Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.4% → 88.2%

Time: 7.8s

Alternatives: 14

Speedup: 0.7×

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: 80.4% 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: 88.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_1 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_2 := \frac{t\_1 \cdot t\_0}{d}\\ \mathsf{copysign}\left(1, w0\right) \cdot \begin{array}{l} \mathbf{if}\;\left|w0\right| \cdot \sqrt{1 - {\left(\frac{t\_0 \cdot t\_1}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq 10^{+164}:\\ \;\;\;\;\left|w0\right| \cdot \sqrt{\mathsf{fma}\left(\frac{h}{\ell}, \left(t\_2 \cdot t\_2\right) \cdot -0.25, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|w0\right| \cdot \sqrt{\mathsf{fma}\left(\frac{t\_0}{d + d} \cdot t\_1, \frac{\frac{t\_1}{d + d} \cdot \left(t\_0 \cdot h\right)}{-\ell}, 1\right)}\\ \end{array} \end{array} \]

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 t_0) d)))
  (*
   (copysign 1.0 w0)
   (if (<=
        (*
         (fabs w0)
         (sqrt
          (- 1.0 (* (pow (/ (* t_0 t_1) (* 2.0 d)) 2.0) (/ h l)))))
        1e+164)
     (* (fabs w0) (sqrt (fma (/ h l) (* (* t_2 t_2) -0.25) 1.0)))
     (*
      (fabs w0)
      (sqrt
       (fma
        (* (/ t_0 (+ d d)) t_1)
        (/ (* (/ t_1 (+ d d)) (* t_0 h)) (- 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 * t_0) / d;
	double tmp;
	if ((fabs(w0) * sqrt((1.0 - (pow(((t_0 * t_1) / (2.0 * d)), 2.0) * (h / l))))) <= 1e+164) {
		tmp = fabs(w0) * sqrt(fma((h / l), ((t_2 * t_2) * -0.25), 1.0));
	} else {
		tmp = fabs(w0) * sqrt(fma(((t_0 / (d + d)) * t_1), (((t_1 / (d + d)) * (t_0 * h)) / -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 * t_0) / d)
	tmp = 0.0
	if (Float64(abs(w0) * sqrt(Float64(1.0 - Float64((Float64(Float64(t_0 * t_1) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))))) <= 1e+164)
		tmp = Float64(abs(w0) * sqrt(fma(Float64(h / l), Float64(Float64(t_2 * t_2) * -0.25), 1.0)));
	else
		tmp = Float64(abs(w0) * sqrt(fma(Float64(Float64(t_0 / Float64(d + d)) * t_1), Float64(Float64(Float64(t_1 / Float64(d + d)) * Float64(t_0 * h)) / Float64(-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 * t$95$0), $MachinePrecision] / d), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[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 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e+164], N[(N[Abs[w0], $MachinePrecision] * N[Sqrt[N[(N[(h / l), $MachinePrecision] * N[(N[(t$95$2 * t$95$2), $MachinePrecision] * -0.25), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Abs[w0], $MachinePrecision] * N[Sqrt[N[(N[(N[(t$95$0 / N[(d + d), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[(N[(t$95$1 / N[(d + d), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * h), $MachinePrecision]), $MachinePrecision] / (-l)), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]]

Derivation

1. Split input into 2 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))))) < 1e164

   1. Initial program 80.4%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. swap-sqr N/A

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

   3. Applied rewrites 64.9%

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

   5. Applied rewrites 65.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 80.4%

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

   7. Applied rewrites 80.4%

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

   if 1e164 < (*.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 80.4%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 84.2%

      \[\leadsto w0 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{M}{d + d} \cdot D, \frac{\frac{D}{d + d} \cdot \left(M \cdot h\right)}{-\ell}, 1\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 87.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(M, \left|D\right|\right)\\ t_1 := \mathsf{max}\left(M, \left|D\right|\right)\\ t_2 := \frac{t\_1 \cdot t\_0}{d}\\ \mathsf{copysign}\left(1, w0\right) \cdot \begin{array}{l} \mathbf{if}\;\left|w0\right| \cdot \sqrt{1 - {\left(\frac{t\_0 \cdot t\_1}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq 2 \cdot 10^{+226}:\\ \;\;\;\;\left|w0\right| \cdot \sqrt{\mathsf{fma}\left(\frac{h}{\ell}, \left(t\_2 \cdot t\_2\right) \cdot -0.25, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|w0\right| \cdot \sqrt{\mathsf{fma}\left(\frac{t\_0}{d + d} \cdot t\_1, \left(h \cdot t\_0\right) \cdot \frac{\frac{t\_1}{-2 \cdot d}}{\ell}, 1\right)}\\ \end{array} \end{array} \]

