Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.1% → 92.6%

Time: 7.6s

Alternatives: 7

Speedup: 0.5×

Specification

Math

\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

FPCore

(FPCore (w0 M D h l d)
  :precision binary64
  (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - (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(w0, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * sqrt((1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))))
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
}

Python

def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))))

Julia

function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))))
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l))));
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 80.1% accurate, 1.0× speedup

Math

\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

FPCore

(FPCore (w0 M D h l d)
  :precision binary64
  (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - (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(w0, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * sqrt((1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))))
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
}

Python

def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))))

Julia

function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))))
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l))));
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 92.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_1 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_2 := \sqrt{1 - {\left(\frac{t\_1 \cdot t\_0}{2 \cdot \left|d\right|}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{if}\;t\_2 \leq 10^{+115}:\\ \;\;\;\;w0 \cdot t\_2\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\left(t\_0 \cdot \left(\left|t\_1\right| \cdot \sqrt{-0.25 \cdot \frac{h}{\ell}}\right)\right) \cdot \frac{w0}{\left|d\right|}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{t\_1}{\left|d\right| + \left|d\right|} \cdot t\_0, -0.5 \cdot \frac{t\_0 \cdot \left(t\_1 \cdot h\right)}{\left|d\right| \cdot \ell}, 1\right)}\\ \end{array} \]

FPCore

(FPCore (w0 M D h l d)
  :precision binary64
  (let* ((t_0 (fmax (fabs M) (fabs D)))
       (t_1 (fmin (fabs M) (fabs D)))
       (t_2
        (sqrt
         (-
          1.0
          (* (pow (/ (* t_1 t_0) (* 2.0 (fabs d))) 2.0) (/ h l))))))
  (if (<= t_2 1e+115)
    (* w0 t_2)
    (if (<= t_2 INFINITY)
      (*
       (* t_0 (* (fabs t_1) (sqrt (* -0.25 (/ h l)))))
       (/ w0 (fabs d)))
      (*
       w0
       (sqrt
        (fma
         (* (/ t_1 (+ (fabs d) (fabs d))) t_0)
         (* -0.5 (/ (* t_0 (* t_1 h)) (* (fabs d) l)))
         1.0)))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = fmax(fabs(M), fabs(D));
	double t_1 = fmin(fabs(M), fabs(D));
	double t_2 = sqrt((1.0 - (pow(((t_1 * t_0) / (2.0 * fabs(d))), 2.0) * (h / l))));
	double tmp;
	if (t_2 <= 1e+115) {
		tmp = w0 * t_2;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (t_0 * (fabs(t_1) * sqrt((-0.25 * (h / l))))) * (w0 / fabs(d));
	} else {
		tmp = w0 * sqrt(fma(((t_1 / (fabs(d) + fabs(d))) * t_0), (-0.5 * ((t_0 * (t_1 * h)) / (fabs(d) * l))), 1.0));
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	t_0 = fmax(abs(M), abs(D))
	t_1 = fmin(abs(M), abs(D))
	t_2 = sqrt(Float64(1.0 - Float64((Float64(Float64(t_1 * t_0) / Float64(2.0 * abs(d))) ^ 2.0) * Float64(h / l))))
	tmp = 0.0
	if (t_2 <= 1e+115)
		tmp = Float64(w0 * t_2);
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(t_0 * Float64(abs(t_1) * sqrt(Float64(-0.25 * Float64(h / l))))) * Float64(w0 / abs(d)));
	else
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(t_1 / Float64(abs(d) + abs(d))) * t_0), Float64(-0.5 * Float64(Float64(t_0 * Float64(t_1 * h)) / Float64(abs(d) * l))), 1.0)));
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[Max[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(t$95$1 * t$95$0), $MachinePrecision] / N[(2.0 * N[Abs[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 1e+115], N[(w0 * t$95$2), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(t$95$0 * N[(N[Abs[t$95$1], $MachinePrecision] * N[Sqrt[N[(-0.25 * N[(h / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(w0 / N[Abs[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(N[(t$95$1 / N[(N[Abs[d], $MachinePrecision] + N[Abs[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(-0.5 * N[(N[(t$95$0 * N[(t$95$1 * h), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[d], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))) < 1e115

   1. Initial program 80.1%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   if 1e115 < (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))) < +inf.0

   1. Initial program 80.1%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   2. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{w0 \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right)}}{d}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{w0 \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right)}}{\color{blue}{d}} \]

   4. Applied rewrites 9.3%

      \[\leadsto \color{blue}{\frac{w0 \cdot \sqrt{-0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}}{d}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{w0 \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}}{\color{blue}{d}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{w0 \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}}{d} \]

      3. *-commutative N/A

         \[\leadsto \frac{\sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}} \cdot w0}{d} \]

      4. associate-/l* N/A

         \[\leadsto \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}} \cdot \color{blue}{\frac{w0}{d}} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}} \cdot \color{blue}{\frac{w0}{d}} \]

   6. Applied rewrites 8.9%

      \[\leadsto \sqrt{\left(-0.25 \cdot \left(D \cdot D\right)\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}} \cdot \color{blue}{\frac{w0}{d}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}} \cdot \frac{\color{blue}{w0}}{d} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}} \cdot \frac{w0}{d} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\frac{\left(M \cdot M\right) \cdot h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)} \cdot \frac{w0}{d} \]

      4. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{\left(M \cdot M\right) \cdot h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)} \cdot \frac{w0}{d} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(M \cdot M\right) \cdot h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)} \cdot \frac{w0}{d} \]

      6. associate-/l* N/A

         \[\leadsto \sqrt{\left(\left(M \cdot M\right) \cdot \frac{h}{\ell}\right) \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)} \cdot \frac{w0}{d} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\left(M \cdot M\right) \cdot \left(\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)\right)} \cdot \frac{w0}{d} \]

      8. sqrt-prod N/A

         \[\leadsto \left(\sqrt{M \cdot M} \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{\color{blue}{w0}}{d} \]

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

         \[\leadsto \left(\sqrt{M \cdot M} \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{M \cdot M} \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{M \cdot M} \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

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

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

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

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{\color{blue}{w0}}{d} \]

      14. lower-fabs.f64 N/A

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

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

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

      16. lower-*.f64 N/A

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

      17. lower-/.f64 10.6%

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(-0.25 \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

      18. lift-*.f64 N/A

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

      19. *-commutative N/A

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\left(D \cdot D\right) \cdot \frac{-1}{4}\right)}\right) \cdot \frac{w0}{d} \]

      20. lower-*.f64 10.6%

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\left(D \cdot D\right) \cdot -0.25\right)}\right) \cdot \frac{w0}{d} \]

   8. Applied rewrites 10.6%

      \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\left(D \cdot D\right) \cdot -0.25\right)}\right) \cdot \frac{\color{blue}{w0}}{d} \]

   9. Taylor expanded in D around 0

      \[\leadsto \left(D \cdot \left(\left|M\right| \cdot \sqrt{\frac{-1}{4} \cdot \frac{h}{\ell}}\right)\right) \cdot \frac{\color{blue}{w0}}{d} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left(D \cdot \left(\left|M\right| \cdot \sqrt{\frac{-1}{4} \cdot \frac{h}{\ell}}\right)\right) \cdot \frac{w0}{d} \]

       2. lower-*.f64 N/A

          \[\leadsto \left(D \cdot \left(\left|M\right| \cdot \sqrt{\frac{-1}{4} \cdot \frac{h}{\ell}}\right)\right) \cdot \frac{w0}{d} \]

       3. lower-fabs.f64 N/A

          \[\leadsto \left(D \cdot \left(\left|M\right| \cdot \sqrt{\frac{-1}{4} \cdot \frac{h}{\ell}}\right)\right) \cdot \frac{w0}{d} \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \left(D \cdot \left(\left|M\right| \cdot \sqrt{\frac{-1}{4} \cdot \frac{h}{\ell}}\right)\right) \cdot \frac{w0}{d} \]

       5. lower-*.f64 N/A

          \[\leadsto \left(D \cdot \left(\left|M\right| \cdot \sqrt{\frac{-1}{4} \cdot \frac{h}{\ell}}\right)\right) \cdot \frac{w0}{d} \]

       6. lower-/.f64 12.4%

          \[\leadsto \left(D \cdot \left(\left|M\right| \cdot \sqrt{-0.25 \cdot \frac{h}{\ell}}\right)\right) \cdot \frac{w0}{d} \]

   11. Applied rewrites 12.4%

       \[\leadsto \left(D \cdot \left(\left|M\right| \cdot \sqrt{-0.25 \cdot \frac{h}{\ell}}\right)\right) \cdot \frac{\color{blue}{w0}}{d} \]

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

   1. Initial program 80.1%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\color{blue}{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto w0 \cdot \sqrt{\color{blue}{1 + \left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)\right) \cdot \frac{h}{\ell}}} \]

      4. +-commutative N/A

         \[\leadsto w0 \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)\right) \cdot \frac{h}{\ell} + 1}} \]

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

         \[\leadsto w0 \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} + 1} \]

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

         \[\leadsto w0 \cdot \sqrt{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\mathsf{neg}\left(\frac{h}{\ell}\right)\right)} + 1} \]

      7. lift-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{h}{\ell}}\right)\right) + 1} \]

      8. distribute-neg-frac2 N/A

         \[\leadsto w0 \cdot \sqrt{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{h}{\mathsf{neg}\left(\ell\right)}} + 1} \]

      9. associate-/l* N/A

         \[\leadsto w0 \cdot \sqrt{\color{blue}{\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\mathsf{neg}\left(\ell\right)}} + 1} \]

      10. lift-pow.f64 N/A

          \[\leadsto w0 \cdot \sqrt{\frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot h}{\mathsf{neg}\left(\ell\right)} + 1} \]

      11. unpow2 N/A

          \[\leadsto w0 \cdot \sqrt{\frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot h}{\mathsf{neg}\left(\ell\right)} + 1} \]

      12. associate-*l* N/A

          \[\leadsto w0 \cdot \sqrt{\frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot h\right)}}{\mathsf{neg}\left(\ell\right)} + 1} \]

      13. associate-/l* N/A

          \[\leadsto w0 \cdot \sqrt{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot h}{\mathsf{neg}\left(\ell\right)}} + 1} \]

      14. lower-fma.f64 N/A

          \[\leadsto w0 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{2 \cdot d}, \frac{\frac{M \cdot D}{2 \cdot d} \cdot h}{\mathsf{neg}\left(\ell\right)}, 1\right)}} \]

   3. Applied rewrites 83.9%

      \[\leadsto w0 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{M}{d + d} \cdot D, \frac{\frac{D}{d + d} \cdot \left(M \cdot h\right)}{-\ell}, 1\right)}} \]

   4. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M}{d + d} \cdot D, \color{blue}{\frac{-1}{2} \cdot \frac{D \cdot \left(M \cdot h\right)}{d \cdot \ell}}, 1\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M}{d + d} \cdot D, \frac{-1}{2} \cdot \color{blue}{\frac{D \cdot \left(M \cdot h\right)}{d \cdot \ell}}, 1\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M}{d + d} \cdot D, \frac{-1}{2} \cdot \frac{D \cdot \left(M \cdot h\right)}{\color{blue}{d \cdot \ell}}, 1\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M}{d + d} \cdot D, \frac{-1}{2} \cdot \frac{D \cdot \left(M \cdot h\right)}{\color{blue}{d} \cdot \ell}, 1\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M}{d + d} \cdot D, \frac{-1}{2} \cdot \frac{D \cdot \left(M \cdot h\right)}{d \cdot \ell}, 1\right)} \]

      5. lower-*.f64 81.3%

         \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M}{d + d} \cdot D, -0.5 \cdot \frac{D \cdot \left(M \cdot h\right)}{d \cdot \color{blue}{\ell}}, 1\right)} \]

