Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.6% → 71.2%

Time: 10.3s

Alternatives: 15

Speedup: 1.5×

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: 71.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.3 \cdot 10^{+39}:\\ \;\;\;\;\left({\left(\ell \cdot h\right)}^{-0.5} \cdot \left(-d\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;d \leq 3.3 \cdot 10^{-264}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\frac{h}{\ell}}, d, \frac{{\left(\frac{h}{\ell}\right)}^{1.5} \cdot {\left(M \cdot D\right)}^{2}}{d} \cdot -0.125\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(\left(\left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)\right) \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -2.3e+39)
   (*
    (* (pow (* l h) -0.5) (- d))
    (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))
   (if (<= d 3.3e-264)
     (/
      (fma
       (sqrt (/ h l))
       d
       (* (/ (* (pow (/ h l) 1.5) (pow (* M D) 2.0)) d) -0.125))
      h)
     (*
      (/ (sqrt d) (sqrt h))
      (*
       (sqrt (/ d l))
       (-
        1.0
        (*
         (/ h l)
         (* (* (* (/ D d) (/ M 2.0)) (* (/ D d) (* 0.5 M))) 0.5))))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -2.3e+39) {
		tmp = (pow((l * h), -0.5) * -d) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	} else if (d <= 3.3e-264) {
		tmp = fma(sqrt((h / l)), d, (((pow((h / l), 1.5) * pow((M * D), 2.0)) / d) * -0.125)) / h;
	} else {
		tmp = (sqrt(d) / sqrt(h)) * (sqrt((d / l)) * (1.0 - ((h / l) * ((((D / d) * (M / 2.0)) * ((D / d) * (0.5 * M))) * 0.5))));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -2.3e+39)
		tmp = Float64(Float64((Float64(l * h) ^ -0.5) * Float64(-d)) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))));
	elseif (d <= 3.3e-264)
		tmp = Float64(fma(sqrt(Float64(h / l)), d, Float64(Float64(Float64((Float64(h / l) ^ 1.5) * (Float64(M * D) ^ 2.0)) / d) * -0.125)) / h);
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(Float64(h / l) * Float64(Float64(Float64(Float64(D / d) * Float64(M / 2.0)) * Float64(Float64(D / d) * Float64(0.5 * M))) * 0.5)))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -2.3e+39], N[(N[(N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision] * (-d)), $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[d, 3.3e-264], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d + N[(N[(N[(N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision] * N[Power[N[(M * D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[(N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[(0.5 * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.30000000000000012e39

   1. Initial program 67.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 56.1

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

      11. lift-/.f64 N/A

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

      12. metadata-eval 56.1

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

   4. Applied rewrites 56.1%

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

   5. Taylor expanded in d around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow-1 N/A

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

      7. sqrt-pow1 N/A

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

      8. *-commutative N/A

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

      9. lower-pow.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. metadata-eval 77.6

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

   7. Applied rewrites 77.6%

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

   if -2.30000000000000012e39 < d < 3.30000000000000013e-264

   1. Initial program 54.9%

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

   4. Applied rewrites 54.7%

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

   5. Taylor expanded in h around 0

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

   7. Applied rewrites 58.6%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. cube-div N/A

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

      8. *-commutative N/A

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

      9. lower-/.f64 N/A

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

   9. Applied rewrites 68.9%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{h}{\ell}}, d, \frac{{\left(\frac{h}{\ell}\right)}^{1.5} \cdot {\left(M \cdot D\right)}^{2}}{d} \cdot -0.125\right)}{h} \]