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_1 t_0) d)))
  (*
   (copysign 1.0 w0)
   (if (<=
        (*
         (fabs w0)
         (sqrt
          (- 1.0 (* (pow (/ (* t_0 t_1) (* 2.0 d)) 2.0) (/ h l)))))
        2e+226)
     (* (fabs w0) (sqrt (fma (/ h l) (* (* t_2 t_2) -0.25) 1.0)))
     (*
      (fabs w0)
      (sqrt
       (fma
        (* (/ t_0 (+ d d)) t_1)
        (* (* h t_0) (/ (/ t_1 (* -2.0 d)) l))
        1.0)))))))

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_1 * t_0) / d;
	double tmp;
	if ((fabs(w0) * sqrt((1.0 - (pow(((t_0 * t_1) / (2.0 * d)), 2.0) * (h / l))))) <= 2e+226) {
		tmp = fabs(w0) * sqrt(fma((h / l), ((t_2 * t_2) * -0.25), 1.0));
	} else {
		tmp = fabs(w0) * sqrt(fma(((t_0 / (d + d)) * t_1), ((h * t_0) * ((t_1 / (-2.0 * d)) / l)), 1.0));
	}
	return 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(Float64(t_1 * t_0) / d)
	tmp = 0.0
	if (Float64(abs(w0) * sqrt(Float64(1.0 - Float64((Float64(Float64(t_0 * t_1) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))))) <= 2e+226)
		tmp = Float64(abs(w0) * sqrt(fma(Float64(h / l), Float64(Float64(t_2 * t_2) * -0.25), 1.0)));
	else
		tmp = Float64(abs(w0) * sqrt(fma(Float64(Float64(t_0 / Float64(d + d)) * t_1), Float64(Float64(h * t_0) * Float64(Float64(t_1 / Float64(-2.0 * 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[M, N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Max[M, N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * t$95$0), $MachinePrecision] / d), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[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 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e+226], N[(N[Abs[w0], $MachinePrecision] * N[Sqrt[N[(N[(h / l), $MachinePrecision] * N[(N[(t$95$2 * t$95$2), $MachinePrecision] * -0.25), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Abs[w0], $MachinePrecision] * N[Sqrt[N[(N[(N[(t$95$0 / N[(d + d), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[(h * t$95$0), $MachinePrecision] * N[(N[(t$95$1 / N[(-2.0 * d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]]

Derivation

1. Split input into 2 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))))) < 1.9999999999999999e226

   1. Initial program 80.4%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. swap-sqr N/A

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

   3. Applied rewrites 64.9%

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

   5. Applied rewrites 65.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 80.4%

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

   7. Applied rewrites 80.4%

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

   if 1.9999999999999999e226 < (*.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 80.4%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 84.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-2neg N/A

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

      11. distribute-frac-neg2 N/A

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

      12. lift-neg.f64 N/A

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

      13. frac-2neg-rev N/A

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

      14. lower-/.f64 N/A

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

   5. Applied rewrites 82.8%

      \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M}{d + d} \cdot D, \color{blue}{\left(h \cdot M\right) \cdot \frac{\frac{D}{-2 \cdot d}}{\ell}}, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 87.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(M, \left|D\right|\right)\\ t_1 := \mathsf{max}\left(M, \left|D\right|\right)\\ t_2 := {\left(\frac{t\_0 \cdot t\_1}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\\ t_3 := t\_1 \cdot t\_0\\ t_4 := \frac{t\_3}{d}\\ t_5 := t\_3 \cdot t\_1\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(\left(t\_5 \cdot t\_0\right) \cdot -0.25, \frac{h}{\ell}, d \cdot d\right)} \cdot w0}{\left|d\right|}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-32}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{h}{\ell}, \left(t\_4 \cdot t\_4\right) \cdot -0.25, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(h, \frac{\left(-0.25 \cdot t\_5\right) \cdot \frac{t\_0}{d \cdot d}}{\ell}, 1\right)}\\ \end{array} \]

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 (* (pow (/ (* t_0 t_1) (* 2.0 d)) 2.0) (/ h l)))
       (t_3 (* t_1 t_0))
       (t_4 (/ t_3 d))
       (t_5 (* t_3 t_1)))
  (if (<= t_2 (- INFINITY))
    (/
     (* (sqrt (fma (* (* t_5 t_0) -0.25) (/ h l) (* d d))) w0)
     (fabs d))
    (if (<= t_2 5e-32)
      (* w0 (sqrt (fma (/ h l) (* (* t_4 t_4) -0.25) 1.0)))
      (*
       w0
       (sqrt (fma h (/ (* (* -0.25 t_5) (/ t_0 (* d d))) l) 1.0)))))))

