}{d}\right) \cdot t\_1\\ \mathbf{elif}\;\ell \leq 2.15 \cdot 10^{+43}:\\ \;\;\;\;\left(-0.125 \cdot \frac{t\_0}{d}\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (pow (* D M) 2.0)) (t_1 (sqrt (/ h (* (* l l) l)))))
   (if (<= l -7.4e-38)
     (* (- d) (pow (* l h) -0.5))
     (if (<= l -1.25e-303)
       (* (* -0.125 (/ (* t_0 -1.0) d)) t_1)
       (if (<= l 2.15e+43)
         (* (* -0.125 (/ t_0 d)) t_1)
         (* (/ 1.0 (* (sqrt l) (sqrt h))) d))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = pow((D * M), 2.0);
	double t_1 = sqrt((h / ((l * l) * l)));
	double tmp;
	if (l <= -7.4e-38) {
		tmp = -d * pow((l * h), -0.5);
	} else if (l <= -1.25e-303) {
		tmp = (-0.125 * ((t_0 * -1.0) / d)) * t_1;
	} else if (l <= 2.15e+43) {
		tmp = (-0.125 * (t_0 / d)) * t_1;
	} else {
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (d_1 * m) ** 2.0d0
    t_1 = sqrt((h / ((l * l) * l)))
    if (l <= (-7.4d-38)) then
        tmp = -d * ((l * h) ** (-0.5d0))
    else if (l <= (-1.25d-303)) then
        tmp = ((-0.125d0) * ((t_0 * (-1.0d0)) / d)) * t_1
    else if (l <= 2.15d+43) then
        tmp = ((-0.125d0) * (t_0 / d)) * t_1
    else
        tmp = (1.0d0 / (sqrt(l) * sqrt(h))) * d
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.pow((D * M), 2.0);
	double t_1 = Math.sqrt((h / ((l * l) * l)));
	double tmp;
	if (l <= -7.4e-38) {
		tmp = -d * Math.pow((l * h), -0.5);
	} else if (l <= -1.25e-303) {
		tmp = (-0.125 * ((t_0 * -1.0) / d)) * t_1;
	} else if (l <= 2.15e+43) {
		tmp = (-0.125 * (t_0 / d)) * t_1;
	} else {
		tmp = (1.0 / (Math.sqrt(l) * Math.sqrt(h))) * d;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.pow((D * M), 2.0)
	t_1 = math.sqrt((h / ((l * l) * l)))
	tmp = 0
	if l <= -7.4e-38:
		tmp = -d * math.pow((l * h), -0.5)
	elif l <= -1.25e-303:
		tmp = (-0.125 * ((t_0 * -1.0) / d)) * t_1
	elif l <= 2.15e+43:
		tmp = (-0.125 * (t_0 / d)) * t_1
	else:
		tmp = (1.0 / (math.sqrt(l) * math.sqrt(h))) * d
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(D * M) ^ 2.0
	t_1 = sqrt(Float64(h / Float64(Float64(l * l) * l)))
	tmp = 0.0
	if (l <= -7.4e-38)
		tmp = Float64(Float64(-d) * (Float64(l * h) ^ -0.5));
	elseif (l <= -1.25e-303)
		tmp = Float64(Float64(-0.125 * Float64(Float64(t_0 * -1.0) / d)) * t_1);
	elseif (l <= 2.15e+43)
		tmp = Float64(Float64(-0.125 * Float64(t_0 / d)) * t_1);
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(l) * sqrt(h))) * d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (D * M) ^ 2.0;
	t_1 = sqrt((h / ((l * l) * l)));
	tmp = 0.0;
	if (l <= -7.4e-38)
		tmp = -d * ((l * h) ^ -0.5);
	elseif (l <= -1.25e-303)
		tmp = (-0.125 * ((t_0 * -1.0) / d)) * t_1;
	elseif (l <= 2.15e+43)
		tmp = (-0.125 * (t_0 / d)) * t_1;
	else
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Power[N[(D * M), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -7.4e-38], N[((-d) * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1.25e-303], N[(N[(-0.125 * N[(N[(t$95$0 * -1.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[l, 2.15e+43], N[(N[(-0.125 * N[(t$95$0 / d), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -7.4e-38

   1. Initial program 53.0%

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

      1. lift-/.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-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-pow.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-pow.f64 N/A

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

      14. lift-/.f64 53.0

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

   4. Applied rewrites 53.0%

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

   5. Taylor expanded in l around -inf

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

      1. pow-prod-up N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow2 N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. inv-pow N/A

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

      12. *-commutative N/A

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

      13. sqrt-pow1 N/A

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

      14. lower-pow.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. metadata-eval 44.2

          \[\leadsto \left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5} \]

   7. Applied rewrites 44.2%

      \[\leadsto \color{blue}{\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}} \]

   if -7.4e-38 < l < -1.25e-303

   1. Initial program 80.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-pow.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-pow.f64 N/A

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

      14. lift-/.f64 80.8

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

   4. Applied rewrites 80.8%

      \[\leadsto \left(\color{blue}{\left({\left(\frac{d}{h}\right)}^{0.25} \cdot {\left(\frac{d}{h}\right)}^{0.25}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. metadata-eval N/A

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

      18. pow1/2 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lift-/.f64 80.8

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

   6. Applied rewrites 80.8%

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

   7. Taylor expanded in h around -inf

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

   9. Applied rewrites 64.5%

      \[\leadsto \color{blue}{\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot -1}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]
   10. Step-by-step derivation

       1. lift-pow.f64 N/A

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

       2. unpow3 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 64.5

          \[\leadsto \left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot -1}{d}\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

   11. Applied rewrites 64.5%

       \[\leadsto \left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot -1}{d}\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

   if -1.25e-303 < l < 2.15e43

   1. Initial program 82.0%

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

   3. Taylor expanded in d around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow-prod-down N/A

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

      6. *-commutative N/A

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

      7. lower-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-pow.f64 56.6

          \[\leadsto \left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

   5. Applied rewrites 56.6%

      \[\leadsto \color{blue}{\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-*.f64 56.6

         \[\leadsto \left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

   7. Applied rewrites 56.6%

      \[\leadsto \left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

   if 2.15e43 < l

   1. Initial program 70.0%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 51.2

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

   5. Applied rewrites 51.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. *-commutative N/A

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

      5. inv-pow N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-*.f64 51.1

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

   7. Applied rewrites 51.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 60.9

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

   9. Applied rewrites 60.9%

      \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]
3. Recombined 4 regimes into one program.

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7.4 \cdot 10^{-38}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq -1.25 \cdot 10^{-303}:\\ \;\;\;\;\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot -1}{d}\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 2.15 \cdot 10^{+43}:\\ \;\;\;\;\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 27.4% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{\ell \cdot h}} \cdot d \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (* (sqrt (/ 1.0 (* l h))) d))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[d_, h_, l_, M_, D_] := N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]

Derivation

1. Initial program 71.3%

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

3. Taylor expanded in d around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. inv-pow N/A

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

   5. lower-pow.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 25.7

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

5. Applied rewrites 25.7%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow-1 N/A

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

   4. lower-/.f64 N/A

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

   5. lift-*.f64 25.7

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

7. Applied rewrites 25.7%

   \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025043 
(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)))))