   6. Applied rewrites 81.3%

      \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M}{d + d} \cdot D, \color{blue}{-0.5 \cdot \frac{D \cdot \left(M \cdot h\right)}{d \cdot \ell}}, 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 92.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_1 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_2 := \frac{t\_1}{\left|d\right| + \left|d\right|} \cdot t\_0\\ t_3 := \sqrt{1 - {\left(\frac{t\_1 \cdot t\_0}{2 \cdot \left|d\right|}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{if}\;t\_3 \leq 10^{+115}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{h}{\ell} \cdot \left(\frac{-0.5}{\left|d\right|} \cdot \left(t\_0 \cdot t\_1\right)\right), t\_2, 1\right)}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\left(t\_0 \cdot \left(\left|t\_1\right| \cdot \sqrt{-0.25 \cdot \frac{h}{\ell}}\right)\right) \cdot \frac{w0}{\left|d\right|}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(t\_2, -0.5 \cdot \frac{t\_0 \cdot \left(t\_1 \cdot h\right)}{\left|d\right| \cdot \ell}, 1\right)}\\ \end{array} \]

FPCore

(FPCore (w0 M D h l d)
  :precision binary64
  (let* ((t_0 (fmax (fabs M) (fabs D)))
       (t_1 (fmin (fabs M) (fabs D)))
       (t_2 (* (/ t_1 (+ (fabs d) (fabs d))) t_0))
       (t_3
        (sqrt
         (-
          1.0
          (* (pow (/ (* t_1 t_0) (* 2.0 (fabs d))) 2.0) (/ h l))))))
  (if (<= t_3 1e+115)
    (*
     w0
     (sqrt
      (fma (* (/ h l) (* (/ -0.5 (fabs d)) (* t_0 t_1))) t_2 1.0)))
    (if (<= t_3 INFINITY)
      (*
       (* t_0 (* (fabs t_1) (sqrt (* -0.25 (/ h l)))))
       (/ w0 (fabs d)))
      (*
       w0
       (sqrt
        (fma
         t_2
         (* -0.5 (/ (* t_0 (* t_1 h)) (* (fabs d) l)))
         1.0)))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = fmax(fabs(M), fabs(D));
	double t_1 = fmin(fabs(M), fabs(D));
	double t_2 = (t_1 / (fabs(d) + fabs(d))) * t_0;
	double t_3 = sqrt((1.0 - (pow(((t_1 * t_0) / (2.0 * fabs(d))), 2.0) * (h / l))));
	double tmp;
	if (t_3 <= 1e+115) {
		tmp = w0 * sqrt(fma(((h / l) * ((-0.5 / fabs(d)) * (t_0 * t_1))), t_2, 1.0));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (t_0 * (fabs(t_1) * sqrt((-0.25 * (h / l))))) * (w0 / fabs(d));
	} else {
		tmp = w0 * sqrt(fma(t_2, (-0.5 * ((t_0 * (t_1 * h)) / (fabs(d) * l))), 1.0));
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	t_0 = fmax(abs(M), abs(D))
	t_1 = fmin(abs(M), abs(D))
	t_2 = Float64(Float64(t_1 / Float64(abs(d) + abs(d))) * t_0)
	t_3 = sqrt(Float64(1.0 - Float64((Float64(Float64(t_1 * t_0) / Float64(2.0 * abs(d))) ^ 2.0) * Float64(h / l))))
	tmp = 0.0
	if (t_3 <= 1e+115)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(h / l) * Float64(Float64(-0.5 / abs(d)) * Float64(t_0 * t_1))), t_2, 1.0)));
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(t_0 * Float64(abs(t_1) * sqrt(Float64(-0.25 * Float64(h / l))))) * Float64(w0 / abs(d)));
	else
		tmp = Float64(w0 * sqrt(fma(t_2, Float64(-0.5 * Float64(Float64(t_0 * Float64(t_1 * h)) / Float64(abs(d) * l))), 1.0)));
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[Max[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 / N[(N[Abs[d], $MachinePrecision] + N[Abs[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(t$95$1 * t$95$0), $MachinePrecision] / N[(2.0 * N[Abs[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 1e+115], N[(w0 * N[Sqrt[N[(N[(N[(h / l), $MachinePrecision] * N[(N[(-0.5 / N[Abs[d], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(t$95$0 * N[(N[Abs[t$95$1], $MachinePrecision] * N[Sqrt[N[(-0.25 * N[(h / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(w0 / N[Abs[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(t$95$2 * N[(-0.5 * N[(N[(t$95$0 * N[(t$95$1 * h), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[d], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))) < 1e115

   1. Initial program 80.1%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\color{blue}{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto w0 \cdot \sqrt{\color{blue}{1 + \left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)\right) \cdot \frac{h}{\ell}}} \]

      4. +-commutative N/A

         \[\leadsto w0 \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)\right) \cdot \frac{h}{\ell} + 1}} \]

      5. *-commutative N/A

         \[\leadsto w0 \cdot \sqrt{\color{blue}{\frac{h}{\ell} \cdot \left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)\right)} + 1} \]

      6. lift-pow.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\frac{h}{\ell} \cdot \left(\mathsf{neg}\left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right)\right) + 1} \]

      7. unpow2 N/A

         \[\leadsto w0 \cdot \sqrt{\frac{h}{\ell} \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}\right)\right) + 1} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto w0 \cdot \sqrt{\frac{h}{\ell} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)} + 1} \]

      9. associate-*r* N/A

         \[\leadsto w0 \cdot \sqrt{\color{blue}{\left(\frac{h}{\ell} \cdot \left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right)\right) \cdot \frac{M \cdot D}{2 \cdot d}} + 1} \]

      10. lower-fma.f64 N/A

          \[\leadsto w0 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{h}{\ell} \cdot \left(\mathsf{neg}\left(\frac{M \cdot D}{2 \cdot d}\right)\right), \frac{M \cdot D}{2 \cdot d}, 1\right)}} \]

   3. Applied rewrites 81.2%

      \[\leadsto w0 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{h}{\ell} \cdot \left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right), \frac{M}{d + d} \cdot D, 1\right)}} \]

   if 1e115 < (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))) < +inf.0

   1. Initial program 80.1%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   2. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{w0 \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right)}}{d}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{w0 \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right)}}{\color{blue}{d}} \]

   4. Applied rewrites 9.3%

      \[\leadsto \color{blue}{\frac{w0 \cdot \sqrt{-0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}}{d}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{w0 \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}}{\color{blue}{d}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{w0 \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}}{d} \]

      3. *-commutative N/A

         \[\leadsto \frac{\sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}} \cdot w0}{d} \]

      4. associate-/l* N/A

         \[\leadsto \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}} \cdot \color{blue}{\frac{w0}{d}} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}} \cdot \color{blue}{\frac{w0}{d}} \]

   6. Applied rewrites 8.9%

      \[\leadsto \sqrt{\left(-0.25 \cdot \left(D \cdot D\right)\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}} \cdot \color{blue}{\frac{w0}{d}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}} \cdot \frac{\color{blue}{w0}}{d} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}} \cdot \frac{w0}{d} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\frac{\left(M \cdot M\right) \cdot h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)} \cdot \frac{w0}{d} \]

      4. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{\left(M \cdot M\right) \cdot h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)} \cdot \frac{w0}{d} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(M \cdot M\right) \cdot h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)} \cdot \frac{w0}{d} \]

      6. associate-/l* N/A

         \[\leadsto \sqrt{\left(\left(M \cdot M\right) \cdot \frac{h}{\ell}\right) \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)} \cdot \frac{w0}{d} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\left(M \cdot M\right) \cdot \left(\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)\right)} \cdot \frac{w0}{d} \]

      8. sqrt-prod N/A

         \[\leadsto \left(\sqrt{M \cdot M} \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{\color{blue}{w0}}{d} \]

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

         \[\leadsto \left(\sqrt{M \cdot M} \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{M \cdot M} \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{M \cdot M} \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

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

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

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

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{\color{blue}{w0}}{d} \]

      14. lower-fabs.f64 N/A

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

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

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

      16. lower-*.f64 N/A

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

      17. lower-/.f64 10.6%

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(-0.25 \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

      18. lift-*.f64 N/A

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

      19. *-commutative N/A

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\left(D \cdot D\right) \cdot \frac{-1}{4}\right)}\right) \cdot \frac{w0}{d} \]

      20. lower-*.f64 10.6%

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\left(D \cdot D\right) \cdot -0.25\right)}\right) \cdot \frac{w0}{d} \]

   8. Applied rewrites 10.6%

      \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\left(D \cdot D\right) \cdot -0.25\right)}\right) \cdot \frac{\color{blue}{w0}}{d} \]

   9. Taylor expanded in D around 0

      \[\leadsto \left(D \cdot \left(\left|M\right| \cdot \sqrt{\frac{-1}{4} \cdot \frac{h}{\ell}}\right)\right) \cdot \frac{\color{blue}{w0}}{d} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left(D \cdot \left(\left|M\right| \cdot \sqrt{\frac{-1}{4} \cdot \frac{h}{\ell}}\right)\right) \cdot \frac{w0}{d} \]

       2. lower-*.f64 N/A

          \[\leadsto \left(D \cdot \left(\left|M\right| \cdot \sqrt{\frac{-1}{4} \cdot \frac{h}{\ell}}\right)\right) \cdot \frac{w0}{d} \]

       3. lower-fabs.f64 N/A

          \[\leadsto \left(D \cdot \left(\left|M\right| \cdot \sqrt{\frac{-1}{4} \cdot \frac{h}{\ell}}\right)\right) \cdot \frac{w0}{d} \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \left(D \cdot \left(\left|M\right| \cdot \sqrt{\frac{-1}{4} \cdot \frac{h}{\ell}}\right)\right) \cdot \frac{w0}{d} \]

       5. lower-*.f64 N/A

          \[\leadsto \left(D \cdot \left(\left|M\right| \cdot \sqrt{\frac{-1}{4} \cdot \frac{h}{\ell}}\right)\right) \cdot \frac{w0}{d} \]

       6. lower-/.f64 12.4%

          \[\leadsto \left(D \cdot \left(\left|M\right| \cdot \sqrt{-0.25 \cdot \frac{h}{\ell}}\right)\right) \cdot \frac{w0}{d} \]

   11. Applied rewrites 12.4%

       \[\leadsto \left(D \cdot \left(\left|M\right| \cdot \sqrt{-0.25 \cdot \frac{h}{\ell}}\right)\right) \cdot \frac{\color{blue}{w0}}{d} \]

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

   1. Initial program 80.1%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\color{blue}{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto w0 \cdot \sqrt{\color{blue}{1 + \left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)\right) \cdot \frac{h}{\ell}}} \]

      4. +-commutative N/A

         \[\leadsto w0 \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)\right) \cdot \frac{h}{\ell} + 1}} \]

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

         \[\leadsto w0 \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} + 1} \]

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

         \[\leadsto w0 \cdot \sqrt{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\mathsf{neg}\left(\frac{h}{\ell}\right)\right)} + 1} \]

      7. lift-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{h}{\ell}}\right)\right) + 1} \]

      8. distribute-neg-frac2 N/A

         \[\leadsto w0 \cdot \sqrt{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{h}{\mathsf{neg}\left(\ell\right)}} + 1} \]

      9. associate-/l* N/A

         \[\leadsto w0 \cdot \sqrt{\color{blue}{\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\mathsf{neg}\left(\ell\right)}} + 1} \]

      10. lift-pow.f64 N/A

          \[\leadsto w0 \cdot \sqrt{\frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot h}{\mathsf{neg}\left(\ell\right)} + 1} \]

      11. unpow2 N/A

          \[\leadsto w0 \cdot \sqrt{\frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot h}{\mathsf{neg}\left(\ell\right)} + 1} \]

      12. associate-*l* N/A

          \[\leadsto w0 \cdot \sqrt{\frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot h\right)}}{\mathsf{neg}\left(\ell\right)} + 1} \]