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 = pow(((t_0 * t_1) / (2.0 * d)), 2.0) * (h / l);
	double t_3 = t_1 * t_0;
	double t_4 = t_3 / d;
	double t_5 = t_3 * t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (sqrt(fma(((t_5 * t_0) * -0.25), (h / l), (d * d))) * w0) / fabs(d);
	} else if (t_2 <= 5e-32) {
		tmp = w0 * sqrt(fma((h / l), ((t_4 * t_4) * -0.25), 1.0));
	} else {
		tmp = w0 * sqrt(fma(h, (((-0.25 * t_5) * (t_0 / (d * d))) / l), 1.0));
	}
	return 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((Float64(Float64(t_0 * t_1) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))
	t_3 = Float64(t_1 * t_0)
	t_4 = Float64(t_3 / d)
	t_5 = Float64(t_3 * t_1)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(fma(Float64(Float64(t_5 * t_0) * -0.25), Float64(h / l), Float64(d * d))) * w0) / abs(d));
	elseif (t_2 <= 5e-32)
		tmp = Float64(w0 * sqrt(fma(Float64(h / l), Float64(Float64(t_4 * t_4) * -0.25), 1.0)));
	else
		tmp = Float64(w0 * sqrt(fma(h, Float64(Float64(Float64(-0.25 * t_5) * Float64(t_0 / Float64(d * d))) / l), 1.0)));
	end
	return 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[(N[Power[N[(N[(t$95$0 * t$95$1), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 / d), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$3 * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[Sqrt[N[(N[(N[(t$95$5 * t$95$0), $MachinePrecision] * -0.25), $MachinePrecision] * N[(h / l), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision] / N[Abs[d], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-32], N[(w0 * N[Sqrt[N[(N[(h / l), $MachinePrecision] * N[(N[(t$95$4 * t$95$4), $MachinePrecision] * -0.25), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(h * N[(N[(N[(-0.25 * t$95$5), $MachinePrecision] * N[(t$95$0 / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision]], $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)) < -inf.0

   1. Initial program 80.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 80.4%

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

   3. Applied rewrites 56.3%

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

   5. Applied rewrites 47.2%

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

   if -inf.0 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < 5.0000000000000004e-32

   1. Initial program 80.4%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. swap-sqr N/A

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

   3. Applied rewrites 64.9%

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

   5. Applied rewrites 65.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 80.4%

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

   7. Applied rewrites 80.4%

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

   if 5.0000000000000004e-32 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 80.4%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. swap-sqr N/A

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

   3. Applied rewrites 64.9%

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

   5. Applied rewrites 65.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 71.0%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 71.0%

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

   7. Applied rewrites 71.0%

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

Alternative 4: 87.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_1 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_2 := t\_1 \cdot t\_0\\ t_3 := t\_2 \cdot t\_1\\ t_4 := \frac{t\_2}{d}\\ t_5 := \left|w0\right| \cdot \sqrt{1 - {\left(\frac{t\_0 \cdot t\_1}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathsf{copysign}\left(1, w0\right) \cdot \begin{array}{l} \mathbf{if}\;t\_5 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\left|w0\right| \cdot \sqrt{\mathsf{fma}\left(\frac{h}{\ell}, \left(t\_4 \cdot t\_4\right) \cdot -0.25, 1\right)}\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(\left(t\_3 \cdot t\_0\right) \cdot -0.25, \frac{h}{\ell}, d \cdot d\right)}}{\left|d\right|} \cdot \left|w0\right|\\ \mathbf{else}:\\ \;\;\;\;\left|w0\right| \cdot \sqrt{\mathsf{fma}\left(h, \frac{\left(-0.25 \cdot t\_3\right) \cdot \frac{t\_0}{d \cdot d}}{\ell}, 1\right)}\\ \end{array} \end{array} \]