   if 3.30000000000000013e-264 < d

   1. Initial program 75.5%

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

   4. Applied rewrites 75.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 75.5

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 75.5

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-/.f64 75.5

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

   6. Applied rewrites 75.5%

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

   7. Taylor expanded in M around 0

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

      1. lower-*.f64 75.5

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

   9. Applied rewrites 75.5%

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

       1. lift-/.f64 N/A

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

       2. lift-sqrt.f64 N/A

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

       3. sqrt-div N/A

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

       4. lower-/.f64 N/A

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

       5. lower-sqrt.f64 N/A

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

       6. lower-sqrt.f64 81.2

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

   11. Applied rewrites 81.2%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.3 \cdot 10^{+39}:\\ \;\;\;\;\left({\left(\ell \cdot h\right)}^{-0.5} \cdot \left(-d\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;d \leq 3.3 \cdot 10^{-264}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\frac{h}{\ell}}, d, \frac{{\left(\frac{h}{\ell}\right)}^{1.5} \cdot {\left(M \cdot D\right)}^{2}}{d} \cdot -0.125\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(\left(\left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)\right) \cdot 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 63.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ t_3 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-171}:\\ \;\;\;\;t\_3 \cdot \left(t\_0 \cdot \left(\frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-170}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+210}:\\ \;\;\;\;t\_3 \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot h\right)}^{-0.5} \cdot \left(-d\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l)))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_2 (/ (* (sqrt (/ h l)) d) h))
        (t_3 (sqrt (/ d h))))
   (if (<= t_1 -5e-171)
     (* t_3 (* t_0 (* (/ (* (* (* M D) (* M D)) h) (* (* d d) l)) -0.125)))
     (if (<= t_1 2e-170)
       t_2
       (if (<= t_1 5e+210)
         (* t_3 t_0)
         (if (<= t_1 INFINITY) t_2 (* (pow (* l h) -0.5) (- d))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / l));
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = (sqrt((h / l)) * d) / h;
	double t_3 = sqrt((d / h));
	double tmp;
	if (t_1 <= -5e-171) {
		tmp = t_3 * (t_0 * (((((M * D) * (M * D)) * h) / ((d * d) * l)) * -0.125));
	} else if (t_1 <= 2e-170) {
		tmp = t_2;
	} else if (t_1 <= 5e+210) {
		tmp = t_3 * t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = pow((l * h), -0.5) * -d;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((d / l));
	double t_1 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = (Math.sqrt((h / l)) * d) / h;
	double t_3 = Math.sqrt((d / h));
	double tmp;
	if (t_1 <= -5e-171) {
		tmp = t_3 * (t_0 * (((((M * D) * (M * D)) * h) / ((d * d) * l)) * -0.125));
	} else if (t_1 <= 2e-170) {
		tmp = t_2;
	} else if (t_1 <= 5e+210) {
		tmp = t_3 * t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else {
		tmp = Math.pow((l * h), -0.5) * -d;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / l))
	t_1 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_2 = (math.sqrt((h / l)) * d) / h
	t_3 = math.sqrt((d / h))
	tmp = 0
	if t_1 <= -5e-171:
		tmp = t_3 * (t_0 * (((((M * D) * (M * D)) * h) / ((d * d) * l)) * -0.125))
	elif t_1 <= 2e-170:
		tmp = t_2
	elif t_1 <= 5e+210:
		tmp = t_3 * t_0
	elif t_1 <= math.inf:
		tmp = t_2
	else:
		tmp = math.pow((l * h), -0.5) * -d
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / l))
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_2 = Float64(Float64(sqrt(Float64(h / l)) * d) / h)
	t_3 = sqrt(Float64(d / h))
	tmp = 0.0
	if (t_1 <= -5e-171)
		tmp = Float64(t_3 * Float64(t_0 * Float64(Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) * h) / Float64(Float64(d * d) * l)) * -0.125)));
	elseif (t_1 <= 2e-170)
		tmp = t_2;
	elseif (t_1 <= 5e+210)
		tmp = Float64(t_3 * t_0);
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64((Float64(l * h) ^ -0.5) * Float64(-d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / l));
	t_1 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_2 = (sqrt((h / l)) * d) / h;
	t_3 = sqrt((d / h));
	tmp = 0.0;
	if (t_1 <= -5e-171)
		tmp = t_3 * (t_0 * (((((M * D) * (M * D)) * h) / ((d * d) * l)) * -0.125));
	elseif (t_1 <= 2e-170)
		tmp = t_2;
	elseif (t_1 <= 5e+210)
		tmp = t_3 * t_0;
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = ((l * h) ^ -0.5) * -d;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] / h), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -5e-171], N[(t$95$3 * N[(t$95$0 * N[(N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-170], t$95$2, If[LessEqual[t$95$1, 5e+210], N[(t$95$3 * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$2, N[(N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision] * (-d)), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -4.99999999999999992e-171

   1. Initial program 88.9%

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

   4. Applied rewrites 88.8%

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

   5. Taylor expanded in d around 0

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

      1. frac-times N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 72.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 72.2

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

   9. Applied rewrites 72.2%

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

   if -4.99999999999999992e-171 < (*.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)))) < 1.99999999999999997e-170 or 4.9999999999999998e210 < (*.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 50.9%

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

   4. Applied rewrites 50.8%

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

   5. Taylor expanded in h around 0

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

   7. Applied rewrites 61.4%

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

   8. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.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 70.3

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

   10. Applied rewrites 70.3%

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

   if 1.99999999999999997e-170 < (*.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)))) < 4.9999999999999998e210

   1. Initial program 99.1%

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in d around inf

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 99.1

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

   7. Applied rewrites 99.1%

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

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

   1. Initial program 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

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 0.0

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

      11. lift-/.f64 N/A

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

      12. metadata-eval 0.0

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in d around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow-1 N/A

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

      7. sqrt-pow1 N/A

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

      8. *-commutative N/A

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

      9. lower-pow.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. metadata-eval 15.8

