Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.7% → 90.5%

Time: 6.8s

Alternatives: 6

Speedup: 0.6×

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: 80.7% 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: 90.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_1 := \left|d\right| + \left|d\right|\\ t_2 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_3 := \frac{t\_0 \cdot t\_2}{2 \cdot \left|d\right|}\\ t_4 := t\_2 \cdot t\_0\\ \mathbf{if}\;t\_3 \leq 10^{-20}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{t\_0}{t\_1} \cdot t\_2, \frac{\frac{t\_2}{t\_1} \cdot \left(t\_0 \cdot h\right)}{-\ell}, 1\right)}\\ \mathbf{elif}\;t\_3 \leq 10^{+213}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{h \cdot t\_4}{\ell \cdot t\_1}, \frac{-0.5}{\left|d\right|} \cdot t\_4, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2 \cdot \left(w0 \cdot \sqrt{-0.5 \cdot \frac{{t\_0}^{2} \cdot h}{\left|d\right| \cdot \ell}}\right)}{\sqrt{t\_1}}\\ \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (fmin (fabs M) (fabs D)))
        (t_1 (+ (fabs d) (fabs d)))
        (t_2 (fmax (fabs M) (fabs D)))
        (t_3 (/ (* t_0 t_2) (* 2.0 (fabs d))))
        (t_4 (* t_2 t_0)))
   (if (<= t_3 1e-20)
     (*
      w0
      (sqrt (fma (* (/ t_0 t_1) t_2) (/ (* (/ t_2 t_1) (* t_0 h)) (- l)) 1.0)))
     (if (<= t_3 1e+213)
       (*
        w0
        (sqrt (fma (/ (* h t_4) (* l t_1)) (* (/ -0.5 (fabs d)) t_4) 1.0)))
       (/
        (*
         t_2
         (* w0 (sqrt (- (* 0.5 (/ (* (pow t_0 2.0) h) (* (fabs d) l)))))))
        (sqrt t_1))))))

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 = fabs(d) + fabs(d);
	double t_2 = fmax(fabs(M), fabs(D));
	double t_3 = (t_0 * t_2) / (2.0 * fabs(d));
	double t_4 = t_2 * t_0;
	double tmp;
	if (t_3 <= 1e-20) {
		tmp = w0 * sqrt(fma(((t_0 / t_1) * t_2), (((t_2 / t_1) * (t_0 * h)) / -l), 1.0));
	} else if (t_3 <= 1e+213) {
		tmp = w0 * sqrt(fma(((h * t_4) / (l * t_1)), ((-0.5 / fabs(d)) * t_4), 1.0));
	} else {
		tmp = (t_2 * (w0 * sqrt(-(0.5 * ((pow(t_0, 2.0) * h) / (fabs(d) * l)))))) / sqrt(t_1);
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	t_0 = fmin(abs(M), abs(D))
	t_1 = Float64(abs(d) + abs(d))
	t_2 = fmax(abs(M), abs(D))
	t_3 = Float64(Float64(t_0 * t_2) / Float64(2.0 * abs(d)))
	t_4 = Float64(t_2 * t_0)
	tmp = 0.0
	if (t_3 <= 1e-20)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(t_0 / t_1) * t_2), Float64(Float64(Float64(t_2 / t_1) * Float64(t_0 * h)) / Float64(-l)), 1.0)));
	elseif (t_3 <= 1e+213)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(h * t_4) / Float64(l * t_1)), Float64(Float64(-0.5 / abs(d)) * t_4), 1.0)));
	else
		tmp = Float64(Float64(t_2 * Float64(w0 * sqrt(Float64(-Float64(0.5 * Float64(Float64((t_0 ^ 2.0) * h) / Float64(abs(d) * l))))))) / sqrt(t_1));
	end
	return 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[(N[Abs[d], $MachinePrecision] + N[Abs[d], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$0 * t$95$2), $MachinePrecision] / N[(2.0 * N[Abs[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$3, 1e-20], N[(w0 * N[Sqrt[N[(N[(N[(t$95$0 / t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(N[(N[(t$95$2 / t$95$1), $MachinePrecision] * N[(t$95$0 * h), $MachinePrecision]), $MachinePrecision] / (-l)), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e+213], N[(w0 * N[Sqrt[N[(N[(N[(h * t$95$4), $MachinePrecision] / N[(l * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.5 / N[Abs[d], $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 * N[(w0 * N[Sqrt[(-N[(0.5 * N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] * h), $MachinePrecision] / N[(N[Abs[d], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 9.9999999999999995e-21