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 t_0))
       (t_3 (* t_2 t_1))
       (t_4 (/ t_2 d))
       (t_5
        (*
         (fabs w0)
         (sqrt
          (- 1.0 (* (pow (/ (* t_0 t_1) (* 2.0 d)) 2.0) (/ h l)))))))
  (*
   (copysign 1.0 w0)
   (if (<= t_5 2e+306)
     (* (fabs w0) (sqrt (fma (/ h l) (* (* t_4 t_4) -0.25) 1.0)))
     (if (<= t_5 INFINITY)
       (*
        (/
         (sqrt (fma (* (* t_3 t_0) -0.25) (/ h l) (* d d)))
         (fabs d))
        (fabs w0))
       (*
        (fabs w0)
        (sqrt
         (fma h (/ (* (* -0.25 t_3) (/ t_0 (* d 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 * t_0;
	double t_3 = t_2 * t_1;
	double t_4 = t_2 / d;
	double t_5 = fabs(w0) * sqrt((1.0 - (pow(((t_0 * t_1) / (2.0 * d)), 2.0) * (h / l))));
	double tmp;
	if (t_5 <= 2e+306) {
		tmp = fabs(w0) * sqrt(fma((h / l), ((t_4 * t_4) * -0.25), 1.0));
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = (sqrt(fma(((t_3 * t_0) * -0.25), (h / l), (d * d))) / fabs(d)) * fabs(w0);
	} else {
		tmp = fabs(w0) * sqrt(fma(h, (((-0.25 * t_3) * (t_0 / (d * 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(t_1 * t_0)
	t_3 = Float64(t_2 * t_1)
	t_4 = Float64(t_2 / d)
	t_5 = Float64(abs(w0) * sqrt(Float64(1.0 - Float64((Float64(Float64(t_0 * t_1) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))))
	tmp = 0.0
	if (t_5 <= 2e+306)
		tmp = Float64(abs(w0) * sqrt(fma(Float64(h / l), Float64(Float64(t_4 * t_4) * -0.25), 1.0)));
	elseif (t_5 <= Inf)
		tmp = Float64(Float64(sqrt(fma(Float64(Float64(t_3 * t_0) * -0.25), Float64(h / l), Float64(d * d))) / abs(d)) * abs(w0));
	else
		tmp = Float64(abs(w0) * sqrt(fma(h, Float64(Float64(Float64(-0.25 * t_3) * Float64(t_0 / Float64(d * 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[(t$95$1 * t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 / d), $MachinePrecision]}, Block[{t$95$5 = 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 * d), $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$5, 2e+306], N[(N[Abs[w0], $MachinePrecision] * N[Sqrt[N[(N[(h / l), $MachinePrecision] * N[(N[(t$95$4 * t$95$4), $MachinePrecision] * -0.25), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, Infinity], N[(N[(N[Sqrt[N[(N[(N[(t$95$3 * t$95$0), $MachinePrecision] * -0.25), $MachinePrecision] * N[(h / l), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[d], $MachinePrecision]), $MachinePrecision] * N[Abs[w0], $MachinePrecision]), $MachinePrecision], N[(N[Abs[w0], $MachinePrecision] * N[Sqrt[N[(h * N[(N[(N[(-0.25 * t$95$3), $MachinePrecision] * N[(t$95$0 / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $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))))) < 2e306

   1. Initial program 80.4%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. swap-sqr N/A

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

   3. Applied rewrites 64.9%

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

   5. Applied rewrites 65.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 80.4%

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

   7. Applied rewrites 80.4%

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

   if 2e306 < (*.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 80.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 80.4%

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

   3. Applied rewrites 56.3%

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

   5. Applied rewrites 50.5%

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

   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 80.4%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. swap-sqr N/A

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

   3. Applied rewrites 64.9%

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

   5. Applied rewrites 65.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 71.0%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 71.0%

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

   7. Applied rewrites 71.0%

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

Alternative 5: 86.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_1 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_2 := t\_0 \cdot t\_1\\ t_3 := \frac{t\_0}{d} \cdot t\_1\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-80}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{h \cdot t\_2}{\ell \cdot \left(d + d\right)}, \frac{-0.5}{d} \cdot t\_2, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot t\_3\right) \cdot \left(t\_3 \cdot 0.25\right)}\\ \end{array} \]

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_0 t_1))
       (t_3 (* (/ t_0 d) t_1)))
  (if (<= t_1 5e-80)
    (*
     w0
     (sqrt (fma (/ (* h t_2) (* l (+ d d))) (* (/ -0.5 d) t_2) 1.0)))
    (* w0 (sqrt (- 1.0 (* (* (/ h l) t_3) (* t_3 0.25))))))))

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_0 * t_1;
	double t_3 = (t_0 / d) * t_1;
	double tmp;
	if (t_1 <= 5e-80) {
		tmp = w0 * sqrt(fma(((h * t_2) / (l * (d + d))), ((-0.5 / d) * t_2), 1.0));
	} else {
		tmp = w0 * sqrt((1.0 - (((h / l) * t_3) * (t_3 * 0.25))));
	}
	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_0 * t_1)
	t_3 = Float64(Float64(t_0 / d) * t_1)
	tmp = 0.0
	if (t_1 <= 5e-80)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(h * t_2) / Float64(l * Float64(d + d))), Float64(Float64(-0.5 / d) * t_2), 1.0)));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(h / l) * t_3) * Float64(t_3 * 0.25)))));
	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[(t$95$0 * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$0 / d), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-80], N[(w0 * N[Sqrt[N[(N[(N[(h * t$95$2), $MachinePrecision] / N[(l * N[(d + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.5 / d), $MachinePrecision] * t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(h / l), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(t$95$3 * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if M < 5e-80

   1. Initial program 80.4%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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-rgt-neg-in N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   if 5e-80 < M

   1. Initial program 80.4%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. swap-sqr N/A

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

   3. Applied rewrites 64.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l/ N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. associate-/l* N/A