      13. associate-/l* N/A

          \[\leadsto w0 \cdot \sqrt{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot h}{\mathsf{neg}\left(\ell\right)}} + 1} \]

      14. lower-fma.f64 N/A

          \[\leadsto w0 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{2 \cdot d}, \frac{\frac{M \cdot D}{2 \cdot d} \cdot h}{\mathsf{neg}\left(\ell\right)}, 1\right)}} \]

   3. Applied rewrites 83.9%

      \[\leadsto w0 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{M}{d + d} \cdot D, \frac{\frac{D}{d + d} \cdot \left(M \cdot h\right)}{-\ell}, 1\right)}} \]

   4. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M}{d + d} \cdot D, \color{blue}{\frac{-1}{2} \cdot \frac{D \cdot \left(M \cdot h\right)}{d \cdot \ell}}, 1\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M}{d + d} \cdot D, \frac{-1}{2} \cdot \color{blue}{\frac{D \cdot \left(M \cdot h\right)}{d \cdot \ell}}, 1\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M}{d + d} \cdot D, \frac{-1}{2} \cdot \frac{D \cdot \left(M \cdot h\right)}{\color{blue}{d \cdot \ell}}, 1\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M}{d + d} \cdot D, \frac{-1}{2} \cdot \frac{D \cdot \left(M \cdot h\right)}{\color{blue}{d} \cdot \ell}, 1\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M}{d + d} \cdot D, \frac{-1}{2} \cdot \frac{D \cdot \left(M \cdot h\right)}{d \cdot \ell}, 1\right)} \]

      5. lower-*.f64 81.3%

         \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M}{d + d} \cdot D, -0.5 \cdot \frac{D \cdot \left(M \cdot h\right)}{d \cdot \color{blue}{\ell}}, 1\right)} \]

   6. Applied rewrites 81.3%

      \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M}{d + d} \cdot D, \color{blue}{-0.5 \cdot \frac{D \cdot \left(M \cdot h\right)}{d \cdot \ell}}, 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 91.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_1 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_2 := \frac{t\_1}{\left|d\right|} \cdot t\_0\\ t_3 := \left|w0\right| \cdot \sqrt{1 - {\left(\frac{t\_0 \cdot t\_1}{2 \cdot \left|d\right|}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathsf{copysign}\left(1, w0\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\left|w0\right| \cdot \sqrt{1 - \frac{t\_2 \cdot t\_2}{4} \cdot \frac{h}{\ell}}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\left(t\_1 \cdot \left(\left|t\_0\right| \cdot \sqrt{-0.25 \cdot \frac{h}{\ell}}\right)\right) \cdot \frac{\left|w0\right|}{\left|d\right|}\\ \mathbf{else}:\\ \;\;\;\;\left|w0\right| \cdot \sqrt{\mathsf{fma}\left(\frac{t\_0}{\left|d\right| + \left|d\right|} \cdot t\_1, -0.5 \cdot \frac{t\_1 \cdot \left(t\_0 \cdot h\right)}{\left|d\right| \cdot \ell}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
  :precision binary64
  (let* ((t_0 (fmin (fabs M) (fabs D)))
       (t_1 (fmax (fabs M) (fabs D)))
       (t_2 (* (/ t_1 (fabs d)) t_0))
       (t_3
        (*
         (fabs w0)
         (sqrt
          (-
           1.0
           (* (pow (/ (* t_0 t_1) (* 2.0 (fabs d))) 2.0) (/ h l)))))))
  (*
   (copysign 1.0 w0)
   (if (<= t_3 5e+306)
     (* (fabs w0) (sqrt (- 1.0 (* (/ (* t_2 t_2) 4.0) (/ h l)))))
     (if (<= t_3 INFINITY)
       (*
        (* t_1 (* (fabs t_0) (sqrt (* -0.25 (/ h l)))))
        (/ (fabs w0) (fabs d)))
       (*
        (fabs w0)
        (sqrt
         (fma
          (* (/ t_0 (+ (fabs d) (fabs d))) t_1)
          (* -0.5 (/ (* t_1 (* t_0 h)) (* (fabs d) l)))
          1.0))))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = fmin(fabs(M), fabs(D));
	double t_1 = fmax(fabs(M), fabs(D));
	double t_2 = (t_1 / fabs(d)) * t_0;
	double t_3 = fabs(w0) * sqrt((1.0 - (pow(((t_0 * t_1) / (2.0 * fabs(d))), 2.0) * (h / l))));
	double tmp;
	if (t_3 <= 5e+306) {
		tmp = fabs(w0) * sqrt((1.0 - (((t_2 * t_2) / 4.0) * (h / l))));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (t_1 * (fabs(t_0) * sqrt((-0.25 * (h / l))))) * (fabs(w0) / fabs(d));
	} else {
		tmp = fabs(w0) * sqrt(fma(((t_0 / (fabs(d) + fabs(d))) * t_1), (-0.5 * ((t_1 * (t_0 * h)) / (fabs(d) * l))), 1.0));
	}
	return copysign(1.0, w0) * tmp;
}

Julia

function code(w0, M, D, h, l, d)
	t_0 = fmin(abs(M), abs(D))
	t_1 = fmax(abs(M), abs(D))
	t_2 = Float64(Float64(t_1 / abs(d)) * t_0)
	t_3 = Float64(abs(w0) * sqrt(Float64(1.0 - Float64((Float64(Float64(t_0 * t_1) / Float64(2.0 * abs(d))) ^ 2.0) * Float64(h / l)))))
	tmp = 0.0
	if (t_3 <= 5e+306)
		tmp = Float64(abs(w0) * sqrt(Float64(1.0 - Float64(Float64(Float64(t_2 * t_2) / 4.0) * Float64(h / l)))));
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(t_1 * Float64(abs(t_0) * sqrt(Float64(-0.25 * Float64(h / l))))) * Float64(abs(w0) / abs(d)));
	else
		tmp = Float64(abs(w0) * sqrt(fma(Float64(Float64(t_0 / Float64(abs(d) + abs(d))) * t_1), Float64(-0.5 * Float64(Float64(t_1 * Float64(t_0 * h)) / Float64(abs(d) * l))), 1.0)));
	end
	return Float64(copysign(1.0, w0) * tmp)
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[Min[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 / N[Abs[d], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Abs[w0], $MachinePrecision] * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(t$95$0 * t$95$1), $MachinePrecision] / N[(2.0 * N[Abs[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, 5e+306], N[(N[Abs[w0], $MachinePrecision] * N[Sqrt[N[(1.0 - N[(N[(N[(t$95$2 * t$95$2), $MachinePrecision] / 4.0), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(t$95$1 * N[(N[Abs[t$95$0], $MachinePrecision] * N[Sqrt[N[(-0.25 * N[(h / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[w0], $MachinePrecision] / N[Abs[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[w0], $MachinePrecision] * N[Sqrt[N[(N[(N[(t$95$0 / N[(N[Abs[d], $MachinePrecision] + N[Abs[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(-0.5 * N[(N[(t$95$1 * N[(t$95$0 * h), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[d], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 w0 (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))))) < 4.9999999999999999e306

   1. Initial program 80.1%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{h}{\ell}} \]

      2. unpow2 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}} \]

      3. lift-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{h}{\ell}} \]

      4. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{M \cdot D}{\color{blue}{2 \cdot d}} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{h}{\ell}} \]

      5. *-commutative N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{M \cdot D}{\color{blue}{d \cdot 2}} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{h}{\ell}} \]

      6. associate-/r* N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\color{blue}{\frac{\frac{M \cdot D}{d}}{2}} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{h}{\ell}} \]

      7. lift-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\frac{M \cdot D}{d}}{2} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \frac{h}{\ell}} \]

      8. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\frac{M \cdot D}{d}}{2} \cdot \frac{M \cdot D}{\color{blue}{2 \cdot d}}\right) \cdot \frac{h}{\ell}} \]

      9. *-commutative N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\frac{M \cdot D}{d}}{2} \cdot \frac{M \cdot D}{\color{blue}{d \cdot 2}}\right) \cdot \frac{h}{\ell}} \]

      10. associate-/r* N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\frac{M \cdot D}{d}}{2} \cdot \color{blue}{\frac{\frac{M \cdot D}{d}}{2}}\right) \cdot \frac{h}{\ell}} \]

      11. frac-times N/A

          \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{M \cdot D}{d} \cdot \frac{M \cdot D}{d}}{2 \cdot 2}} \cdot \frac{h}{\ell}} \]

      12. metadata-eval N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{M \cdot D}{d} \cdot \frac{M \cdot D}{d}}{2 \cdot \color{blue}{\left(1 + 1\right)}} \cdot \frac{h}{\ell}} \]

      13. cosh-0-rev N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{M \cdot D}{d} \cdot \frac{M \cdot D}{d}}{2 \cdot \left(\color{blue}{\cosh 0} + 1\right)} \cdot \frac{h}{\ell}} \]

      14. lower-/.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{M \cdot D}{d} \cdot \frac{M \cdot D}{d}}{2 \cdot \left(\cosh 0 + 1\right)}} \cdot \frac{h}{\ell}} \]

   3. Applied rewrites 80.1%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\left(\frac{D}{d} \cdot M\right) \cdot \left(\frac{D}{d} \cdot M\right)}{4}} \cdot \frac{h}{\ell}} \]

   if 4.9999999999999999e306 < (*.f64 w0 (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))))) < +inf.0

   1. Initial program 80.1%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   2. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{w0 \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right)}}{d}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{w0 \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right)}}{\color{blue}{d}} \]

   4. Applied rewrites 9.3%

      \[\leadsto \color{blue}{\frac{w0 \cdot \sqrt{-0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}}{d}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{w0 \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}}{\color{blue}{d}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{w0 \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}}{d} \]

      3. *-commutative N/A

         \[\leadsto \frac{\sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}} \cdot w0}{d} \]

      4. associate-/l* N/A

         \[\leadsto \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}} \cdot \color{blue}{\frac{w0}{d}} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}} \cdot \color{blue}{\frac{w0}{d}} \]

   6. Applied rewrites 8.9%

      \[\leadsto \sqrt{\left(-0.25 \cdot \left(D \cdot D\right)\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}} \cdot \color{blue}{\frac{w0}{d}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}} \cdot \frac{\color{blue}{w0}}{d} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}} \cdot \frac{w0}{d} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\frac{\left(M \cdot M\right) \cdot h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)} \cdot \frac{w0}{d} \]

      4. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{\left(M \cdot M\right) \cdot h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)} \cdot \frac{w0}{d} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(M \cdot M\right) \cdot h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)} \cdot \frac{w0}{d} \]

      6. associate-/l* N/A

         \[\leadsto \sqrt{\left(\left(M \cdot M\right) \cdot \frac{h}{\ell}\right) \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)} \cdot \frac{w0}{d} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\left(M \cdot M\right) \cdot \left(\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)\right)} \cdot \frac{w0}{d} \]

      8. sqrt-prod N/A

         \[\leadsto \left(\sqrt{M \cdot M} \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{\color{blue}{w0}}{d} \]

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

         \[\leadsto \left(\sqrt{M \cdot M} \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{M \cdot M} \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{M \cdot M} \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

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

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

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

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{\color{blue}{w0}}{d} \]

      14. lower-fabs.f64 N/A

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

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

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

      16. lower-*.f64 N/A

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

      17. lower-/.f64 10.6%

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(-0.25 \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

      18. lift-*.f64 N/A

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

      19. *-commutative N/A

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\left(D \cdot D\right) \cdot \frac{-1}{4}\right)}\right) \cdot \frac{w0}{d} \]

      20. lower-*.f64 10.6%

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\left(D \cdot D\right) \cdot -0.25\right)}\right) \cdot \frac{w0}{d} \]

   8. Applied rewrites 10.6%

      \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\left(D \cdot D\right) \cdot -0.25\right)}\right) \cdot \frac{\color{blue}{w0}}{d} \]