   1. Initial program 80.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac2 N/A

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

      9. associate-/l* N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. associate-/l* N/A

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

      14. lower-fma.f64 N/A

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

   3. Applied rewrites 83.8%

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

   if 9.9999999999999995e-21 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 9.9999999999999998e212

   1. Initial program 80.7%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 84.0%

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

   if 9.9999999999999998e212 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 80.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac2 N/A

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

      9. associate-/l* N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. associate-/l* N/A

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

      14. lower-fma.f64 N/A

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

   3. Applied rewrites 83.8%

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

   5. Applied rewrites 41.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 38.4%

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

   8. Taylor expanded in D around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-*.f64 5.9%

         \[\leadsto \frac{D \cdot \left(w0 \cdot \sqrt{-0.5 \cdot \frac{{M}^{2} \cdot h}{d \cdot \ell}}\right)}{\sqrt{d + d}} \]

   10. Applied rewrites 5.9%

       \[\leadsto \frac{\color{blue}{D \cdot \left(w0 \cdot \sqrt{-0.5 \cdot \frac{{M}^{2} \cdot h}{d \cdot \ell}}\right)}}{\sqrt{d + d}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 89.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_1 := \left|d\right| + \left|d\right|\\ t_2 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ \mathbf{if}\;{\left(\frac{t\_2 \cdot t\_0}{2 \cdot \left|d\right|}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+47}:\\ \;\;\;\;w0 \cdot \left(t\_2 \cdot \left(-1 \cdot \left(t\_0 \cdot \left(-1 \cdot \frac{\sqrt{-0.25 \cdot \frac{h}{\ell}}}{\left|d\right|}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{t\_2}{t\_1} \cdot t\_0, \frac{\frac{t\_0}{t\_1} \cdot \left(t\_2 \cdot h\right)}{-\ell}, 1\right)}\\ \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (fmax (fabs M) (fabs D)))
        (t_1 (+ (fabs d) (fabs d)))
        (t_2 (fmin (fabs M) (fabs D))))
   (if (<= (* (pow (/ (* t_2 t_0) (* 2.0 (fabs d))) 2.0) (/ h l)) -5e+47)
     (*
      w0
      (*
       t_2
       (* -1.0 (* t_0 (* -1.0 (/ (sqrt (- (* 0.25 (/ h l)))) (fabs d)))))))
     (*
      w0
      (sqrt
       (fma (* (/ t_2 t_1) t_0) (/ (* (/ t_0 t_1) (* t_2 h)) (- 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 = fabs(d) + fabs(d);
	double t_2 = fmin(fabs(M), fabs(D));
	double tmp;
	if ((pow(((t_2 * t_0) / (2.0 * fabs(d))), 2.0) * (h / l)) <= -5e+47) {
		tmp = w0 * (t_2 * (-1.0 * (t_0 * (-1.0 * (sqrt(-(0.25 * (h / l))) / fabs(d))))));
	} else {
		tmp = w0 * sqrt(fma(((t_2 / t_1) * t_0), (((t_0 / t_1) * (t_2 * h)) / -l), 1.0));
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	t_0 = fmax(abs(M), abs(D))
	t_1 = Float64(abs(d) + abs(d))
	t_2 = fmin(abs(M), abs(D))
	tmp = 0.0
	if (Float64((Float64(Float64(t_2 * t_0) / Float64(2.0 * abs(d))) ^ 2.0) * Float64(h / l)) <= -5e+47)
		tmp = Float64(w0 * Float64(t_2 * Float64(-1.0 * Float64(t_0 * Float64(-1.0 * Float64(sqrt(Float64(-Float64(0.25 * Float64(h / l)))) / abs(d)))))));
	else
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(t_2 / t_1) * t_0), Float64(Float64(Float64(t_0 / t_1) * Float64(t_2 * h)) / Float64(-l)), 1.0)));
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[Max[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[d], $MachinePrecision] + N[Abs[d], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(N[(t$95$2 * t$95$0), $MachinePrecision] / N[(2.0 * N[Abs[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+47], N[(w0 * N[(t$95$2 * N[(-1.0 * N[(t$95$0 * N[(-1.0 * N[(N[Sqrt[(-N[(0.25 * N[(h / l), $MachinePrecision]), $MachinePrecision])], $MachinePrecision] / N[Abs[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(N[(t$95$2 / t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[(N[(t$95$0 / t$95$1), $MachinePrecision] * N[(t$95$2 * h), $MachinePrecision]), $MachinePrecision] / (-l)), $MachinePrecision] + 1.0), $MachinePrecision]], $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)) < -5.0000000000000002e47