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

      17. associate-*r* N/A

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

      18. lift-*.f64 N/A

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

      19. associate-/l* N/A

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

   5. Applied rewrites 82.7%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{h}{\ell} \cdot \left(\frac{D}{d} \cdot M\right)\right) \cdot \left(\left(\frac{D}{d} \cdot M\right) \cdot 0.25\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 85.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_1 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ w0 \cdot \sqrt{\mathsf{fma}\left(\frac{t\_1}{d + d} \cdot t\_0, \frac{\frac{\left(h \cdot t\_1\right) \cdot t\_0}{d} \cdot -0.5}{\ell}, 1\right)} \end{array} \]

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))))
  (*
   w0
   (sqrt
    (fma
     (* (/ t_1 (+ d d)) t_0)
     (/ (* (/ (* (* h t_1) t_0) d) -0.5) 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));
	return w0 * sqrt(fma(((t_1 / (d + d)) * t_0), (((((h * t_1) * t_0) / d) * -0.5) / l), 1.0));
}

Julia

function code(w0, M, D, h, l, d)
	t_0 = fmax(abs(M), abs(D))
	t_1 = fmin(abs(M), abs(D))
	return Float64(w0 * sqrt(fma(Float64(Float64(t_1 / Float64(d + d)) * t_0), Float64(Float64(Float64(Float64(Float64(h * t_1) * t_0) / d) * -0.5) / l), 1.0)))
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]}, N[(w0 * N[Sqrt[N[(N[(N[(t$95$1 / N[(d + d), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[(N[(N[(N[(h * t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision] / d), $MachinePrecision] * -0.5), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 80.4%

   \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
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 84.2%

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

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 81.2%

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

6. Applied rewrites 81.2%

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

   1. lift-*.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. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/r* N/A

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

   6. associate-*l/ N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 84.1%

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 84.1%

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 84.1%

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

8. Applied rewrites 84.1%

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

Alternative 7: 85.4% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (w0 M D h l d)
  :precision binary64
  (let* ((t_0 (/ (* D M) d)))
  (if (<=
       (*
        w0
        (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)))))
       INFINITY)
    (* w0 (sqrt (fma (/ h l) (* (* t_0 t_0) -0.25) 1.0)))
    (*
     w0
     (sqrt
      (fma h (/ (* (* -0.25 (* (* D M) D)) (/ M (* d d))) l) 1.0))))))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 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))))) < +inf.0

   1. Initial program 80.4%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. swap-sqr N/A

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

   3. Applied rewrites 64.9%

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

   5. Applied rewrites 65.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 80.4%

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

   7. Applied rewrites 80.4%

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

   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 80.4%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. swap-sqr N/A

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

   3. Applied rewrites 64.9%

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

   5. Applied rewrites 65.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 71.0%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 71.0%

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

   7. Applied rewrites 71.0%

      \[\leadsto w0 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(h, \frac{\left(-0.25 \cdot \left(\left(D \cdot M\right) \cdot D\right)\right) \cdot \frac{M}{d \cdot d}}{\ell}, 1\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 83.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_1 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ \mathbf{if}\;{\left(\frac{t\_1 \cdot t\_0}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+54}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(\left(\left(\left(t\_0 \cdot t\_1\right) \cdot t\_0\right) \cdot t\_1\right) \cdot \frac{0.25}{d}\right) \cdot h}{d \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \end{array} \]

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))))
  (if (<= (* (pow (/ (* t_1 t_0) (* 2.0 d)) 2.0) (/ h l)) -2e+54)
    (*
     w0
     (sqrt
      (-
       1.0
       (/ (* (* (* (* (* t_0 t_1) t_0) t_1) (/ 0.25 d)) h) (* 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 tmp;
	if ((pow(((t_1 * t_0) / (2.0 * d)), 2.0) * (h / l)) <= -2e+54) {
		tmp = w0 * sqrt((1.0 - ((((((t_0 * t_1) * t_0) * t_1) * (0.25 / d)) * h) / (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) :: tmp
    t_0 = fmax(abs(m), abs(d))
    t_1 = fmin(abs(m), abs(d))
    if (((((t_1 * t_0) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-2d+54)) then
        tmp = w0 * sqrt((1.0d0 - ((((((t_0 * t_1) * t_0) * t_1) * (0.25d0 / d_1)) * h) / (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 tmp;
	if ((Math.pow(((t_1 * t_0) / (2.0 * d)), 2.0) * (h / l)) <= -2e+54) {
		tmp = w0 * Math.sqrt((1.0 - ((((((t_0 * t_1) * t_0) * t_1) * (0.25 / d)) * h) / (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))
	tmp = 0
	if (math.pow(((t_1 * t_0) / (2.0 * d)), 2.0) * (h / l)) <= -2e+54:
		tmp = w0 * math.sqrt((1.0 - ((((((t_0 * t_1) * t_0) * t_1) * (0.25 / d)) * h) / (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))
	tmp = 0.0
	if (Float64((Float64(Float64(t_1 * t_0) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e+54)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(t_0 * t_1) * t_0) * t_1) * Float64(0.25 / d)) * h) / Float64(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));
	tmp = 0.0;
	if (((((t_1 * t_0) / (2.0 * d)) ^ 2.0) * (h / l)) <= -2e+54)
		tmp = w0 * sqrt((1.0 - ((((((t_0 * t_1) * t_0) * t_1) * (0.25 / d)) * h) / (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]}, If[LessEqual[N[(N[Power[N[(N[(t$95$1 * t$95$0), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+54], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(N[(N[(t$95$0 * t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(0.25 / d), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $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)) < -2.0000000000000002e54

