Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.6% → 91.8%

Time: 7.6s

Alternatives: 8

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.6% 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: 91.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_1 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_2 := \frac{t\_0}{\left|d\right|} \cdot t\_1\\ \mathbf{if}\;{\left(\frac{t\_1 \cdot t\_0}{2 \cdot \left|d\right|}\right)}^{2} \cdot \frac{h}{\ell} \leq -100000000:\\ \;\;\;\;w0 \cdot \left(\left(\frac{\sqrt{-0.25 \cdot \frac{h}{\ell}}}{\left|d\right|} \cdot t\_1\right) \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(t\_2 \cdot 0.25\right) \cdot \frac{t\_2 \cdot h}{\ell}}\\ \end{array} \]

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 (fabs d)) t_1)))
   (if (<= (* (pow (/ (* t_1 t_0) (* 2.0 (fabs d))) 2.0) (/ h l)) -100000000.0)
     (* w0 (* (* (/ (sqrt (* -0.25 (/ h l))) (fabs d)) t_1) t_0))
     (* w0 (sqrt (- 1.0 (* (* t_2 0.25) (/ (* t_2 h) l))))))))

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 / fabs(d)) * t_1;
	double tmp;
	if ((pow(((t_1 * t_0) / (2.0 * fabs(d))), 2.0) * (h / l)) <= -100000000.0) {
		tmp = w0 * (((sqrt((-0.25 * (h / l))) / fabs(d)) * t_1) * t_0);
	} else {
		tmp = w0 * sqrt((1.0 - ((t_2 * 0.25) * ((t_2 * h) / l))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(w0, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = fmax(abs(m), abs(d))
    t_1 = fmin(abs(m), abs(d))
    t_2 = (t_0 / abs(d_1)) * t_1
    if (((((t_1 * t_0) / (2.0d0 * abs(d_1))) ** 2.0d0) * (h / l)) <= (-100000000.0d0)) then
        tmp = w0 * (((sqrt(((-0.25d0) * (h / l))) / abs(d_1)) * t_1) * t_0)
    else
        tmp = w0 * sqrt((1.0d0 - ((t_2 * 0.25d0) * ((t_2 * h) / l))))
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = fmax(Math.abs(M), Math.abs(D));
	double t_1 = fmin(Math.abs(M), Math.abs(D));
	double t_2 = (t_0 / Math.abs(d)) * t_1;
	double tmp;
	if ((Math.pow(((t_1 * t_0) / (2.0 * Math.abs(d))), 2.0) * (h / l)) <= -100000000.0) {
		tmp = w0 * (((Math.sqrt((-0.25 * (h / l))) / Math.abs(d)) * t_1) * t_0);
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((t_2 * 0.25) * ((t_2 * h) / l))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	t_0 = fmax(math.fabs(M), math.fabs(D))
	t_1 = fmin(math.fabs(M), math.fabs(D))
	t_2 = (t_0 / math.fabs(d)) * t_1
	tmp = 0
	if (math.pow(((t_1 * t_0) / (2.0 * math.fabs(d))), 2.0) * (h / l)) <= -100000000.0:
		tmp = w0 * (((math.sqrt((-0.25 * (h / l))) / math.fabs(d)) * t_1) * t_0)
	else:
		tmp = w0 * math.sqrt((1.0 - ((t_2 * 0.25) * ((t_2 * h) / l))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	t_0 = fmax(abs(M), abs(D))
	t_1 = fmin(abs(M), abs(D))
	t_2 = Float64(Float64(t_0 / abs(d)) * t_1)
	tmp = 0.0
	if (Float64((Float64(Float64(t_1 * t_0) / Float64(2.0 * abs(d))) ^ 2.0) * Float64(h / l)) <= -100000000.0)
		tmp = Float64(w0 * Float64(Float64(Float64(sqrt(Float64(-0.25 * Float64(h / l))) / abs(d)) * t_1) * t_0));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(t_2 * 0.25) * Float64(Float64(t_2 * h) / l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = max(abs(M), abs(D));
	t_1 = min(abs(M), abs(D));
	t_2 = (t_0 / abs(d)) * t_1;
	tmp = 0.0;
	if (((((t_1 * t_0) / (2.0 * abs(d))) ^ 2.0) * (h / l)) <= -100000000.0)
		tmp = w0 * (((sqrt((-0.25 * (h / l))) / abs(d)) * t_1) * t_0);
	else
		tmp = w0 * sqrt((1.0 - ((t_2 * 0.25) * ((t_2 * h) / l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[Max[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 / N[Abs[d], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(N[(t$95$1 * t$95$0), $MachinePrecision] / N[(2.0 * N[Abs[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -100000000.0], N[(w0 * N[(N[(N[(N[Sqrt[N[(-0.25 * N[(h / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[d], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(t$95$2 * 0.25), $MachinePrecision] * N[(N[(t$95$2 * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $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)) < -1e8