   1. Initial program 80.7%

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

   2. Taylor expanded in M around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 8.8%

         \[\leadsto w0 \cdot \left(M \cdot \sqrt{-0.25 \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right) \]

   4. Applied rewrites 8.8%

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

   5. Taylor expanded in D around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 11.4%

         \[\leadsto w0 \cdot \left(M \cdot \left(-1 \cdot \left(D \cdot \sqrt{-0.25 \cdot \frac{h}{{d}^{2} \cdot \ell}}\right)\right)\right) \]

   7. Applied rewrites 11.4%

      \[\leadsto w0 \cdot \left(M \cdot \left(-1 \cdot \color{blue}{\left(D \cdot \sqrt{-0.25 \cdot \frac{h}{{d}^{2} \cdot \ell}}\right)}\right)\right) \]

   8. Taylor expanded in d around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 13.3%

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

   10. Applied rewrites 13.3%

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

   if -5.0000000000000002e47 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 80.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac2 N/A

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

      9. associate-/l* N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. associate-/l* N/A

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

      14. lower-fma.f64 N/A

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

   3. Applied rewrites 83.8%

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

Alternative 3: 88.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;{\left(\frac{\left|M\right| \cdot \left|D\right|}{2 \cdot \left|d\right|}\right)}^{2} \cdot \frac{h}{\ell} \leq -2000:\\ \;\;\;\;w0 \cdot \left(\left|M\right| \cdot \left(-1 \cdot \left(\left|D\right| \cdot \left(-1 \cdot \frac{\sqrt{-0.25 \cdot \frac{h}{\ell}}}{\left|d\right|}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]

FPCore

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

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((fabs(M) * fabs(D)) / (2.0 * fabs(d))), 2.0) * (h / l)) <= -2000.0) {
		tmp = w0 * (fabs(M) * (-1.0 * (fabs(D) * (-1.0 * (sqrt(-(0.25 * (h / l))) / fabs(d))))));
	} else {
		tmp = w0 * 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 (((((abs(m) * abs(d)) / (2.0d0 * abs(d_1))) ** 2.0d0) * (h / l)) <= (-2000.0d0)) then
        tmp = w0 * (abs(m) * ((-1.0d0) * (abs(d) * ((-1.0d0) * (sqrt(-(0.25d0 * (h / l))) / abs(d_1))))))
    else
        tmp = w0 * 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(((Math.abs(M) * Math.abs(D)) / (2.0 * Math.abs(d))), 2.0) * (h / l)) <= -2000.0) {
		tmp = w0 * (Math.abs(M) * (-1.0 * (Math.abs(D) * (-1.0 * (Math.sqrt(-(0.25 * (h / l))) / Math.abs(d))))));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(N[Abs[M], $MachinePrecision] * N[Abs[D], $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[Abs[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2000.0], N[(w0 * N[(N[Abs[M], $MachinePrecision] * N[(-1.0 * N[(N[Abs[D], $MachinePrecision] * N[(-1.0 * N[(N[Sqrt[(-N[(0.25 * N[(h / l), $MachinePrecision]), $MachinePrecision])], $MachinePrecision] / N[Abs[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 80.7%

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

   2. Taylor expanded in M around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 8.8%

         \[\leadsto w0 \cdot \left(M \cdot \sqrt{-0.25 \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right) \]