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

   7. Applied rewrites 15.8%

      \[\leadsto \color{blue}{-{\left(\ell \cdot h\right)}^{-0.5} \cdot d} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-171}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2 \cdot 10^{-170}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+210}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot h\right)}^{-0.5} \cdot \left(-d\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-171}:\\ \;\;\;\;t\_2 \cdot \left(t\_0 \cdot \left(\frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-170} \lor \neg \left(t\_1 \leq 5 \cdot 10^{+210}\right):\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l)))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_2 (sqrt (/ d h))))
   (if (<= t_1 -5e-171)
     (* t_2 (* t_0 (* (/ (* (* (* M D) (* M D)) h) (* (* d d) l)) -0.125)))
     (if (or (<= t_1 2e-170) (not (<= t_1 5e+210)))
       (/ (* (sqrt (/ h l)) d) h)
       (* t_2 t_0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / l));
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = sqrt((d / h));
	double tmp;
	if (t_1 <= -5e-171) {
		tmp = t_2 * (t_0 * (((((M * D) * (M * D)) * h) / ((d * d) * l)) * -0.125));
	} else if ((t_1 <= 2e-170) || !(t_1 <= 5e+210)) {
		tmp = (sqrt((h / l)) * d) / h;
	} else {
		tmp = t_2 * t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt((d / l))
    t_1 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    t_2 = sqrt((d / h))
    if (t_1 <= (-5d-171)) then
        tmp = t_2 * (t_0 * (((((m * d_1) * (m * d_1)) * h) / ((d * d) * l)) * (-0.125d0)))
    else if ((t_1 <= 2d-170) .or. (.not. (t_1 <= 5d+210))) then
        tmp = (sqrt((h / l)) * d) / h
    else
        tmp = t_2 * t_0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((d / l));
	double t_1 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = Math.sqrt((d / h));
	double tmp;
	if (t_1 <= -5e-171) {
		tmp = t_2 * (t_0 * (((((M * D) * (M * D)) * h) / ((d * d) * l)) * -0.125));
	} else if ((t_1 <= 2e-170) || !(t_1 <= 5e+210)) {
		tmp = (Math.sqrt((h / l)) * d) / h;
	} else {
		tmp = t_2 * t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / l))
	t_1 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_2 = math.sqrt((d / h))
	tmp = 0
	if t_1 <= -5e-171:
		tmp = t_2 * (t_0 * (((((M * D) * (M * D)) * h) / ((d * d) * l)) * -0.125))
	elif (t_1 <= 2e-170) or not (t_1 <= 5e+210):
		tmp = (math.sqrt((h / l)) * d) / h
	else:
		tmp = t_2 * t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / l))
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_2 = sqrt(Float64(d / h))
	tmp = 0.0
	if (t_1 <= -5e-171)
		tmp = Float64(t_2 * Float64(t_0 * Float64(Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) * h) / Float64(Float64(d * d) * l)) * -0.125)));
	elseif ((t_1 <= 2e-170) || !(t_1 <= 5e+210))
		tmp = Float64(Float64(sqrt(Float64(h / l)) * d) / h);
	else
		tmp = Float64(t_2 * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / l));
	t_1 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_2 = sqrt((d / h));
	tmp = 0.0;
	if (t_1 <= -5e-171)
		tmp = t_2 * (t_0 * (((((M * D) * (M * D)) * h) / ((d * d) * l)) * -0.125));
	elseif ((t_1 <= 2e-170) || ~((t_1 <= 5e+210)))
		tmp = (sqrt((h / l)) * d) / h;
	else
		tmp = t_2 * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -5e-171], N[(t$95$2 * N[(t$95$0 * N[(N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$1, 2e-170], N[Not[LessEqual[t$95$1, 5e+210]], $MachinePrecision]], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] / h), $MachinePrecision], N[(t$95$2 * t$95$0), $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)))) < -4.99999999999999992e-171

   1. Initial program 88.9%

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

   4. Applied rewrites 88.8%

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

   5. Taylor expanded in d around 0

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

      1. frac-times N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 72.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 72.2

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

   9. Applied rewrites 72.2%

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

   if -4.99999999999999992e-171 < (*.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)))) < 1.99999999999999997e-170 or 4.9999999999999998e210 < (*.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 30.2%

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

   4. Applied rewrites 30.1%

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

   5. Taylor expanded in h around 0

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

   7. Applied rewrites 46.1%

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

   8. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.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 43.7

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

   10. Applied rewrites 43.7%

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

   if 1.99999999999999997e-170 < (*.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)))) < 4.9999999999999998e210

   1. Initial program 99.1%

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in d around inf

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 99.1

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

   7. Applied rewrites 99.1%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-171}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2 \cdot 10^{-170} \lor \neg \left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+210}\right):\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 48.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(-d\right) \cdot t\_0}{h}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-170} \lor \neg \left(t\_1 \leq 5 \cdot 10^{+210}\right):\\ \;\;\;\;\frac{t\_0 \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \end{array} \end{array} \]