   1. Initial program 80.4%

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

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      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. lift-*.f64 N/A

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

      7. unswap-sqr N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 77.2%

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

   5. Applied rewrites 77.2%

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

   if -2.0000000000000002e54 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 80.4%

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

   2. Taylor expanded in M around 0

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

   4. Applied rewrites 67.5%

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

Alternative 9: 83.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -0.04:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{0.25}{d \cdot d} \cdot \left(D \cdot M\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \end{array} \]

FPCore

(FPCore (w0 M D h l d)
  :precision binary64
  (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -0.04)
  (*
   w0
   (sqrt
    (- 1.0 (* (* (/ h l) (* D M)) (* (/ 0.25 (* d d)) (* D M))))))
  (* w0 (sqrt 1.0))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -0.04) {
		tmp = w0 * sqrt((1.0 - (((h / l) * (D * M)) * ((0.25 / (d * d)) * (D * M)))));
	} 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) :: tmp
    if (((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-0.04d0)) then
        tmp = w0 * sqrt((1.0d0 - (((h / l) * (d * m)) * ((0.25d0 / (d_1 * d_1)) * (d * m)))))
    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 tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -0.04) {
		tmp = w0 * Math.sqrt((1.0 - (((h / l) * (D * M)) * ((0.25 / (d * d)) * (D * M)))));
	} else {
		tmp = w0 * Math.sqrt(1.0);
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -0.04:
		tmp = w0 * math.sqrt((1.0 - (((h / l) * (D * M)) * ((0.25 / (d * d)) * (D * M)))))
	else:
		tmp = w0 * math.sqrt(1.0)
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -0.04)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(h / l) * Float64(D * M)) * Float64(Float64(0.25 / Float64(d * d)) * Float64(D * M))))));
	else
		tmp = Float64(w0 * sqrt(1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -0.04)
		tmp = w0 * sqrt((1.0 - (((h / l) * (D * M)) * ((0.25 / (d * d)) * (D * M)))));
	else
		tmp = w0 * sqrt(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -0.04], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(h / l), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision] * N[(N[(0.25 / N[(d * d), $MachinePrecision]), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $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)) < -0.040000000000000001

   1. Initial program 80.4%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. mult-flip N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 71.4%

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

   if -0.040000000000000001 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 80.4%

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

   2. Taylor expanded in M around 0

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

   4. Applied rewrites 67.5%

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

Alternative 10: 82.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -0.04:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{h}{\ell}, \left(\left(\frac{M}{d \cdot d} \cdot \left(D \cdot M\right)\right) \cdot D\right) \cdot -0.25, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \end{array} \]

FPCore

(FPCore (w0 M D h l d)
  :precision binary64
  (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -0.04)
  (*
   w0
   (sqrt (fma (/ h l) (* (* (* (/ M (* d d)) (* D M)) D) -0.25) 1.0)))
  (* w0 (sqrt 1.0))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -0.04) {
		tmp = w0 * sqrt(fma((h / l), ((((M / (d * d)) * (D * M)) * D) * -0.25), 1.0));
	} else {
		tmp = w0 * sqrt(1.0);
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -0.04)
		tmp = Float64(w0 * sqrt(fma(Float64(h / l), Float64(Float64(Float64(Float64(M / Float64(d * d)) * Float64(D * M)) * D) * -0.25), 1.0)));
	else
		tmp = Float64(w0 * sqrt(1.0));
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -0.04], N[(w0 * N[Sqrt[N[(N[(h / l), $MachinePrecision] * N[(N[(N[(N[(M / N[(d * d), $MachinePrecision]), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision] * -0.25), $MachinePrecision] + 1.0), $MachinePrecision]], $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)) < -0.040000000000000001

   1. Initial program 80.4%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. swap-sqr N/A

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

   3. Applied rewrites 64.9%

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

   5. Applied rewrites 65.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 69.6%

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

   7. Applied rewrites 69.6%

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

   if -0.040000000000000001 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 80.4%