   9. Taylor expanded in D around 0

      \[\leadsto \left(D \cdot \left(\left|M\right| \cdot \sqrt{\frac{-1}{4} \cdot \frac{h}{\ell}}\right)\right) \cdot \frac{\color{blue}{w0}}{d} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left(D \cdot \left(\left|M\right| \cdot \sqrt{\frac{-1}{4} \cdot \frac{h}{\ell}}\right)\right) \cdot \frac{w0}{d} \]

       2. lower-*.f64 N/A

          \[\leadsto \left(D \cdot \left(\left|M\right| \cdot \sqrt{\frac{-1}{4} \cdot \frac{h}{\ell}}\right)\right) \cdot \frac{w0}{d} \]

       3. lower-fabs.f64 N/A

          \[\leadsto \left(D \cdot \left(\left|M\right| \cdot \sqrt{\frac{-1}{4} \cdot \frac{h}{\ell}}\right)\right) \cdot \frac{w0}{d} \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \left(D \cdot \left(\left|M\right| \cdot \sqrt{\frac{-1}{4} \cdot \frac{h}{\ell}}\right)\right) \cdot \frac{w0}{d} \]

       5. lower-*.f64 N/A

          \[\leadsto \left(D \cdot \left(\left|M\right| \cdot \sqrt{\frac{-1}{4} \cdot \frac{h}{\ell}}\right)\right) \cdot \frac{w0}{d} \]

       6. lower-/.f64 12.4%

          \[\leadsto \left(D \cdot \left(\left|M\right| \cdot \sqrt{-0.25 \cdot \frac{h}{\ell}}\right)\right) \cdot \frac{w0}{d} \]

   11. Applied rewrites 12.4%

       \[\leadsto \left(D \cdot \left(\left|M\right| \cdot \sqrt{-0.25 \cdot \frac{h}{\ell}}\right)\right) \cdot \frac{\color{blue}{w0}}{d} \]

   if +inf.0 < (*.f64 w0 (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))))

   1. Initial program 80.1%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\color{blue}{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto w0 \cdot \sqrt{\color{blue}{1 + \left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)\right) \cdot \frac{h}{\ell}}} \]

      4. +-commutative N/A

         \[\leadsto w0 \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)\right) \cdot \frac{h}{\ell} + 1}} \]

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

         \[\leadsto w0 \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} + 1} \]

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

         \[\leadsto w0 \cdot \sqrt{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\mathsf{neg}\left(\frac{h}{\ell}\right)\right)} + 1} \]

      7. lift-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{h}{\ell}}\right)\right) + 1} \]

      8. distribute-neg-frac2 N/A

         \[\leadsto w0 \cdot \sqrt{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{h}{\mathsf{neg}\left(\ell\right)}} + 1} \]

      9. associate-/l* N/A

         \[\leadsto w0 \cdot \sqrt{\color{blue}{\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\mathsf{neg}\left(\ell\right)}} + 1} \]

      10. lift-pow.f64 N/A

          \[\leadsto w0 \cdot \sqrt{\frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot h}{\mathsf{neg}\left(\ell\right)} + 1} \]

      11. unpow2 N/A

          \[\leadsto w0 \cdot \sqrt{\frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot h}{\mathsf{neg}\left(\ell\right)} + 1} \]

      12. associate-*l* N/A

          \[\leadsto w0 \cdot \sqrt{\frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot h\right)}}{\mathsf{neg}\left(\ell\right)} + 1} \]

      13. associate-/l* N/A

          \[\leadsto w0 \cdot \sqrt{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot h}{\mathsf{neg}\left(\ell\right)}} + 1} \]

      14. lower-fma.f64 N/A

          \[\leadsto w0 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{2 \cdot d}, \frac{\frac{M \cdot D}{2 \cdot d} \cdot h}{\mathsf{neg}\left(\ell\right)}, 1\right)}} \]

   3. Applied rewrites 83.9%

      \[\leadsto w0 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{M}{d + d} \cdot D, \frac{\frac{D}{d + d} \cdot \left(M \cdot h\right)}{-\ell}, 1\right)}} \]

   4. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M}{d + d} \cdot D, \color{blue}{\frac{-1}{2} \cdot \frac{D \cdot \left(M \cdot h\right)}{d \cdot \ell}}, 1\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M}{d + d} \cdot D, \frac{-1}{2} \cdot \color{blue}{\frac{D \cdot \left(M \cdot h\right)}{d \cdot \ell}}, 1\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M}{d + d} \cdot D, \frac{-1}{2} \cdot \frac{D \cdot \left(M \cdot h\right)}{\color{blue}{d \cdot \ell}}, 1\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M}{d + d} \cdot D, \frac{-1}{2} \cdot \frac{D \cdot \left(M \cdot h\right)}{\color{blue}{d} \cdot \ell}, 1\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M}{d + d} \cdot D, \frac{-1}{2} \cdot \frac{D \cdot \left(M \cdot h\right)}{d \cdot \ell}, 1\right)} \]

      5. lower-*.f64 81.3%

         \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M}{d + d} \cdot D, -0.5 \cdot \frac{D \cdot \left(M \cdot h\right)}{d \cdot \color{blue}{\ell}}, 1\right)} \]

   6. Applied rewrites 81.3%

      \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M}{d + d} \cdot D, \color{blue}{-0.5 \cdot \frac{D \cdot \left(M \cdot h\right)}{d \cdot \ell}}, 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 91.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(M, \left|D\right|\right)\\ t_1 := \mathsf{max}\left(M, \left|D\right|\right)\\ t_2 := t\_0 \cdot t\_1\\ t_3 := \frac{t\_1}{\left|d\right|} \cdot t\_0\\ t_4 := \left|w0\right| \cdot \sqrt{1 - {\left(\frac{t\_2}{2 \cdot \left|d\right|}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathsf{copysign}\left(1, w0\right) \cdot \begin{array}{l} \mathbf{if}\;t\_4 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\left|w0\right| \cdot \sqrt{1 - \frac{t\_3 \cdot t\_3}{4} \cdot \frac{h}{\ell}}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\left(t\_1 \cdot \left(\left|t\_0\right| \cdot \sqrt{-0.25 \cdot \frac{h}{\ell}}\right)\right) \cdot \frac{\left|w0\right|}{\left|d\right|}\\ \mathbf{else}:\\ \;\;\;\;\left|w0\right| \cdot \sqrt{1 - t\_2 \cdot \frac{\frac{0.25}{\left|d\right|} \cdot \left(t\_2 \cdot h\right)}{\left|d\right| \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
  :precision binary64
  (let* ((t_0 (fmin M (fabs D)))
       (t_1 (fmax M (fabs D)))
       (t_2 (* t_0 t_1))
       (t_3 (* (/ t_1 (fabs d)) t_0))
       (t_4
        (*
         (fabs w0)
         (sqrt
          (- 1.0 (* (pow (/ t_2 (* 2.0 (fabs d))) 2.0) (/ h l)))))))
  (*
   (copysign 1.0 w0)
   (if (<= t_4 5e+306)
     (* (fabs w0) (sqrt (- 1.0 (* (/ (* t_3 t_3) 4.0) (/ h l)))))
     (if (<= t_4 INFINITY)
       (*
        (* t_1 (* (fabs t_0) (sqrt (* -0.25 (/ h l)))))
        (/ (fabs w0) (fabs d)))
       (*
        (fabs w0)
        (sqrt
         (-
          1.0
          (*
           t_2
           (/ (* (/ 0.25 (fabs d)) (* t_2 h)) (* (fabs d) l)))))))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = fmin(M, fabs(D));
	double t_1 = fmax(M, fabs(D));
	double t_2 = t_0 * t_1;
	double t_3 = (t_1 / fabs(d)) * t_0;
	double t_4 = fabs(w0) * sqrt((1.0 - (pow((t_2 / (2.0 * fabs(d))), 2.0) * (h / l))));
	double tmp;
	if (t_4 <= 5e+306) {
		tmp = fabs(w0) * sqrt((1.0 - (((t_3 * t_3) / 4.0) * (h / l))));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = (t_1 * (fabs(t_0) * sqrt((-0.25 * (h / l))))) * (fabs(w0) / fabs(d));
	} else {
		tmp = fabs(w0) * sqrt((1.0 - (t_2 * (((0.25 / fabs(d)) * (t_2 * h)) / (fabs(d) * l)))));
	}
	return copysign(1.0, w0) * tmp;
}

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = fmin(M, Math.abs(D));
	double t_1 = fmax(M, Math.abs(D));
	double t_2 = t_0 * t_1;
	double t_3 = (t_1 / Math.abs(d)) * t_0;
	double t_4 = Math.abs(w0) * Math.sqrt((1.0 - (Math.pow((t_2 / (2.0 * Math.abs(d))), 2.0) * (h / l))));
	double tmp;
	if (t_4 <= 5e+306) {
		tmp = Math.abs(w0) * Math.sqrt((1.0 - (((t_3 * t_3) / 4.0) * (h / l))));
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = (t_1 * (Math.abs(t_0) * Math.sqrt((-0.25 * (h / l))))) * (Math.abs(w0) / Math.abs(d));
	} else {
		tmp = Math.abs(w0) * Math.sqrt((1.0 - (t_2 * (((0.25 / Math.abs(d)) * (t_2 * h)) / (Math.abs(d) * l)))));
	}
	return Math.copySign(1.0, w0) * tmp;
}

Python

def code(w0, M, D, h, l, d):
	t_0 = fmin(M, math.fabs(D))
	t_1 = fmax(M, math.fabs(D))
	t_2 = t_0 * t_1
	t_3 = (t_1 / math.fabs(d)) * t_0
	t_4 = math.fabs(w0) * math.sqrt((1.0 - (math.pow((t_2 / (2.0 * math.fabs(d))), 2.0) * (h / l))))
	tmp = 0
	if t_4 <= 5e+306:
		tmp = math.fabs(w0) * math.sqrt((1.0 - (((t_3 * t_3) / 4.0) * (h / l))))
	elif t_4 <= math.inf:
		tmp = (t_1 * (math.fabs(t_0) * math.sqrt((-0.25 * (h / l))))) * (math.fabs(w0) / math.fabs(d))
	else:
		tmp = math.fabs(w0) * math.sqrt((1.0 - (t_2 * (((0.25 / math.fabs(d)) * (t_2 * h)) / (math.fabs(d) * l)))))
	return math.copysign(1.0, w0) * tmp

Julia

function code(w0, M, D, h, l, d)
	t_0 = fmin(M, abs(D))
	t_1 = fmax(M, abs(D))
	t_2 = Float64(t_0 * t_1)
	t_3 = Float64(Float64(t_1 / abs(d)) * t_0)
	t_4 = Float64(abs(w0) * sqrt(Float64(1.0 - Float64((Float64(t_2 / Float64(2.0 * abs(d))) ^ 2.0) * Float64(h / l)))))
	tmp = 0.0
	if (t_4 <= 5e+306)
		tmp = Float64(abs(w0) * sqrt(Float64(1.0 - Float64(Float64(Float64(t_3 * t_3) / 4.0) * Float64(h / l)))));
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(t_1 * Float64(abs(t_0) * sqrt(Float64(-0.25 * Float64(h / l))))) * Float64(abs(w0) / abs(d)));
	else
		tmp = Float64(abs(w0) * sqrt(Float64(1.0 - Float64(t_2 * Float64(Float64(Float64(0.25 / abs(d)) * Float64(t_2 * h)) / Float64(abs(d) * l))))));
	end
	return Float64(copysign(1.0, w0) * tmp)
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = min(M, abs(D));
	t_1 = max(M, abs(D));
	t_2 = t_0 * t_1;
	t_3 = (t_1 / abs(d)) * t_0;
	t_4 = abs(w0) * sqrt((1.0 - (((t_2 / (2.0 * abs(d))) ^ 2.0) * (h / l))));
	tmp = 0.0;
	if (t_4 <= 5e+306)
		tmp = abs(w0) * sqrt((1.0 - (((t_3 * t_3) / 4.0) * (h / l))));
	elseif (t_4 <= Inf)
		tmp = (t_1 * (abs(t_0) * sqrt((-0.25 * (h / l))))) * (abs(w0) / abs(d));
	else
		tmp = abs(w0) * sqrt((1.0 - (t_2 * (((0.25 / abs(d)) * (t_2 * h)) / (abs(d) * l)))));
	end
	tmp_2 = (sign(w0) * abs(1.0)) * tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[Min[M, N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Max[M, N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$1 / N[Abs[d], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[(N[Abs[w0], $MachinePrecision] * N[Sqrt[N[(1.0 - N[(N[Power[N[(t$95$2 / N[(2.0 * N[Abs[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$4, 5e+306], N[(N[Abs[w0], $MachinePrecision] * N[Sqrt[N[(1.0 - N[(N[(N[(t$95$3 * t$95$3), $MachinePrecision] / 4.0), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[(t$95$1 * N[(N[Abs[t$95$0], $MachinePrecision] * N[Sqrt[N[(-0.25 * N[(h / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[w0], $MachinePrecision] / N[Abs[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[w0], $MachinePrecision] * N[Sqrt[N[(1.0 - N[(t$95$2 * N[(N[(N[(0.25 / N[Abs[d], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * h), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[d], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 w0 (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))))) < 4.9999999999999999e306