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-171)
     (/ (* (- d) t_0) h)
     (if (or (<= t_1 2e-170) (not (<= t_1 5e+210)))
       (/ (* t_0 d) h)
       (* (sqrt (/ d h)) (sqrt (/ d 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-171) {
		tmp = (-d * t_0) / h;
	} else if ((t_1 <= 2e-170) || !(t_1 <= 5e+210)) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = sqrt((d / h)) * sqrt((d / l));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((h / l))
    t_1 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_1 <= (-5d-171)) then
        tmp = (-d * t_0) / h
    else if ((t_1 <= 2d-170) .or. (.not. (t_1 <= 5d+210))) then
        tmp = (t_0 * d) / h
    else
        tmp = sqrt((d / h)) * sqrt((d / l))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((h / l));
	double 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-171) {
		tmp = (-d * t_0) / h;
	} else if ((t_1 <= 2e-170) || !(t_1 <= 5e+210)) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / 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-171:
		tmp = (-d * t_0) / h
	elif (t_1 <= 2e-170) or not (t_1 <= 5e+210):
		tmp = (t_0 * d) / h
	else:
		tmp = math.sqrt((d / h)) * math.sqrt((d / 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-171)
		tmp = Float64(Float64(Float64(-d) * t_0) / h);
	elseif ((t_1 <= 2e-170) || !(t_1 <= 5e+210))
		tmp = Float64(Float64(t_0 * d) / h);
	else
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / 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-171)
		tmp = (-d * t_0) / h;
	elseif ((t_1 <= 2e-170) || ~((t_1 <= 5e+210)))
		tmp = (t_0 * d) / h;
	else
		tmp = sqrt((d / h)) * sqrt((d / 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-171], N[(N[((-d) * t$95$0), $MachinePrecision] / h), $MachinePrecision], If[Or[LessEqual[t$95$1, 2e-170], N[Not[LessEqual[t$95$1, 5e+210]], $MachinePrecision]], N[(N[(t$95$0 * d), $MachinePrecision] / h), $MachinePrecision], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -4.99999999999999992e-171

   1. Initial program 88.9%

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

   4. Applied rewrites 88.8%

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

   5. Taylor expanded in h around 0

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

   7. Applied rewrites 64.1%

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

   8. Taylor expanded in l around -inf

      \[\leadsto \frac{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   9. 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. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-/.f64 21.9

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

   10. Applied rewrites 21.9%

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

   if -4.99999999999999992e-171 < (*.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)))) < 1.99999999999999997e-170 or 4.9999999999999998e210 < (*.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 30.2%

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

   4. Applied rewrites 30.1%

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

   5. Taylor expanded in h around 0

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

   7. Applied rewrites 46.1%

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

   8. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.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 43.7

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

   10. Applied rewrites 43.7%

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

   if 1.99999999999999997e-170 < (*.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)))) < 4.9999999999999998e210

   1. Initial program 99.1%

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in d around inf

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 99.1

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

   7. Applied rewrites 99.1%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(-d\right) \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2 \cdot 10^{-170} \lor \neg \left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+210}\right):\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+210}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(\left(\frac{D}{d} \cdot \left(\frac{M}{2} \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right)\right) \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{t\_0 \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot \frac{M \cdot M}{d}, {\left(\frac{h}{\ell}\right)}^{1.5}, \frac{d}{D \cdot D} \cdot t\_0\right) \cdot \left(D \cdot D\right)}{h}\\ \end{array} \end{array} \]

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 (<= t_1 5e+210)
     (*
      (sqrt (/ d h))
      (*
       (sqrt (/ d l))
       (-
        1.0
        (* (/ h l) (* (* (/ D d) (* (/ M 2.0) (* (/ M 2.0) (/ D d)))) 0.5)))))
     (if (<= t_1 INFINITY)
       (/ (* t_0 d) h)
       (/
        (*
         (fma (* -0.125 (/ (* M M) d)) (pow (/ h l) 1.5) (* (/ d (* D D)) t_0))
         (* D 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+210) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - ((h / l) * (((D / d) * ((M / 2.0) * ((M / 2.0) * (D / d)))) * 0.5))));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = (fma((-0.125 * ((M * M) / d)), pow((h / l), 1.5), ((d / (D * D)) * t_0)) * (D * 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+210)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(Float64(h / l) * Float64(Float64(Float64(D / d) * Float64(Float64(M / 2.0) * Float64(Float64(M / 2.0) * Float64(D / d)))) * 0.5)))));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(t_0 * d) / h);
	else
		tmp = Float64(Float64(fma(Float64(-0.125 * Float64(Float64(M * M) / d)), (Float64(h / l) ^ 1.5), Float64(Float64(d / Float64(D * D)) * t_0)) * Float64(D * D)) / h);
	end
	return 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+210], 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[(N[(D / d), $MachinePrecision] * N[(N[(M / 2.0), $MachinePrecision] * N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(t$95$0 * d), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(N[(-0.125 * N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision] + N[(N[(d / N[(D * D), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(D * D), $MachinePrecision]), $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)))) < 4.9999999999999998e210

   1. Initial program 86.0%

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

   4. Applied rewrites 85.9%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 85.9

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 85.9

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-/.f64 85.9

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

   6. Applied rewrites 85.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 84.4

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 84.4

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

   8. Applied rewrites 84.4%

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

   if 4.9999999999999998e210 < (*.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 51.0%

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in h around 0

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

   7. Applied rewrites 52.9%

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

   8. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.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 73.7

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

   10. Applied rewrites 73.7%

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

   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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in h around 0

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

   7. Applied rewrites 23.9%

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

   8. Taylor expanded in D around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 23.8%

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

Alternative 6: 70.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+210}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(\left(\frac{D}{d} \cdot \left(\frac{M}{2} \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right)\right) \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{t\_0 \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, d, \left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot -0.125\right)}{h}\\ \end{array} \end{array} \]