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

   2. Taylor expanded in M around 0

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

   4. Applied rewrites 67.5%

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

Alternative 11: 81.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_1 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ \mathbf{if}\;{\left(\frac{t\_1 \cdot t\_0}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+54}:\\ \;\;\;\;\sqrt{\frac{\left(\left(\left(\left(t\_0 \cdot t\_1\right) \cdot t\_0\right) \cdot t\_1\right) \cdot -0.25\right) \cdot h}{\left(\ell \cdot d\right) \cdot d} + 1} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \end{array} \]

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))))
  (if (<= (* (pow (/ (* t_1 t_0) (* 2.0 d)) 2.0) (/ h l)) -2e+54)
    (*
     (sqrt
      (+
       (/ (* (* (* (* (* t_0 t_1) t_0) t_1) -0.25) h) (* (* l d) d))
       1.0))
     w0)
    (* 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 tmp;
	if ((pow(((t_1 * t_0) / (2.0 * d)), 2.0) * (h / l)) <= -2e+54) {
		tmp = sqrt((((((((t_0 * t_1) * t_0) * t_1) * -0.25) * h) / ((l * d) * d)) + 1.0)) * w0;
	} 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), abs(d))
    t_1 = fmin(abs(m), abs(d))
    if (((((t_1 * t_0) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-2d+54)) then
        tmp = sqrt((((((((t_0 * t_1) * t_0) * t_1) * (-0.25d0)) * h) / ((l * d_1) * d_1)) + 1.0d0)) * w0
    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 tmp;
	if ((Math.pow(((t_1 * t_0) / (2.0 * d)), 2.0) * (h / l)) <= -2e+54) {
		tmp = Math.sqrt((((((((t_0 * t_1) * t_0) * t_1) * -0.25) * h) / ((l * d) * d)) + 1.0)) * w0;
	} 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))
	tmp = 0
	if (math.pow(((t_1 * t_0) / (2.0 * d)), 2.0) * (h / l)) <= -2e+54:
		tmp = math.sqrt((((((((t_0 * t_1) * t_0) * t_1) * -0.25) * h) / ((l * d) * d)) + 1.0)) * w0
	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))
	tmp = 0.0
	if (Float64((Float64(Float64(t_1 * t_0) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e+54)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(Float64(Float64(Float64(t_0 * t_1) * t_0) * t_1) * -0.25) * h) / Float64(Float64(l * d) * d)) + 1.0)) * w0);
	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));
	tmp = 0.0;
	if (((((t_1 * t_0) / (2.0 * d)) ^ 2.0) * (h / l)) <= -2e+54)
		tmp = sqrt((((((((t_0 * t_1) * t_0) * t_1) * -0.25) * h) / ((l * d) * d)) + 1.0)) * w0;
	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]}, If[LessEqual[N[(N[Power[N[(N[(t$95$1 * t$95$0), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+54], N[(N[Sqrt[N[(N[(N[(N[(N[(N[(N[(t$95$0 * t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision] * -0.25), $MachinePrecision] * h), $MachinePrecision] / N[(N[(l * d), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $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)) < -2.0000000000000002e54

   1. Initial program 80.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 80.4%

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

   3. Applied rewrites 56.3%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

   5. Applied rewrites 71.5%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

   7. Applied rewrites 71.5%

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

   if -2.0000000000000002e54 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 80.4%

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

   2. Taylor expanded in M around 0

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

   4. Applied rewrites 67.5%

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

Alternative 12: 81.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(M, \left|D\right|\right)\\ t_1 := \mathsf{min}\left(M, \left|D\right|\right)\\ \mathbf{if}\;{\left(\frac{t\_1 \cdot t\_0}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+17}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\left(t\_0 \cdot t\_1\right) \cdot t\_0, \frac{\left(-0.25 \cdot t\_1\right) \cdot h}{\left(d \cdot d\right) \cdot \ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \end{array} \]

FPCore

(FPCore (w0 M D h l d)
  :precision binary64
  (let* ((t_0 (fmax M (fabs D))) (t_1 (fmin M (fabs D))))
  (if (<= (* (pow (/ (* t_1 t_0) (* 2.0 d)) 2.0) (/ h l)) -1e+17)
    (*
     w0
     (sqrt
      (fma
       (* (* t_0 t_1) t_0)
       (/ (* (* -0.25 t_1) h) (* (* d d) l))
       1.0)))
    (* w0 (sqrt 1.0)))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = fmax(M, fabs(D));
	double t_1 = fmin(M, fabs(D));
	double tmp;
	if ((pow(((t_1 * t_0) / (2.0 * d)), 2.0) * (h / l)) <= -1e+17) {
		tmp = w0 * sqrt(fma(((t_0 * t_1) * t_0), (((-0.25 * t_1) * h) / ((d * d) * l)), 1.0));
	} else {
		tmp = w0 * sqrt(1.0);
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	t_0 = fmax(M, abs(D))
	t_1 = fmin(M, abs(D))
	tmp = 0.0
	if (Float64((Float64(Float64(t_1 * t_0) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -1e+17)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(t_0 * t_1) * t_0), Float64(Float64(Float64(-0.25 * t_1) * h) / Float64(Float64(d * d) * l)), 1.0)));
	else
		tmp = Float64(w0 * sqrt(1.0));
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[Max[M, N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Min[M, N[Abs[D], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(N[(t$95$1 * t$95$0), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -1e+17], N[(w0 * N[Sqrt[N[(N[(N[(t$95$0 * t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[(N[(-0.25 * t$95$1), $MachinePrecision] * h), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $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)) < -1e17