   1. Initial program 80.1%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{h}{\ell}} \]

      2. unpow2 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}} \]

      3. lift-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{h}{\ell}} \]

      4. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{M \cdot D}{\color{blue}{2 \cdot d}} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{h}{\ell}} \]

      5. *-commutative N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{M \cdot D}{\color{blue}{d \cdot 2}} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{h}{\ell}} \]

      6. associate-/r* N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\color{blue}{\frac{\frac{M \cdot D}{d}}{2}} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{h}{\ell}} \]

      7. lift-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\frac{M \cdot D}{d}}{2} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \frac{h}{\ell}} \]

      8. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\frac{M \cdot D}{d}}{2} \cdot \frac{M \cdot D}{\color{blue}{2 \cdot d}}\right) \cdot \frac{h}{\ell}} \]

      9. *-commutative N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\frac{M \cdot D}{d}}{2} \cdot \frac{M \cdot D}{\color{blue}{d \cdot 2}}\right) \cdot \frac{h}{\ell}} \]

      10. associate-/r* N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\frac{M \cdot D}{d}}{2} \cdot \color{blue}{\frac{\frac{M \cdot D}{d}}{2}}\right) \cdot \frac{h}{\ell}} \]

      11. frac-times N/A

          \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{M \cdot D}{d} \cdot \frac{M \cdot D}{d}}{2 \cdot 2}} \cdot \frac{h}{\ell}} \]

      12. metadata-eval N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{M \cdot D}{d} \cdot \frac{M \cdot D}{d}}{2 \cdot \color{blue}{\left(1 + 1\right)}} \cdot \frac{h}{\ell}} \]

      13. cosh-0-rev N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{M \cdot D}{d} \cdot \frac{M \cdot D}{d}}{2 \cdot \left(\color{blue}{\cosh 0} + 1\right)} \cdot \frac{h}{\ell}} \]

      14. lower-/.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{M \cdot D}{d} \cdot \frac{M \cdot D}{d}}{2 \cdot \left(\cosh 0 + 1\right)}} \cdot \frac{h}{\ell}} \]

   3. Applied rewrites 80.1%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\left(\frac{D}{d} \cdot M\right) \cdot \left(\frac{D}{d} \cdot M\right)}{4}} \cdot \frac{h}{\ell}} \]

   if 4.9999999999999999e306 < (*.f64 w0 (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))))) < +inf.0

   1. Initial program 80.1%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   2. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{w0 \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right)}}{d}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{w0 \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right)}}{\color{blue}{d}} \]

   4. Applied rewrites 9.3%

      \[\leadsto \color{blue}{\frac{w0 \cdot \sqrt{-0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}}{d}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{w0 \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}}{\color{blue}{d}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{w0 \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}}{d} \]

      3. *-commutative N/A

         \[\leadsto \frac{\sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}} \cdot w0}{d} \]

      4. associate-/l* N/A

         \[\leadsto \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}} \cdot \color{blue}{\frac{w0}{d}} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}} \cdot \color{blue}{\frac{w0}{d}} \]

   6. Applied rewrites 8.9%

      \[\leadsto \sqrt{\left(-0.25 \cdot \left(D \cdot D\right)\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}} \cdot \color{blue}{\frac{w0}{d}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}} \cdot \frac{\color{blue}{w0}}{d} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}} \cdot \frac{w0}{d} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\frac{\left(M \cdot M\right) \cdot h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)} \cdot \frac{w0}{d} \]

      4. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{\left(M \cdot M\right) \cdot h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)} \cdot \frac{w0}{d} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(M \cdot M\right) \cdot h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)} \cdot \frac{w0}{d} \]

      6. associate-/l* N/A

         \[\leadsto \sqrt{\left(\left(M \cdot M\right) \cdot \frac{h}{\ell}\right) \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)} \cdot \frac{w0}{d} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\left(M \cdot M\right) \cdot \left(\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)\right)} \cdot \frac{w0}{d} \]

      8. sqrt-prod N/A

         \[\leadsto \left(\sqrt{M \cdot M} \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{\color{blue}{w0}}{d} \]

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

         \[\leadsto \left(\sqrt{M \cdot M} \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{M \cdot M} \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{M \cdot M} \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

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

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

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

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{\color{blue}{w0}}{d} \]

      14. lower-fabs.f64 N/A

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

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

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

      16. lower-*.f64 N/A

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

      17. lower-/.f64 10.6%

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(-0.25 \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

      18. lift-*.f64 N/A

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

      19. *-commutative N/A

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\left(D \cdot D\right) \cdot \frac{-1}{4}\right)}\right) \cdot \frac{w0}{d} \]

      20. lower-*.f64 10.6%

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\left(D \cdot D\right) \cdot -0.25\right)}\right) \cdot \frac{w0}{d} \]

   8. Applied rewrites 10.6%

      \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\left(D \cdot D\right) \cdot -0.25\right)}\right) \cdot \frac{\color{blue}{w0}}{d} \]

   9. Taylor expanded in D around 0

      \[\leadsto \left(D \cdot \left(\left|M\right| \cdot \sqrt{\frac{-1}{4} \cdot \frac{h}{\ell}}\right)\right) \cdot \frac{\color{blue}{w0}}{d} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left(D \cdot \left(\left|M\right| \cdot \sqrt{\frac{-1}{4} \cdot \frac{h}{\ell}}\right)\right) \cdot \frac{w0}{d} \]

       2. lower-*.f64 N/A

          \[\leadsto \left(D \cdot \left(\left|M\right| \cdot \sqrt{\frac{-1}{4} \cdot \frac{h}{\ell}}\right)\right) \cdot \frac{w0}{d} \]

       3. lower-fabs.f64 N/A

          \[\leadsto \left(D \cdot \left(\left|M\right| \cdot \sqrt{\frac{-1}{4} \cdot \frac{h}{\ell}}\right)\right) \cdot \frac{w0}{d} \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \left(D \cdot \left(\left|M\right| \cdot \sqrt{\frac{-1}{4} \cdot \frac{h}{\ell}}\right)\right) \cdot \frac{w0}{d} \]

       5. lower-*.f64 N/A

          \[\leadsto \left(D \cdot \left(\left|M\right| \cdot \sqrt{\frac{-1}{4} \cdot \frac{h}{\ell}}\right)\right) \cdot \frac{w0}{d} \]

       6. lower-/.f64 12.4%

          \[\leadsto \left(D \cdot \left(\left|M\right| \cdot \sqrt{-0.25 \cdot \frac{h}{\ell}}\right)\right) \cdot \frac{w0}{d} \]

   11. Applied rewrites 12.4%

       \[\leadsto \left(D \cdot \left(\left|M\right| \cdot \sqrt{-0.25 \cdot \frac{h}{\ell}}\right)\right) \cdot \frac{\color{blue}{w0}}{d} \]

   if +inf.0 < (*.f64 w0 (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))))

   1. Initial program 80.1%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{h}{\ell}} \]

      3. unpow2 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}} \]

      4. lift-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \frac{h}{\ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{\color{blue}{2 \cdot d}}\right) \cdot \frac{h}{\ell}} \]

      6. associate-/r* N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\frac{\frac{M \cdot D}{2}}{d}}\right) \cdot \frac{h}{\ell}} \]

      7. associate-*r/ N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2}}{d}} \cdot \frac{h}{\ell}} \]

      8. lift-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2}}{d} \cdot \color{blue}{\frac{h}{\ell}}} \]

      9. frac-times N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2}\right) \cdot h}{d \cdot \ell}}} \]

      10. lower-/.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2}\right) \cdot h}{d \cdot \ell}}} \]

   3. Applied rewrites 64.4%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)}{d} \cdot h}{d \cdot \ell}}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{\color{blue}{\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)}}{d} \cdot h}{d \cdot \ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{\left(D \cdot D\right) \cdot \color{blue}{\left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)}}{d} \cdot h}{d \cdot \ell}} \]

      3. associate-*r* N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{\color{blue}{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{1}{4}}}{d} \cdot h}{d \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{\color{blue}{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{1}{4}}}{d} \cdot h}{d \cdot \ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot \frac{1}{4}}{d} \cdot h}{d \cdot \ell}} \]

      6. associate-*r* N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{\color{blue}{\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right)} \cdot \frac{1}{4}}{d} \cdot h}{d \cdot \ell}} \]

      7. lower-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{\color{blue}{\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right)} \cdot \frac{1}{4}}{d} \cdot h}{d \cdot \ell}} \]

      8. lower-*.f64 70.9%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{\left(\color{blue}{\left(\left(D \cdot D\right) \cdot M\right)} \cdot M\right) \cdot 0.25}{d} \cdot h}{d \cdot \ell}} \]

   5. Applied rewrites 70.9%

      \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{\color{blue}{\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right) \cdot 0.25}}{d} \cdot h}{d \cdot \ell}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right) \cdot \frac{1}{4}}{d}} \cdot h}{d \cdot \ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{\color{blue}{\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right) \cdot \frac{1}{4}}}{d} \cdot h}{d \cdot \ell}} \]

      3. associate-/l* N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right) \cdot \frac{\frac{1}{4}}{d}\right)} \cdot h}{d \cdot \ell}} \]

      4. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\color{blue}{\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right)} \cdot \frac{\frac{1}{4}}{d}\right) \cdot h}{d \cdot \ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\left(\color{blue}{\left(\left(D \cdot D\right) \cdot M\right)} \cdot M\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot h}{d \cdot \ell}} \]

      6. associate-*l* N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\color{blue}{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right)} \cdot \frac{\frac{1}{4}}{d}\right) \cdot h}{d \cdot \ell}} \]

      7. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\left(\color{blue}{\left(D \cdot D\right)} \cdot \left(M \cdot M\right)\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot h}{d \cdot \ell}} \]

      8. unswap-sqr N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot \frac{\frac{1}{4}}{d}\right) \cdot h}{d \cdot \ell}} \]

      9. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\left(\color{blue}{\left(D \cdot M\right)} \cdot \left(D \cdot M\right)\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot h}{d \cdot \ell}} \]

      10. lift-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\left(\left(D \cdot M\right) \cdot \color{blue}{\left(D \cdot M\right)}\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot h}{d \cdot \ell}} \]

      11. associate-*l* N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\left(D \cdot M\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{\frac{1}{4}}{d}\right)\right)} \cdot h}{d \cdot \ell}} \]

      12. lower-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\left(D \cdot M\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{\frac{1}{4}}{d}\right)\right)} \cdot h}{d \cdot \ell}} \]