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 (<= t_1 5e+210)
     (*
      (sqrt (/ d h))
      (*
       (sqrt (/ d l))
       (-
        1.0
        (* (/ h l) (* (* (/ D d) (* (/ M 2.0) (* (/ M 2.0) (/ D d)))) 0.5)))))
     (if (<= t_1 INFINITY)
       (/ (* t_0 d) h)
       (/
        (fma t_0 d (* (* (pow (/ h l) 1.5) (* (* D D) (/ (* M M) d))) -0.125))
        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+210) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - ((h / l) * (((D / d) * ((M / 2.0) * ((M / 2.0) * (D / d)))) * 0.5))));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = fma(t_0, d, ((pow((h / l), 1.5) * ((D * D) * ((M * M) / d))) * -0.125)) / 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+210)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(Float64(h / l) * Float64(Float64(Float64(D / d) * Float64(Float64(M / 2.0) * Float64(Float64(M / 2.0) * Float64(D / d)))) * 0.5)))));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(t_0 * d) / h);
	else
		tmp = Float64(fma(t_0, d, Float64(Float64((Float64(h / l) ^ 1.5) * Float64(Float64(D * D) * Float64(Float64(M * M) / d))) * -0.125)) / h);
	end
	return 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+210], 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[(N[(D / d), $MachinePrecision] * N[(N[(M / 2.0), $MachinePrecision] * N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(t$95$0 * d), $MachinePrecision] / h), $MachinePrecision], N[(N[(t$95$0 * d + N[(N[(N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision] * N[(N[(D * D), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $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)))) < 4.9999999999999998e210

   1. Initial program 86.0%

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

   4. Applied rewrites 85.9%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 85.9

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 85.9

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-/.f64 85.9

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

   6. Applied rewrites 85.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 84.4

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 84.4

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

   8. Applied rewrites 84.4%

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

   if 4.9999999999999998e210 < (*.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 51.0%

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in h around 0

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

   7. Applied rewrites 52.9%

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

   8. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.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 73.7

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

   10. Applied rewrites 73.7%

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

   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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in h around 0

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

   7. Applied rewrites 23.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow-prod-down N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 21.3

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

   9. Applied rewrites 21.3%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-pow.f64 N/A

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

       4. sqrt-pow1 N/A

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

       5. lower-pow.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. metadata-eval 23.6

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{h}{\ell}}, d, \left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot -0.125\right)}{h} \]

   11. Applied rewrites 23.6%

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

Alternative 7: 70.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+210}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(\left(\frac{D}{d} \cdot \left(\frac{M}{2} \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right)\right) \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{t\_0 \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, d, \left(\sqrt{\frac{h}{\ell} \cdot \left(\frac{h}{\ell} \cdot \frac{h}{\ell}\right)} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot -0.125\right)}{h}\\ \end{array} \end{array} \]

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 (<= t_1 5e+210)
     (*
      (sqrt (/ d h))
      (*
       (sqrt (/ d l))
       (-
        1.0
        (* (/ h l) (* (* (/ D d) (* (/ M 2.0) (* (/ M 2.0) (/ D d)))) 0.5)))))
     (if (<= t_1 INFINITY)
       (/ (* t_0 d) h)
       (/
        (fma
         t_0
         d
         (*
          (* (sqrt (* (/ h l) (* (/ h l) (/ h l)))) (* (* D D) (/ (* M M) d)))
          -0.125))
        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+210) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - ((h / l) * (((D / d) * ((M / 2.0) * ((M / 2.0) * (D / d)))) * 0.5))));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = fma(t_0, d, ((sqrt(((h / l) * ((h / l) * (h / l)))) * ((D * D) * ((M * M) / d))) * -0.125)) / 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+210)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(Float64(h / l) * Float64(Float64(Float64(D / d) * Float64(Float64(M / 2.0) * Float64(Float64(M / 2.0) * Float64(D / d)))) * 0.5)))));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(t_0 * d) / h);
	else
		tmp = Float64(fma(t_0, d, Float64(Float64(sqrt(Float64(Float64(h / l) * Float64(Float64(h / l) * Float64(h / l)))) * Float64(Float64(D * D) * Float64(Float64(M * M) / d))) * -0.125)) / h);
	end
	return 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+210], 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[(N[(D / d), $MachinePrecision] * N[(N[(M / 2.0), $MachinePrecision] * N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(t$95$0 * d), $MachinePrecision] / h), $MachinePrecision], N[(N[(t$95$0 * d + N[(N[(N[Sqrt[N[(N[(h / l), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(D * D), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $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)))) < 4.9999999999999998e210

   1. Initial program 86.0%

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

   4. Applied rewrites 85.9%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 85.9

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 85.9

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-/.f64 85.9

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

   6. Applied rewrites 85.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 84.4

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 84.4

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

   8. Applied rewrites 84.4%

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

   if 4.9999999999999998e210 < (*.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 51.0%

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in h around 0

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

   7. Applied rewrites 52.9%

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

   8. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.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 73.7

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

   10. Applied rewrites 73.7%

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

   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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in h around 0

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

   7. Applied rewrites 23.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow-prod-down N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 21.3