   1. Initial program 80.4%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. swap-sqr N/A

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

   3. Applied rewrites 64.9%

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

   5. Applied rewrites 65.9%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 67.3%

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

   7. Applied rewrites 67.3%

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

   if -1e17 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 80.4%

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

   2. Taylor expanded in M around 0

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

   4. Applied rewrites 67.5%

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

Alternative 13: 71.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot \left|d\right|}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+142}:\\ \;\;\;\;\frac{w0}{\left|\left|d\right|\right|} \cdot \left|d\right|\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \end{array} \]

FPCore

(FPCore (w0 M D h l d)
  :precision binary64
  (if (<= (* (pow (/ (* M D) (* 2.0 (fabs d))) 2.0) (/ h l)) -5e+142)
  (* (/ w0 (fabs (fabs d))) (fabs d))
  (* w0 (sqrt 1.0))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * fabs(d))), 2.0) * (h / l)) <= -5e+142) {
		tmp = (w0 / fabs(fabs(d))) * 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) :: tmp
    if (((((m * d) / (2.0d0 * abs(d_1))) ** 2.0d0) * (h / l)) <= (-5d+142)) then
        tmp = (w0 / abs(abs(d_1))) * 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 tmp;
	if ((Math.pow(((M * D) / (2.0 * Math.abs(d))), 2.0) * (h / l)) <= -5e+142) {
		tmp = (w0 / Math.abs(Math.abs(d))) * Math.abs(d);
	} else {
		tmp = w0 * Math.sqrt(1.0);
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * math.fabs(d))), 2.0) * (h / l)) <= -5e+142:
		tmp = (w0 / math.fabs(math.fabs(d))) * math.fabs(d)
	else:
		tmp = w0 * math.sqrt(1.0)
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * abs(d))) ^ 2.0) * Float64(h / l)) <= -5e+142)
		tmp = Float64(Float64(w0 / abs(abs(d))) * abs(d));
	else
		tmp = Float64(w0 * sqrt(1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * abs(d))) ^ 2.0) * (h / l)) <= -5e+142)
		tmp = (w0 / abs(abs(d))) * abs(d);
	else
		tmp = w0 * sqrt(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * N[Abs[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+142], N[(N[(w0 / N[Abs[N[Abs[d], $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)) < -5.0000000000000001e142

   1. Initial program 80.4%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. swap-sqr N/A

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

   3. Applied rewrites 64.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. sub-to-fraction N/A

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

      7. sqrt-div N/A

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

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

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

      9. lower-sqrt.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. lower-unsound-/.f64 N/A

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

   5. Applied rewrites 50.5%

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

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{d \cdot w0}{\left|d\right|}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{d \cdot w0}{\color{blue}{\left|d\right|}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{d \cdot w0}{\left|\color{blue}{d}\right|} \]

      3. lower-fabs.f64 27.4%

         \[\leadsto \frac{d \cdot w0}{\left|d\right|} \]

   8. Applied rewrites 27.4%

      \[\leadsto \color{blue}{\frac{d \cdot w0}{\left|d\right|}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{d \cdot w0}{\color{blue}{\left|d\right|}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{d \cdot w0}{\left|\color{blue}{d}\right|} \]

      3. associate-/l* N/A

         \[\leadsto d \cdot \color{blue}{\frac{w0}{\left|d\right|}} \]

      4. *-commutative N/A

         \[\leadsto \frac{w0}{\left|d\right|} \cdot \color{blue}{d} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{w0}{\left|d\right|} \cdot \color{blue}{d} \]

      6. lower-/.f64 29.9%

         \[\leadsto \frac{w0}{\left|d\right|} \cdot d \]

   10. Applied rewrites 29.9%

       \[\leadsto \frac{w0}{\left|d\right|} \cdot \color{blue}{d} \]

   if -5.0000000000000001e142 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 80.4%

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

   2. Taylor expanded in M around 0

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

   4. Applied rewrites 67.5%

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

Alternative 14: 67.5% 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 80.4%

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

2. Taylor expanded in M around 0

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

4. Applied rewrites 67.5%

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

Reproduce

herbie shell --seed 2025214 
(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))))))