   4. Applied rewrites 8.8%

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

   5. Taylor expanded in D around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 11.4%

         \[\leadsto w0 \cdot \left(M \cdot \left(-1 \cdot \left(D \cdot \sqrt{-0.25 \cdot \frac{h}{{d}^{2} \cdot \ell}}\right)\right)\right) \]

   7. Applied rewrites 11.4%

      \[\leadsto w0 \cdot \left(M \cdot \left(-1 \cdot \color{blue}{\left(D \cdot \sqrt{-0.25 \cdot \frac{h}{{d}^{2} \cdot \ell}}\right)}\right)\right) \]

   8. Taylor expanded in d around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 13.3%

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

   10. Applied rewrites 13.3%

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

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

   1. Initial program 80.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac2 N/A

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

      9. associate-/l* N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. associate-/l* N/A

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

      14. lower-fma.f64 N/A

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

   3. Applied rewrites 83.8%

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

   5. Applied rewrites 41.7%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 68.6%

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

Alternative 4: 83.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_1 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ \mathbf{if}\;{\left(\frac{t\_1 \cdot t\_0}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+22}:\\ \;\;\;\;\left(\frac{\sqrt{-0.25 \cdot \frac{\left(t\_0 \cdot t\_0\right) \cdot h}{\ell}}}{\left|d\right|} \cdot t\_1\right) \cdot w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]

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+22)
     (* (* (/ (sqrt (* -0.25 (/ (* (* t_0 t_0) h) l))) (fabs d)) t_1) w0)
     (* w0 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+22) {
		tmp = ((sqrt((-0.25 * (((t_0 * t_0) * h) / l))) / fabs(d)) * t_1) * w0;
	} else {
		tmp = w0 * 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+22)) then
        tmp = ((sqrt(((-0.25d0) * (((t_0 * t_0) * h) / l))) / abs(d_1)) * t_1) * w0
    else
        tmp = w0 * 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+22) {
		tmp = ((Math.sqrt((-0.25 * (((t_0 * t_0) * h) / l))) / Math.abs(d)) * t_1) * w0;
	} else {
		tmp = w0 * 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+22:
		tmp = ((math.sqrt((-0.25 * (((t_0 * t_0) * h) / l))) / math.fabs(d)) * t_1) * w0
	else:
		tmp = w0 * 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+22)
		tmp = Float64(Float64(Float64(sqrt(Float64(-0.25 * Float64(Float64(Float64(t_0 * t_0) * h) / l))) / abs(d)) * t_1) * w0);
	else
		tmp = Float64(w0 * 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+22)
		tmp = ((sqrt((-0.25 * (((t_0 * t_0) * h) / l))) / abs(d)) * t_1) * w0;
	else
		tmp = w0 * 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+22], N[(N[(N[(N[Sqrt[N[(-0.25 * N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[d], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * w0), $MachinePrecision], N[(w0 * 1.0), $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)) < -2e22

   1. Initial program 80.7%

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

   2. Taylor expanded in M around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 8.8%

         \[\leadsto w0 \cdot \left(M \cdot \sqrt{-0.25 \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right) \]

   4. Applied rewrites 8.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 8.8%

         \[\leadsto w0 \cdot \left(M \cdot \sqrt{\frac{-0.25 \cdot \left({D}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 8.8%

          \[\leadsto w0 \cdot \left(M \cdot \sqrt{\frac{-0.25 \cdot \left(\left(D \cdot D\right) \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]

      12. lift-pow.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 8.8%

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

   6. Applied rewrites 8.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 8.8%

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

   8. Applied rewrites 10.7%

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

   if -2e22 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 80.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac2 N/A

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

      9. associate-/l* N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. associate-/l* N/A