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

   9. Applied rewrites 21.3%

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

       1. lift-/.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. cube-mult N/A

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

       4. lower-*.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lift-/.f64 N/A

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

       8. lift-/.f64 21.3

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

   11. Applied rewrites 21.3%

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

Alternative 8: 71.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+210}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(\left(\left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)\right) \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{t\_0 \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, d, \left(\sqrt{\frac{h}{\ell} \cdot \left(\frac{h}{\ell} \cdot \frac{h}{\ell}\right)} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot -0.125\right)}{h}\\ \end{array} \end{array} \]

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 (<= t_1 5e+210)
     (*
      (sqrt (/ d h))
      (*
       (sqrt (/ d l))
       (-
        1.0
        (* (/ h l) (* (* (* (/ D d) (/ M 2.0)) (* (/ D d) (* 0.5 M))) 0.5)))))
     (if (<= t_1 INFINITY)
       (/ (* t_0 d) h)
       (/
        (fma
         t_0
         d
         (*
          (* (sqrt (* (/ h l) (* (/ h l) (/ h l)))) (* (* D D) (/ (* M M) d)))
          -0.125))
        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+210) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - ((h / l) * ((((D / d) * (M / 2.0)) * ((D / d) * (0.5 * M))) * 0.5))));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = fma(t_0, d, ((sqrt(((h / l) * ((h / l) * (h / l)))) * ((D * D) * ((M * M) / d))) * -0.125)) / 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+210)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(Float64(h / l) * Float64(Float64(Float64(Float64(D / d) * Float64(M / 2.0)) * Float64(Float64(D / d) * Float64(0.5 * M))) * 0.5)))));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(t_0 * d) / h);
	else
		tmp = Float64(fma(t_0, d, Float64(Float64(sqrt(Float64(Float64(h / l) * Float64(Float64(h / l) * Float64(h / l)))) * Float64(Float64(D * D) * Float64(Float64(M * M) / d))) * -0.125)) / h);
	end
	return 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+210], 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[(N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[(0.5 * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(t$95$0 * d), $MachinePrecision] / h), $MachinePrecision], N[(N[(t$95$0 * d + N[(N[(N[Sqrt[N[(N[(h / l), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(D * D), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $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)))) < 4.9999999999999998e210

   1. Initial program 86.0%

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

   4. Applied rewrites 85.9%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 85.9

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 85.9

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-/.f64 85.9

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

   6. Applied rewrites 85.9%

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

   7. Taylor expanded in M around 0

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

      1. lower-*.f64 85.9

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

   9. Applied rewrites 85.9%

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

   if 4.9999999999999998e210 < (*.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 51.0%

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in h around 0

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

   7. Applied rewrites 52.9%

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

   8. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.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 73.7

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

   10. Applied rewrites 73.7%

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

   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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in h around 0

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

   7. Applied rewrites 23.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow-prod-down N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 21.3

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

   9. Applied rewrites 21.3%

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

       1. lift-/.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. cube-mult N/A

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

       4. lower-*.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lift-/.f64 N/A

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

       8. lift-/.f64 21.3

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

   11. Applied rewrites 21.3%

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

Alternative 9: 64.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+210}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(\left(\frac{D}{d} \cdot \left(\frac{\left(M \cdot M\right) \cdot D}{d} \cdot 0.25\right)\right) \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{t\_0 \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, d, \left(\sqrt{\frac{h}{\ell} \cdot \left(\frac{h}{\ell} \cdot \frac{h}{\ell}\right)} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot -0.125\right)}{h}\\ \end{array} \end{array} \]

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 (<= t_1 5e+210)
     (*
      (sqrt (/ d h))
      (*
       (sqrt (/ d l))
       (- 1.0 (* (/ h l) (* (* (/ D d) (* (/ (* (* M M) D) d) 0.25)) 0.5)))))
     (if (<= t_1 INFINITY)
       (/ (* t_0 d) h)
       (/
        (fma
         t_0
         d
         (*
          (* (sqrt (* (/ h l) (* (/ h l) (/ h l)))) (* (* D D) (/ (* M M) d)))
          -0.125))
        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+210) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - ((h / l) * (((D / d) * ((((M * M) * D) / d) * 0.25)) * 0.5))));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = fma(t_0, d, ((sqrt(((h / l) * ((h / l) * (h / l)))) * ((D * D) * ((M * M) / d))) * -0.125)) / 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+210)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(Float64(h / l) * Float64(Float64(Float64(D / d) * Float64(Float64(Float64(Float64(M * M) * D) / d) * 0.25)) * 0.5)))));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(t_0 * d) / h);
	else
		tmp = Float64(fma(t_0, d, Float64(Float64(sqrt(Float64(Float64(h / l) * Float64(Float64(h / l) * Float64(h / l)))) * Float64(Float64(D * D) * Float64(Float64(M * M) / d))) * -0.125)) / h);
	end
	return 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+210], 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[(N[(D / d), $MachinePrecision] * N[(N[(N[(N[(M * M), $MachinePrecision] * D), $MachinePrecision] / d), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(t$95$0 * d), $MachinePrecision] / h), $MachinePrecision], N[(N[(t$95$0 * d + N[(N[(N[Sqrt[N[(N[(h / l), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(D * D), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $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)))) < 4.9999999999999998e210