      13. lower-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\left(D \cdot M\right) \cdot \color{blue}{\left(\left(D \cdot M\right) \cdot \frac{\frac{1}{4}}{d}\right)}\right) \cdot h}{d \cdot \ell}} \]

      14. lower-/.f64 83.2%

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\left(D \cdot M\right) \cdot \left(\left(D \cdot M\right) \cdot \color{blue}{\frac{0.25}{d}}\right)\right) \cdot h}{d \cdot \ell}} \]

   7. Applied rewrites 83.2%

      \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\left(D \cdot M\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{0.25}{d}\right)\right)} \cdot h}{d \cdot \ell}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\left(\left(D \cdot M\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{\frac{1}{4}}{d}\right)\right) \cdot h}{d \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\left(D \cdot M\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{\frac{1}{4}}{d}\right)\right) \cdot h}}{d \cdot \ell}} \]

      3. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\left(D \cdot M\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{\frac{1}{4}}{d}\right)\right)} \cdot h}{d \cdot \ell}} \]

      4. associate-*l* N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(D \cdot M\right) \cdot \left(\left(\left(D \cdot M\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot h\right)}}{d \cdot \ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(D \cdot M\right) \cdot \left(\left(\left(D \cdot M\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot h\right)}{\color{blue}{d \cdot \ell}}} \]

      6. *-commutative N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(D \cdot M\right) \cdot \left(\left(\left(D \cdot M\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot h\right)}{\color{blue}{\ell \cdot d}}} \]

      7. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(D \cdot M\right) \cdot \left(\left(\left(D \cdot M\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot h\right)}{\color{blue}{\ell \cdot d}}} \]

      8. associate-/l* N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(D \cdot M\right) \cdot \frac{\left(\left(D \cdot M\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot h}{\ell \cdot d}}} \]

      9. lower-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(D \cdot M\right) \cdot \frac{\left(\left(D \cdot M\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot h}{\ell \cdot d}}} \]

      10. lift-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(D \cdot M\right)} \cdot \frac{\left(\left(D \cdot M\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot h}{\ell \cdot d}} \]

      11. *-commutative N/A

          \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(M \cdot D\right)} \cdot \frac{\left(\left(D \cdot M\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot h}{\ell \cdot d}} \]

      12. lower-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(M \cdot D\right)} \cdot \frac{\left(\left(D \cdot M\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot h}{\ell \cdot d}} \]

      13. lower-/.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(M \cdot D\right) \cdot \color{blue}{\frac{\left(\left(D \cdot M\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot h}{\ell \cdot d}}} \]

      14. lift-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(M \cdot D\right) \cdot \frac{\color{blue}{\left(\left(D \cdot M\right) \cdot \frac{\frac{1}{4}}{d}\right)} \cdot h}{\ell \cdot d}} \]

      15. *-commutative N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(M \cdot D\right) \cdot \frac{\color{blue}{\left(\frac{\frac{1}{4}}{d} \cdot \left(D \cdot M\right)\right)} \cdot h}{\ell \cdot d}} \]

      16. associate-*l* N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(M \cdot D\right) \cdot \frac{\color{blue}{\frac{\frac{1}{4}}{d} \cdot \left(\left(D \cdot M\right) \cdot h\right)}}{\ell \cdot d}} \]

      17. lower-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(M \cdot D\right) \cdot \frac{\color{blue}{\frac{\frac{1}{4}}{d} \cdot \left(\left(D \cdot M\right) \cdot h\right)}}{\ell \cdot d}} \]

      18. lower-*.f64 84.2%

          \[\leadsto w0 \cdot \sqrt{1 - \left(M \cdot D\right) \cdot \frac{\frac{0.25}{d} \cdot \color{blue}{\left(\left(D \cdot M\right) \cdot h\right)}}{\ell \cdot d}} \]

      19. lift-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(M \cdot D\right) \cdot \frac{\frac{\frac{1}{4}}{d} \cdot \left(\color{blue}{\left(D \cdot M\right)} \cdot h\right)}{\ell \cdot d}} \]

      20. *-commutative N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(M \cdot D\right) \cdot \frac{\frac{\frac{1}{4}}{d} \cdot \left(\color{blue}{\left(M \cdot D\right)} \cdot h\right)}{\ell \cdot d}} \]

      21. lower-*.f64 84.2%

          \[\leadsto w0 \cdot \sqrt{1 - \left(M \cdot D\right) \cdot \frac{\frac{0.25}{d} \cdot \left(\color{blue}{\left(M \cdot D\right)} \cdot h\right)}{\ell \cdot d}} \]

      22. lift-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(M \cdot D\right) \cdot \frac{\frac{\frac{1}{4}}{d} \cdot \left(\left(M \cdot D\right) \cdot h\right)}{\color{blue}{\ell \cdot d}}} \]

      23. *-commutative N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(M \cdot D\right) \cdot \frac{\frac{\frac{1}{4}}{d} \cdot \left(\left(M \cdot D\right) \cdot h\right)}{\color{blue}{d \cdot \ell}}} \]

      24. lift-*.f64 84.2%

          \[\leadsto w0 \cdot \sqrt{1 - \left(M \cdot D\right) \cdot \frac{\frac{0.25}{d} \cdot \left(\left(M \cdot D\right) \cdot h\right)}{\color{blue}{d \cdot \ell}}} \]

   9. Applied rewrites 84.2%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(M \cdot D\right) \cdot \frac{\frac{0.25}{d} \cdot \left(\left(M \cdot D\right) \cdot h\right)}{d \cdot \ell}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 90.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_1 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_2 := t\_1 \cdot t\_0\\ t_3 := {\left(\frac{t\_2}{2 \cdot \left|d\right|}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+240}:\\ \;\;\;\;\left(t\_0 \cdot \left(\left|t\_1\right| \cdot \sqrt{-0.25 \cdot \frac{h}{\ell}}\right)\right) \cdot \frac{w0}{\left|d\right|}\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-12}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t\_2 \cdot \frac{\frac{0.25}{\left|d\right|} \cdot \left(t\_2 \cdot h\right)}{\left|d\right| \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \end{array} \]

FPCore

(FPCore (w0 M D h l d)
  :precision binary64
  (let* ((t_0 (fmax (fabs M) (fabs D)))
       (t_1 (fmin (fabs M) (fabs D)))
       (t_2 (* t_1 t_0))
       (t_3 (* (pow (/ t_2 (* 2.0 (fabs d))) 2.0) (/ h l))))
  (if (<= t_3 -2e+240)
    (*
     (* t_0 (* (fabs t_1) (sqrt (* -0.25 (/ h l)))))
     (/ w0 (fabs d)))
    (if (<= t_3 -5e-12)
      (*
       w0
       (sqrt
        (-
         1.0
         (* t_2 (/ (* (/ 0.25 (fabs d)) (* t_2 h)) (* (fabs d) l))))))
      (* w0 (sqrt 1.0))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = fmax(fabs(M), fabs(D));
	double t_1 = fmin(fabs(M), fabs(D));
	double t_2 = t_1 * t_0;
	double t_3 = pow((t_2 / (2.0 * fabs(d))), 2.0) * (h / l);
	double tmp;
	if (t_3 <= -2e+240) {
		tmp = (t_0 * (fabs(t_1) * sqrt((-0.25 * (h / l))))) * (w0 / fabs(d));
	} else if (t_3 <= -5e-12) {
		tmp = w0 * sqrt((1.0 - (t_2 * (((0.25 / fabs(d)) * (t_2 * h)) / (fabs(d) * l)))));
	} else {
		tmp = w0 * sqrt(1.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(w0, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = fmax(abs(m), abs(d))
    t_1 = fmin(abs(m), abs(d))
    t_2 = t_1 * t_0
    t_3 = ((t_2 / (2.0d0 * abs(d_1))) ** 2.0d0) * (h / l)
    if (t_3 <= (-2d+240)) then
        tmp = (t_0 * (abs(t_1) * sqrt(((-0.25d0) * (h / l))))) * (w0 / abs(d_1))
    else if (t_3 <= (-5d-12)) then
        tmp = w0 * sqrt((1.0d0 - (t_2 * (((0.25d0 / abs(d_1)) * (t_2 * h)) / (abs(d_1) * l)))))
    else
        tmp = w0 * sqrt(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = fmax(Math.abs(M), Math.abs(D));
	double t_1 = fmin(Math.abs(M), Math.abs(D));
	double t_2 = t_1 * t_0;
	double t_3 = Math.pow((t_2 / (2.0 * Math.abs(d))), 2.0) * (h / l);
	double tmp;
	if (t_3 <= -2e+240) {
		tmp = (t_0 * (Math.abs(t_1) * Math.sqrt((-0.25 * (h / l))))) * (w0 / Math.abs(d));
	} else if (t_3 <= -5e-12) {
		tmp = w0 * Math.sqrt((1.0 - (t_2 * (((0.25 / Math.abs(d)) * (t_2 * h)) / (Math.abs(d) * l)))));
	} else {
		tmp = w0 * Math.sqrt(1.0);
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	t_0 = fmax(math.fabs(M), math.fabs(D))
	t_1 = fmin(math.fabs(M), math.fabs(D))
	t_2 = t_1 * t_0
	t_3 = math.pow((t_2 / (2.0 * math.fabs(d))), 2.0) * (h / l)
	tmp = 0
	if t_3 <= -2e+240:
		tmp = (t_0 * (math.fabs(t_1) * math.sqrt((-0.25 * (h / l))))) * (w0 / math.fabs(d))
	elif t_3 <= -5e-12:
		tmp = w0 * math.sqrt((1.0 - (t_2 * (((0.25 / math.fabs(d)) * (t_2 * h)) / (math.fabs(d) * l)))))
	else:
		tmp = w0 * math.sqrt(1.0)
	return tmp

Julia

function code(w0, M, D, h, l, d)
	t_0 = fmax(abs(M), abs(D))
	t_1 = fmin(abs(M), abs(D))
	t_2 = Float64(t_1 * t_0)
	t_3 = Float64((Float64(t_2 / Float64(2.0 * abs(d))) ^ 2.0) * Float64(h / l))
	tmp = 0.0
	if (t_3 <= -2e+240)
		tmp = Float64(Float64(t_0 * Float64(abs(t_1) * sqrt(Float64(-0.25 * Float64(h / l))))) * Float64(w0 / abs(d)));
	elseif (t_3 <= -5e-12)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(t_2 * Float64(Float64(Float64(0.25 / abs(d)) * Float64(t_2 * h)) / Float64(abs(d) * l))))));
	else
		tmp = Float64(w0 * sqrt(1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = max(abs(M), abs(D));
	t_1 = min(abs(M), abs(D));
	t_2 = t_1 * t_0;
	t_3 = ((t_2 / (2.0 * abs(d))) ^ 2.0) * (h / l);
	tmp = 0.0;
	if (t_3 <= -2e+240)
		tmp = (t_0 * (abs(t_1) * sqrt((-0.25 * (h / l))))) * (w0 / abs(d));
	elseif (t_3 <= -5e-12)
		tmp = w0 * sqrt((1.0 - (t_2 * (((0.25 / abs(d)) * (t_2 * h)) / (abs(d) * l)))));
	else
		tmp = w0 * sqrt(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[Max[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[N[(t$95$2 / N[(2.0 * N[Abs[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -2e+240], N[(N[(t$95$0 * N[(N[Abs[t$95$1], $MachinePrecision] * N[Sqrt[N[(-0.25 * N[(h / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(w0 / N[Abs[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -5e-12], N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$2 * N[(N[(N[(0.25 / N[Abs[d], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * h), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[d], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[1.0], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2e240

   1. Initial program 80.1%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   2. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{w0 \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right)}}{d}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{w0 \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right)}}{\color{blue}{d}} \]

   4. Applied rewrites 9.3%

      \[\leadsto \color{blue}{\frac{w0 \cdot \sqrt{-0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}}{d}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{w0 \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}}{\color{blue}{d}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{w0 \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}}{d} \]