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

      14. lower-fma.f64 N/A

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

   3. Applied rewrites 83.8%

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

   5. Applied rewrites 41.7%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 68.6%

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

Alternative 5: 71.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot \left|d\right|}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+163}:\\ \;\;\;\;\frac{\left|d\right| \cdot \left(w0 \cdot \sqrt{\frac{2}{\left|d\right|}}\right)}{\sqrt{\left|d\right| + \left|d\right|}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 (fabs d))) 2.0) (/ h l)) -2e+163)
   (/ (* (fabs d) (* w0 (sqrt (/ 2.0 (fabs d))))) (sqrt (+ (fabs d) (fabs d))))
   (* w0 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)) <= -2e+163) {
		tmp = (fabs(d) * (w0 * sqrt((2.0 / fabs(d))))) / sqrt((fabs(d) + fabs(d)));
	} else {
		tmp = w0 * 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)) <= (-2d+163)) then
        tmp = (abs(d_1) * (w0 * sqrt((2.0d0 / abs(d_1))))) / sqrt((abs(d_1) + abs(d_1)))
    else
        tmp = w0 * 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)) <= -2e+163) {
		tmp = (Math.abs(d) * (w0 * Math.sqrt((2.0 / Math.abs(d))))) / Math.sqrt((Math.abs(d) + Math.abs(d)));
	} else {
		tmp = w0 * 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)) <= -2e+163:
		tmp = (math.fabs(d) * (w0 * math.sqrt((2.0 / math.fabs(d))))) / math.sqrt((math.fabs(d) + math.fabs(d)))
	else:
		tmp = w0 * 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)) <= -2e+163)
		tmp = Float64(Float64(abs(d) * Float64(w0 * sqrt(Float64(2.0 / abs(d))))) / sqrt(Float64(abs(d) + abs(d))));
	else
		tmp = Float64(w0 * 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)) <= -2e+163)
		tmp = (abs(d) * (w0 * sqrt((2.0 / abs(d))))) / sqrt((abs(d) + abs(d)));
	else
		tmp = w0 * 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], -2e+163], N[(N[(N[Abs[d], $MachinePrecision] * N[(w0 * N[Sqrt[N[(2.0 / N[Abs[d], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[Abs[d], $MachinePrecision] + N[Abs[d], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $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)) < -1.9999999999999999e163

   1. Initial program 80.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac2 N/A

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

      9. associate-/l* N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. associate-/l* N/A

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

      14. lower-fma.f64 N/A

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

   3. Applied rewrites 83.8%

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

   5. Applied rewrites 41.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 38.4%

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

   8. Taylor expanded in d around inf

      \[\leadsto \frac{\color{blue}{d \cdot \left(w0 \cdot \sqrt{\frac{2}{d}}\right)}}{\sqrt{d + d}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{d \cdot \color{blue}{\left(w0 \cdot \sqrt{\frac{2}{d}}\right)}}{\sqrt{d + d}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{d \cdot \left(w0 \cdot \color{blue}{\sqrt{\frac{2}{d}}}\right)}{\sqrt{d + d}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{d \cdot \left(w0 \cdot \sqrt{\frac{2}{d}}\right)}{\sqrt{d + d}} \]

      4. lower-/.f64 28.9%

         \[\leadsto \frac{d \cdot \left(w0 \cdot \sqrt{\frac{2}{d}}\right)}{\sqrt{d + d}} \]

   10. Applied rewrites 28.9%

       \[\leadsto \frac{\color{blue}{d \cdot \left(w0 \cdot \sqrt{\frac{2}{d}}\right)}}{\sqrt{d + d}} \]

   if -1.9999999999999999e163 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 80.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac2 N/A

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

      9. associate-/l* N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. associate-/l* N/A

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

      14. lower-fma.f64 N/A

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

   3. Applied rewrites 83.8%

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

   5. Applied rewrites 41.7%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 68.6%

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

Alternative 6: 68.6% accurate, 10.6× speedup

Math

\[w0 \cdot 1 \]

FPCore

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

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * 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 * 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.7%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. +-commutative N/A

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

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

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

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

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

   7. lift-/.f64 N/A

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

   8. distribute-neg-frac2 N/A

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

   9. associate-/l* N/A

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

   10. lift-pow.f64 N/A

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

   11. unpow2 N/A

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

   12. associate-*l* N/A

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

   13. associate-/l* N/A

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

   14. lower-fma.f64 N/A

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

3. Applied rewrites 83.8%

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

5. Applied rewrites 41.7%

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

6. Taylor expanded in M around 0

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

8. Applied rewrites 68.6%

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

Reproduce

herbie shell --seed 2025204 
(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))))))