   1. Initial program 86.0%

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

   4. Applied rewrites 85.9%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 85.9

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 85.9

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-/.f64 85.9

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

   6. Applied rewrites 85.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 84.4

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 84.4

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

   8. Applied rewrites 84.4%

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

   9. Taylor expanded in d around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 77.7

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

   11. Applied rewrites 77.7%

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

   if 4.9999999999999998e210 < (*.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 51.0%

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in h around 0

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

   7. Applied rewrites 52.9%

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

   8. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.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 73.7

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

   10. Applied rewrites 73.7%

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

   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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in h around 0

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

   7. Applied rewrites 23.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow-prod-down N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 21.3

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

   9. Applied rewrites 21.3%

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

       1. lift-/.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. cube-mult N/A

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

       4. lower-*.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lift-/.f64 N/A

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

       8. lift-/.f64 21.3

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

   11. Applied rewrites 21.3%

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

Alternative 10: 64.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+210}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(\left(\frac{D}{d} \cdot \left(\frac{\left(M \cdot M\right) \cdot D}{d} \cdot 0.25\right)\right) \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot h\right)}^{-0.5} \cdot \left(-d\right)\\ \end{array} \end{array} \]

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+210)
     (*
      (sqrt (/ d h))
      (*
       (sqrt (/ d l))
       (- 1.0 (* (/ h l) (* (* (/ D d) (* (/ (* (* M M) D) d) 0.25)) 0.5)))))
     (if (<= t_0 INFINITY)
       (/ (* (sqrt (/ h l)) d) h)
       (* (pow (* l h) -0.5) (- d))))))

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+210) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - ((h / l) * (((D / d) * ((((M * M) * D) / d) * 0.25)) * 0.5))));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (sqrt((h / l)) * d) / h;
	} else {
		tmp = pow((l * h), -0.5) * -d;
	}
	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+210) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 - ((h / l) * (((D / d) * ((((M * M) * D) / d) * 0.25)) * 0.5))));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (Math.sqrt((h / l)) * d) / h;
	} else {
		tmp = Math.pow((l * h), -0.5) * -d;
	}
	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+210:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 - ((h / l) * (((D / d) * ((((M * M) * D) / d) * 0.25)) * 0.5))))
	elif t_0 <= math.inf:
		tmp = (math.sqrt((h / l)) * d) / h
	else:
		tmp = math.pow((l * h), -0.5) * -d
	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+210)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(Float64(h / l) * Float64(Float64(Float64(D / d) * Float64(Float64(Float64(Float64(M * M) * D) / d) * 0.25)) * 0.5)))));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * d) / h);
	else
		tmp = Float64((Float64(l * h) ^ -0.5) * Float64(-d));
	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+210)
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - ((h / l) * (((D / d) * ((((M * M) * D) / d) * 0.25)) * 0.5))));
	elseif (t_0 <= Inf)
		tmp = (sqrt((h / l)) * d) / h;
	else
		tmp = ((l * h) ^ -0.5) * -d;
	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+210], 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[(N[(D / d), $MachinePrecision] * N[(N[(N[(N[(M * M), $MachinePrecision] * D), $MachinePrecision] / d), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $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[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision] * (-d)), $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)))) < 4.9999999999999998e210

   1. Initial program 86.0%

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

   4. Applied rewrites 85.9%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 85.9

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 85.9

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-/.f64 85.9

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

   6. Applied rewrites 85.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 84.4

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 84.4

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

   8. Applied rewrites 84.4%

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

   9. Taylor expanded in d around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 77.7

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

   11. Applied rewrites 77.7%

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

   if 4.9999999999999998e210 < (*.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 51.0%

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in h around 0

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

   7. Applied rewrites 52.9%

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

   8. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.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 73.7

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

   10. Applied rewrites 73.7%

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

   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

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 0.0

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

      11. lift-/.f64 N/A

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

      12. metadata-eval 0.0

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in d around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow-1 N/A

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

      7. sqrt-pow1 N/A

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

      8. *-commutative N/A

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

      9. lower-pow.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. metadata-eval 15.8

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

   7. Applied rewrites 15.8%

      \[\leadsto \color{blue}{-{\left(\ell \cdot h\right)}^{-0.5} \cdot d} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+210}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(\left(\frac{D}{d} \cdot \left(\frac{\left(M \cdot M\right) \cdot D}{d} \cdot 0.25\right)\right) \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot h\right)}^{-0.5} \cdot \left(-d\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 44.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(-d\right) \cdot t\_0}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot d}{h}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ h l))))
   (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-171)
     (/ (* (- 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 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-171) {
		tmp = (-d * t_0) / h;
	} else {
		tmp = (t_0 * d) / h;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((h / l))
    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-171)) then
        tmp = (-d * t_0) / h
    else
        tmp = (t_0 * d) / h
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((h / l));
	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-171) {
		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))
	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-171:
		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))
	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-171)
		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));
	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-171)
		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]}, 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-171], 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)))) < -4.99999999999999992e-171

   1. Initial program 88.9%

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

   4. Applied rewrites 88.8%

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

   5. Taylor expanded in h around 0

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

   7. Applied rewrites 64.1%

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

   8. Taylor expanded in l around -inf

      \[\leadsto \frac{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   9. 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. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-/.f64 21.9