      3. *-commutative N/A

         \[\leadsto \frac{\sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}} \cdot w0}{d} \]

      4. associate-/l* N/A

         \[\leadsto \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}} \cdot \color{blue}{\frac{w0}{d}} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}} \cdot \color{blue}{\frac{w0}{d}} \]

   6. Applied rewrites 8.9%

      \[\leadsto \sqrt{\left(-0.25 \cdot \left(D \cdot D\right)\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}} \cdot \color{blue}{\frac{w0}{d}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}} \cdot \frac{\color{blue}{w0}}{d} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}} \cdot \frac{w0}{d} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\frac{\left(M \cdot M\right) \cdot h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)} \cdot \frac{w0}{d} \]

      4. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{\left(M \cdot M\right) \cdot h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)} \cdot \frac{w0}{d} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(M \cdot M\right) \cdot h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)} \cdot \frac{w0}{d} \]

      6. associate-/l* N/A

         \[\leadsto \sqrt{\left(\left(M \cdot M\right) \cdot \frac{h}{\ell}\right) \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)} \cdot \frac{w0}{d} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\left(M \cdot M\right) \cdot \left(\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)\right)} \cdot \frac{w0}{d} \]

      8. sqrt-prod N/A

         \[\leadsto \left(\sqrt{M \cdot M} \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{\color{blue}{w0}}{d} \]

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

         \[\leadsto \left(\sqrt{M \cdot M} \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{M \cdot M} \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{M \cdot M} \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

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

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

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

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{\color{blue}{w0}}{d} \]

      14. lower-fabs.f64 N/A

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

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

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

      16. lower-*.f64 N/A

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

      17. lower-/.f64 10.6%

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(-0.25 \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

      18. lift-*.f64 N/A

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

      19. *-commutative N/A

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\left(D \cdot D\right) \cdot \frac{-1}{4}\right)}\right) \cdot \frac{w0}{d} \]

      20. lower-*.f64 10.6%

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\left(D \cdot D\right) \cdot -0.25\right)}\right) \cdot \frac{w0}{d} \]

   8. Applied rewrites 10.6%

      \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\left(D \cdot D\right) \cdot -0.25\right)}\right) \cdot \frac{\color{blue}{w0}}{d} \]

   9. Taylor expanded in D around 0

      \[\leadsto \left(D \cdot \left(\left|M\right| \cdot \sqrt{\frac{-1}{4} \cdot \frac{h}{\ell}}\right)\right) \cdot \frac{\color{blue}{w0}}{d} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left(D \cdot \left(\left|M\right| \cdot \sqrt{\frac{-1}{4} \cdot \frac{h}{\ell}}\right)\right) \cdot \frac{w0}{d} \]

       2. lower-*.f64 N/A

          \[\leadsto \left(D \cdot \left(\left|M\right| \cdot \sqrt{\frac{-1}{4} \cdot \frac{h}{\ell}}\right)\right) \cdot \frac{w0}{d} \]

       3. lower-fabs.f64 N/A

          \[\leadsto \left(D \cdot \left(\left|M\right| \cdot \sqrt{\frac{-1}{4} \cdot \frac{h}{\ell}}\right)\right) \cdot \frac{w0}{d} \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \left(D \cdot \left(\left|M\right| \cdot \sqrt{\frac{-1}{4} \cdot \frac{h}{\ell}}\right)\right) \cdot \frac{w0}{d} \]

       5. lower-*.f64 N/A

          \[\leadsto \left(D \cdot \left(\left|M\right| \cdot \sqrt{\frac{-1}{4} \cdot \frac{h}{\ell}}\right)\right) \cdot \frac{w0}{d} \]

       6. lower-/.f64 12.4%

          \[\leadsto \left(D \cdot \left(\left|M\right| \cdot \sqrt{-0.25 \cdot \frac{h}{\ell}}\right)\right) \cdot \frac{w0}{d} \]

   11. Applied rewrites 12.4%

       \[\leadsto \left(D \cdot \left(\left|M\right| \cdot \sqrt{-0.25 \cdot \frac{h}{\ell}}\right)\right) \cdot \frac{\color{blue}{w0}}{d} \]

   if -2e240 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -4.9999999999999997e-12

   1. Initial program 80.1%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{h}{\ell}} \]

      3. unpow2 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}} \]

      4. lift-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \frac{h}{\ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{\color{blue}{2 \cdot d}}\right) \cdot \frac{h}{\ell}} \]

      6. associate-/r* N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\frac{\frac{M \cdot D}{2}}{d}}\right) \cdot \frac{h}{\ell}} \]

      7. associate-*r/ N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2}}{d}} \cdot \frac{h}{\ell}} \]

      8. lift-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2}}{d} \cdot \color{blue}{\frac{h}{\ell}}} \]

      9. frac-times N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2}\right) \cdot h}{d \cdot \ell}}} \]

      10. lower-/.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2}\right) \cdot h}{d \cdot \ell}}} \]

   3. Applied rewrites 64.4%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)}{d} \cdot h}{d \cdot \ell}}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{\color{blue}{\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)}}{d} \cdot h}{d \cdot \ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{\left(D \cdot D\right) \cdot \color{blue}{\left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)}}{d} \cdot h}{d \cdot \ell}} \]

      3. associate-*r* N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{\color{blue}{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{1}{4}}}{d} \cdot h}{d \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{\color{blue}{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{1}{4}}}{d} \cdot h}{d \cdot \ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot \frac{1}{4}}{d} \cdot h}{d \cdot \ell}} \]

      6. associate-*r* N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{\color{blue}{\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right)} \cdot \frac{1}{4}}{d} \cdot h}{d \cdot \ell}} \]

      7. lower-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{\color{blue}{\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right)} \cdot \frac{1}{4}}{d} \cdot h}{d \cdot \ell}} \]

      8. lower-*.f64 70.9%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{\left(\color{blue}{\left(\left(D \cdot D\right) \cdot M\right)} \cdot M\right) \cdot 0.25}{d} \cdot h}{d \cdot \ell}} \]

   5. Applied rewrites 70.9%

      \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{\color{blue}{\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right) \cdot 0.25}}{d} \cdot h}{d \cdot \ell}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right) \cdot \frac{1}{4}}{d}} \cdot h}{d \cdot \ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{\color{blue}{\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right) \cdot \frac{1}{4}}}{d} \cdot h}{d \cdot \ell}} \]

      3. associate-/l* N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right) \cdot \frac{\frac{1}{4}}{d}\right)} \cdot h}{d \cdot \ell}} \]

      4. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\color{blue}{\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right)} \cdot \frac{\frac{1}{4}}{d}\right) \cdot h}{d \cdot \ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\left(\color{blue}{\left(\left(D \cdot D\right) \cdot M\right)} \cdot M\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot h}{d \cdot \ell}} \]

      6. associate-*l* N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\color{blue}{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right)} \cdot \frac{\frac{1}{4}}{d}\right) \cdot h}{d \cdot \ell}} \]

      7. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\left(\color{blue}{\left(D \cdot D\right)} \cdot \left(M \cdot M\right)\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot h}{d \cdot \ell}} \]

      8. unswap-sqr N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot \frac{\frac{1}{4}}{d}\right) \cdot h}{d \cdot \ell}} \]

      9. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\left(\color{blue}{\left(D \cdot M\right)} \cdot \left(D \cdot M\right)\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot h}{d \cdot \ell}} \]

      10. lift-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\left(\left(D \cdot M\right) \cdot \color{blue}{\left(D \cdot M\right)}\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot h}{d \cdot \ell}} \]

      11. associate-*l* N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\left(D \cdot M\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{\frac{1}{4}}{d}\right)\right)} \cdot h}{d \cdot \ell}} \]

      12. lower-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\left(D \cdot M\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{\frac{1}{4}}{d}\right)\right)} \cdot h}{d \cdot \ell}} \]

      13. lower-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\left(D \cdot M\right) \cdot \color{blue}{\left(\left(D \cdot M\right) \cdot \frac{\frac{1}{4}}{d}\right)}\right) \cdot h}{d \cdot \ell}} \]

      14. lower-/.f64 83.2%

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\left(D \cdot M\right) \cdot \left(\left(D \cdot M\right) \cdot \color{blue}{\frac{0.25}{d}}\right)\right) \cdot h}{d \cdot \ell}} \]

   7. Applied rewrites 83.2%

      \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\left(D \cdot M\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{0.25}{d}\right)\right)} \cdot h}{d \cdot \ell}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\left(\left(D \cdot M\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{\frac{1}{4}}{d}\right)\right) \cdot h}{d \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\left(D \cdot M\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{\frac{1}{4}}{d}\right)\right) \cdot h}}{d \cdot \ell}} \]

      3. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\left(D \cdot M\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{\frac{1}{4}}{d}\right)\right)} \cdot h}{d \cdot \ell}} \]

      4. associate-*l* N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(D \cdot M\right) \cdot \left(\left(\left(D \cdot M\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot h\right)}}{d \cdot \ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(D \cdot M\right) \cdot \left(\left(\left(D \cdot M\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot h\right)}{\color{blue}{d \cdot \ell}}} \]

      6. *-commutative N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(D \cdot M\right) \cdot \left(\left(\left(D \cdot M\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot h\right)}{\color{blue}{\ell \cdot d}}} \]

      7. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(D \cdot M\right) \cdot \left(\left(\left(D \cdot M\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot h\right)}{\color{blue}{\ell \cdot d}}} \]

      8. associate-/l* N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(D \cdot M\right) \cdot \frac{\left(\left(D \cdot M\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot h}{\ell \cdot d}}} \]

      9. lower-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(D \cdot M\right) \cdot \frac{\left(\left(D \cdot M\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot h}{\ell \cdot d}}} \]

      10. lift-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(D \cdot M\right)} \cdot \frac{\left(\left(D \cdot M\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot h}{\ell \cdot d}} \]

      11. *-commutative N/A

          \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(M \cdot D\right)} \cdot \frac{\left(\left(D \cdot M\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot h}{\ell \cdot d}} \]

      12. lower-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(M \cdot D\right)} \cdot \frac{\left(\left(D \cdot M\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot h}{\ell \cdot d}} \]

      13. lower-/.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(M \cdot D\right) \cdot \color{blue}{\frac{\left(\left(D \cdot M\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot h}{\ell \cdot d}}} \]

      14. lift-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(M \cdot D\right) \cdot \frac{\color{blue}{\left(\left(D \cdot M\right) \cdot \frac{\frac{1}{4}}{d}\right)} \cdot h}{\ell \cdot d}} \]

      15. *-commutative N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(M \cdot D\right) \cdot \frac{\color{blue}{\left(\frac{\frac{1}{4}}{d} \cdot \left(D \cdot M\right)\right)} \cdot h}{\ell \cdot d}} \]

      16. associate-*l* N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(M \cdot D\right) \cdot \frac{\color{blue}{\frac{\frac{1}{4}}{d} \cdot \left(\left(D \cdot M\right) \cdot h\right)}}{\ell \cdot d}} \]

      17. lower-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(M \cdot D\right) \cdot \frac{\color{blue}{\frac{\frac{1}{4}}{d} \cdot \left(\left(D \cdot M\right) \cdot h\right)}}{\ell \cdot d}} \]

      18. lower-*.f64 84.2%

          \[\leadsto w0 \cdot \sqrt{1 - \left(M \cdot D\right) \cdot \frac{\frac{0.25}{d} \cdot \color{blue}{\left(\left(D \cdot M\right) \cdot h\right)}}{\ell \cdot d}} \]

      19. lift-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(M \cdot D\right) \cdot \frac{\frac{\frac{1}{4}}{d} \cdot \left(\color{blue}{\left(D \cdot M\right)} \cdot h\right)}{\ell \cdot d}} \]

      20. *-commutative N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(M \cdot D\right) \cdot \frac{\frac{\frac{1}{4}}{d} \cdot \left(\color{blue}{\left(M \cdot D\right)} \cdot h\right)}{\ell \cdot d}} \]

      21. lower-*.f64 84.2%

          \[\leadsto w0 \cdot \sqrt{1 - \left(M \cdot D\right) \cdot \frac{\frac{0.25}{d} \cdot \left(\color{blue}{\left(M \cdot D\right)} \cdot h\right)}{\ell \cdot d}} \]

      22. lift-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(M \cdot D\right) \cdot \frac{\frac{\frac{1}{4}}{d} \cdot \left(\left(M \cdot D\right) \cdot h\right)}{\color{blue}{\ell \cdot d}}} \]

      23. *-commutative N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(M \cdot D\right) \cdot \frac{\frac{\frac{1}{4}}{d} \cdot \left(\left(M \cdot D\right) \cdot h\right)}{\color{blue}{d \cdot \ell}}} \]

      24. lift-*.f64 84.2%

          \[\leadsto w0 \cdot \sqrt{1 - \left(M \cdot D\right) \cdot \frac{\frac{0.25}{d} \cdot \left(\left(M \cdot D\right) \cdot h\right)}{\color{blue}{d \cdot \ell}}} \]

   9. Applied rewrites 84.2%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(M \cdot D\right) \cdot \frac{\frac{0.25}{d} \cdot \left(\left(M \cdot D\right) \cdot h\right)}{d \cdot \ell}}} \]

   if -4.9999999999999997e-12 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 80.1%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   2. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \sqrt{\color{blue}{1}} \]
   3. Step-by-step derivation

   4. Applied rewrites 68.8%

      \[\leadsto w0 \cdot \sqrt{\color{blue}{1}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 90.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|M\right|, D\right)\\ t_1 := \mathsf{min}\left(\left|M\right|, D\right)\\ \mathbf{if}\;{\left(\frac{t\_1 \cdot t\_0}{2 \cdot \left|d\right|}\right)}^{2} \cdot \frac{h}{\ell} \leq -2000:\\ \;\;\;\;\frac{t\_0 \cdot \left(w0 \cdot \left(\left|t\_1\right| \cdot \sqrt{-0.25 \cdot \frac{h}{\ell}}\right)\right)}{\left|d\right|}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \end{array} \]

FPCore

(FPCore (w0 M D h l d)
  :precision binary64
  (let* ((t_0 (fmax (fabs M) D)) (t_1 (fmin (fabs M) D)))
  (if (<=
       (* (pow (/ (* t_1 t_0) (* 2.0 (fabs d))) 2.0) (/ h l))
       -2000.0)
    (/
     (* t_0 (* w0 (* (fabs t_1) (sqrt (* -0.25 (/ h l))))))
     (fabs d))
    (* w0 (sqrt 1.0)))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = fmax(fabs(M), D);
	double t_1 = fmin(fabs(M), D);
	double tmp;
	if ((pow(((t_1 * t_0) / (2.0 * fabs(d))), 2.0) * (h / l)) <= -2000.0) {
		tmp = (t_0 * (w0 * (fabs(t_1) * sqrt((-0.25 * (h / l)))))) / fabs(d);
	} else {
		tmp = w0 * sqrt(1.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(w0, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = fmax(abs(m), d)
    t_1 = fmin(abs(m), d)
    if (((((t_1 * t_0) / (2.0d0 * abs(d_1))) ** 2.0d0) * (h / l)) <= (-2000.0d0)) then
        tmp = (t_0 * (w0 * (abs(t_1) * sqrt(((-0.25d0) * (h / l)))))) / abs(d_1)
    else
        tmp = w0 * sqrt(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = fmax(Math.abs(M), D);
	double t_1 = fmin(Math.abs(M), D);
	double tmp;
	if ((Math.pow(((t_1 * t_0) / (2.0 * Math.abs(d))), 2.0) * (h / l)) <= -2000.0) {
		tmp = (t_0 * (w0 * (Math.abs(t_1) * Math.sqrt((-0.25 * (h / l)))))) / Math.abs(d);
	} else {
		tmp = w0 * Math.sqrt(1.0);
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	t_0 = fmax(math.fabs(M), D)
	t_1 = fmin(math.fabs(M), D)
	tmp = 0
	if (math.pow(((t_1 * t_0) / (2.0 * math.fabs(d))), 2.0) * (h / l)) <= -2000.0:
		tmp = (t_0 * (w0 * (math.fabs(t_1) * math.sqrt((-0.25 * (h / l)))))) / math.fabs(d)
	else:
		tmp = w0 * math.sqrt(1.0)
	return tmp

Julia

function code(w0, M, D, h, l, d)
	t_0 = fmax(abs(M), D)
	t_1 = fmin(abs(M), D)
	tmp = 0.0
	if (Float64((Float64(Float64(t_1 * t_0) / Float64(2.0 * abs(d))) ^ 2.0) * Float64(h / l)) <= -2000.0)
		tmp = Float64(Float64(t_0 * Float64(w0 * Float64(abs(t_1) * sqrt(Float64(-0.25 * Float64(h / l)))))) / abs(d));
	else
		tmp = Float64(w0 * sqrt(1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = max(abs(M), D);
	t_1 = min(abs(M), D);
	tmp = 0.0;
	if (((((t_1 * t_0) / (2.0 * abs(d))) ^ 2.0) * (h / l)) <= -2000.0)
		tmp = (t_0 * (w0 * (abs(t_1) * sqrt((-0.25 * (h / l)))))) / abs(d);
	else
		tmp = w0 * sqrt(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[Max[N[Abs[M], $MachinePrecision], D], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[M], $MachinePrecision], D], $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(N[(t$95$1 * t$95$0), $MachinePrecision] / N[(2.0 * N[Abs[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2000.0], N[(N[(t$95$0 * N[(w0 * N[(N[Abs[t$95$1], $MachinePrecision] * N[Sqrt[N[(-0.25 * N[(h / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[d], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[1.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2e3

   1. Initial program 80.1%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   2. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{w0 \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right)}}{d}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{w0 \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right)}}{\color{blue}{d}} \]

   4. Applied rewrites 9.3%

      \[\leadsto \color{blue}{\frac{w0 \cdot \sqrt{-0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}}{d}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{w0 \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}}{\color{blue}{d}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{w0 \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}}{d} \]

      3. *-commutative N/A

         \[\leadsto \frac{\sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}} \cdot w0}{d} \]

      4. associate-/l* N/A

         \[\leadsto \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}} \cdot \color{blue}{\frac{w0}{d}} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}} \cdot \color{blue}{\frac{w0}{d}} \]

   6. Applied rewrites 8.9%

      \[\leadsto \sqrt{\left(-0.25 \cdot \left(D \cdot D\right)\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}} \cdot \color{blue}{\frac{w0}{d}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}} \cdot \frac{\color{blue}{w0}}{d} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}} \cdot \frac{w0}{d} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\frac{\left(M \cdot M\right) \cdot h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)} \cdot \frac{w0}{d} \]

      4. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{\left(M \cdot M\right) \cdot h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)} \cdot \frac{w0}{d} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(M \cdot M\right) \cdot h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)} \cdot \frac{w0}{d} \]

      6. associate-/l* N/A

         \[\leadsto \sqrt{\left(\left(M \cdot M\right) \cdot \frac{h}{\ell}\right) \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)} \cdot \frac{w0}{d} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\left(M \cdot M\right) \cdot \left(\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)\right)} \cdot \frac{w0}{d} \]

      8. sqrt-prod N/A

         \[\leadsto \left(\sqrt{M \cdot M} \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{\color{blue}{w0}}{d} \]

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

         \[\leadsto \left(\sqrt{M \cdot M} \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{M \cdot M} \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{M \cdot M} \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

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

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

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

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{\color{blue}{w0}}{d} \]

      14. lower-fabs.f64 N/A

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

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

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

      16. lower-*.f64 N/A

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

      17. lower-/.f64 10.6%

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(-0.25 \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

      18. lift-*.f64 N/A

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot D\right)\right)}\right) \cdot \frac{w0}{d} \]

      19. *-commutative N/A

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\left(D \cdot D\right) \cdot \frac{-1}{4}\right)}\right) \cdot \frac{w0}{d} \]

      20. lower-*.f64 10.6%

          \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\left(D \cdot D\right) \cdot -0.25\right)}\right) \cdot \frac{w0}{d} \]

   8. Applied rewrites 10.6%

      \[\leadsto \left(\left|M\right| \cdot \sqrt{\frac{h}{\ell} \cdot \left(\left(D \cdot D\right) \cdot -0.25\right)}\right) \cdot \frac{\color{blue}{w0}}{d} \]

   9. Taylor expanded in D around 0

      \[\leadsto \frac{D \cdot \left(w0 \cdot \left(\left|M\right| \cdot \sqrt{\frac{-1}{4} \cdot \frac{h}{\ell}}\right)\right)}{\color{blue}{d}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{D \cdot \left(w0 \cdot \left(\left|M\right| \cdot \sqrt{\frac{-1}{4} \cdot \frac{h}{\ell}}\right)\right)}{d} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{D \cdot \left(w0 \cdot \left(\left|M\right| \cdot \sqrt{\frac{-1}{4} \cdot \frac{h}{\ell}}\right)\right)}{d} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{D \cdot \left(w0 \cdot \left(\left|M\right| \cdot \sqrt{\frac{-1}{4} \cdot \frac{h}{\ell}}\right)\right)}{d} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{D \cdot \left(w0 \cdot \left(\left|M\right| \cdot \sqrt{\frac{-1}{4} \cdot \frac{h}{\ell}}\right)\right)}{d} \]

       5. lower-fabs.f64 N/A

          \[\leadsto \frac{D \cdot \left(w0 \cdot \left(\left|M\right| \cdot \sqrt{\frac{-1}{4} \cdot \frac{h}{\ell}}\right)\right)}{d} \]

       6. lower-sqrt.f64 N/A

          \[\leadsto \frac{D \cdot \left(w0 \cdot \left(\left|M\right| \cdot \sqrt{\frac{-1}{4} \cdot \frac{h}{\ell}}\right)\right)}{d} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{D \cdot \left(w0 \cdot \left(\left|M\right| \cdot \sqrt{\frac{-1}{4} \cdot \frac{h}{\ell}}\right)\right)}{d} \]

       8. lower-/.f64 12.8%

          \[\leadsto \frac{D \cdot \left(w0 \cdot \left(\left|M\right| \cdot \sqrt{-0.25 \cdot \frac{h}{\ell}}\right)\right)}{d} \]

   11. Applied rewrites 12.8%

       \[\leadsto \frac{D \cdot \left(w0 \cdot \left(\left|M\right| \cdot \sqrt{-0.25 \cdot \frac{h}{\ell}}\right)\right)}{\color{blue}{d}} \]

   if -2e3 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 80.1%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   2. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \sqrt{\color{blue}{1}} \]
   3. Step-by-step derivation

   4. Applied rewrites 68.8%

      \[\leadsto w0 \cdot \sqrt{\color{blue}{1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 68.8% accurate, 7.0× speedup

Math

\[w0 \cdot \sqrt{1} \]

FPCore

(FPCore (w0 M D h l d)
  :precision binary64
  (* w0 (sqrt 1.0)))

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt(1.0);
}

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(w0, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * sqrt(1.0d0)
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * Math.sqrt(1.0);
}

Python

def code(w0, M, D, h, l, d):
	return w0 * math.sqrt(1.0)

Julia

function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(1.0))
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt(1.0);
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[1.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.1%

   \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

2. Taylor expanded in M around 0

   \[\leadsto w0 \cdot \sqrt{\color{blue}{1}} \]
3. Step-by-step derivation

4. Applied rewrites 68.8%

   \[\leadsto w0 \cdot \sqrt{\color{blue}{1}} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025213 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  :precision binary64
  (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))