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

   10. Applied rewrites 21.9%

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

   if -4.99999999999999992e-171 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 53.4%

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

   4. Applied rewrites 53.4%

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

   5. Taylor expanded in h around 0

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

   7. Applied rewrites 50.5%

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

   8. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.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 54.5

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

   10. Applied rewrites 54.5%

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

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(-d\right) \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 41.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-171}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot d\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \end{array} \end{array} \]

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-171)
   (* (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-171) {
		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-171)) 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-171) {
		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-171:
		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-171)
		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-171)
		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-171], 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)))) < -4.99999999999999992e-171

   1. Initial program 88.9%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      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 10.7

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

   5. Applied rewrites 10.7%

      \[\leadsto \color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d} \]
   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. *-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 10.7

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

   7. Applied rewrites 10.7%

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

   if -4.99999999999999992e-171 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 53.4%

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

   4. Applied rewrites 53.4%

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

   5. Taylor expanded in h around 0

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

   7. Applied rewrites 50.5%

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

   8. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.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 54.5

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

   10. Applied rewrites 54.5%

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

Alternative 13: 72.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left({\left(\ell \cdot h\right)}^{-0.5} \cdot \left(-d\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(\left(\left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)\right) \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -2e-310)
   (*
    (* (pow (* l h) -0.5) (- d))
    (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))
   (*
    (/ (sqrt d) (sqrt h))
    (*
     (sqrt (/ d l))
     (-
      1.0
      (* (/ h l) (* (* (* (/ D d) (/ M 2.0)) (* (/ D d) (* 0.5 M))) 0.5)))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -2e-310) {
		tmp = (pow((l * h), -0.5) * -d) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	} else {
		tmp = (sqrt(d) / sqrt(h)) * (sqrt((d / l)) * (1.0 - ((h / l) * ((((D / d) * (M / 2.0)) * ((D / d) * (0.5 * M))) * 0.5))));
	}
	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 <= (-2d-310)) then
        tmp = (((l * h) ** (-0.5d0)) * -d) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    else
        tmp = (sqrt(d) / sqrt(h)) * (sqrt((d / l)) * (1.0d0 - ((h / l) * ((((d_1 / d) * (m / 2.0d0)) * ((d_1 / d) * (0.5d0 * m))) * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -2e-310) {
		tmp = (Math.pow((l * h), -0.5) * -d) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * (Math.sqrt((d / l)) * (1.0 - ((h / l) * ((((D / d) * (M / 2.0)) * ((D / d) * (0.5 * M))) * 0.5))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= -2e-310:
		tmp = (math.pow((l * h), -0.5) * -d) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * (math.sqrt((d / l)) * (1.0 - ((h / l) * ((((D / d) * (M / 2.0)) * ((D / d) * (0.5 * M))) * 0.5))))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -2e-310)
		tmp = (((l * h) ^ -0.5) * -d) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	else
		tmp = (sqrt(d) / sqrt(h)) * (sqrt((d / l)) * (1.0 - ((h / l) * ((((D / d) * (M / 2.0)) * ((D / d) * (0.5 * M))) * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -2e-310], N[(N[(N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision] * (-d)), $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], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[(N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[(0.5 * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.999999999999994e-310

   1. Initial program 63.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 50.8

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

      11. lift-/.f64 N/A

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

      12. metadata-eval 50.8

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

   4. Applied rewrites 50.8%

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

   5. Taylor expanded in d around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow-1 N/A

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

      7. sqrt-pow1 N/A

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

      8. *-commutative N/A

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

      9. lower-pow.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. metadata-eval 69.6

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

   7. Applied rewrites 69.6%

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

   if -1.999999999999994e-310 < d

   1. Initial program 70.3%

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

   4. Applied rewrites 70.3%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 70.3

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 70.3

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-/.f64 70.3

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

   6. Applied rewrites 70.3%

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

   7. Taylor expanded in M around 0

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

      1. lower-*.f64 70.3

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

   9. Applied rewrites 70.3%

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

       1. lift-/.f64 N/A

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

       2. lift-sqrt.f64 N/A

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

       3. sqrt-div N/A

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

       4. lower-/.f64 N/A

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

       5. lower-sqrt.f64 N/A

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

       6. lower-sqrt.f64 77.5

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

   11. Applied rewrites 77.5%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left({\left(\ell \cdot h\right)}^{-0.5} \cdot \left(-d\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(\left(\left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)\right) \cdot 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 26.9% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d \end{array} \]

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(Float64(1.0 / 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[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]

Derivation

1. Initial program 66.3%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. 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 20.6

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

5. Applied rewrites 20.6%

   \[\leadsto \color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d} \]
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. *-commutative N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 20.6

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

7. Applied rewrites 20.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 20.6

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

9. Applied rewrites 20.6%

   \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d \]
10. Add Preprocessing

Alternative 15: 26.7% 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 66.3%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. 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 20.6

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

5. Applied rewrites 20.6%

   \[\leadsto \color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d} \]
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. *-commutative N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 20.6

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

7. Applied rewrites 20.6%

   \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025061 
(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)))))