   1. Initial program 80.6%

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

   2. Taylor expanded in D around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 9.5%

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

   4. Applied rewrites 9.5%

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

   5. Taylor expanded in M around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 11.7%

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

   7. Applied rewrites 11.7%

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

   8. Taylor expanded in d around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 13.5%

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

   10. Applied rewrites 13.5%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lower-*.f64 N/A

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

   12. Applied rewrites 13.6%

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

   if -1e8 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 80.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. frac-times N/A

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

      12. metadata-eval N/A

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

      13. cosh-0-rev N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 80.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. mult-flip N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 88.8%

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

   5. Applied rewrites 88.8%

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

Alternative 2: 91.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_1 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_2 := {\left(\frac{t\_1 \cdot t\_0}{2 \cdot \left|d\right|}\right)}^{2} \cdot \frac{h}{\ell}\\ t_3 := t\_1 \cdot \frac{t\_0}{\left|d\right|}\\ \mathbf{if}\;t\_2 \leq -100000000:\\ \;\;\;\;w0 \cdot \left(\left(\frac{\sqrt{-0.25 \cdot \frac{h}{\ell}}}{\left|d\right|} \cdot t\_1\right) \cdot t\_0\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(t\_3 \cdot t\_3\right) \cdot -0.25, \frac{h}{\ell}, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(t\_3 \cdot \frac{\left(h \cdot t\_1\right) \cdot t\_0}{\left|d\right|}\right)}{\ell}}\\ \end{array} \]

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 (* (pow (/ (* t_1 t_0) (* 2.0 (fabs d))) 2.0) (/ h l)))
        (t_3 (* t_1 (/ t_0 (fabs d)))))
   (if (<= t_2 -100000000.0)
     (* w0 (* (* (/ (sqrt (* -0.25 (/ h l))) (fabs d)) t_1) t_0))
     (if (<= t_2 2e-18)
       (* (sqrt (fma (* (* t_3 t_3) -0.25) (/ h l) 1.0)) w0)
       (*
        w0
        (sqrt
         (- 1.0 (/ (* 0.25 (* t_3 (/ (* (* h t_1) t_0) (fabs d)))) l))))))))

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 = pow(((t_1 * t_0) / (2.0 * fabs(d))), 2.0) * (h / l);
	double t_3 = t_1 * (t_0 / fabs(d));
	double tmp;
	if (t_2 <= -100000000.0) {
		tmp = w0 * (((sqrt((-0.25 * (h / l))) / fabs(d)) * t_1) * t_0);
	} else if (t_2 <= 2e-18) {
		tmp = sqrt(fma(((t_3 * t_3) * -0.25), (h / l), 1.0)) * w0;
	} else {
		tmp = w0 * sqrt((1.0 - ((0.25 * (t_3 * (((h * t_1) * t_0) / fabs(d)))) / l)));
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	t_0 = fmax(abs(M), abs(D))
	t_1 = fmin(abs(M), abs(D))
	t_2 = Float64((Float64(Float64(t_1 * t_0) / Float64(2.0 * abs(d))) ^ 2.0) * Float64(h / l))
	t_3 = Float64(t_1 * Float64(t_0 / abs(d)))
	tmp = 0.0
	if (t_2 <= -100000000.0)
		tmp = Float64(w0 * Float64(Float64(Float64(sqrt(Float64(-0.25 * Float64(h / l))) / abs(d)) * t_1) * t_0));
	elseif (t_2 <= 2e-18)
		tmp = Float64(sqrt(fma(Float64(Float64(t_3 * t_3) * -0.25), Float64(h / l), 1.0)) * w0);
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.25 * Float64(t_3 * Float64(Float64(Float64(h * t_1) * t_0) / abs(d)))) / l))));
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[Max[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[(N[(t$95$1 * t$95$0), $MachinePrecision] / N[(2.0 * N[Abs[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[(t$95$0 / N[Abs[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -100000000.0], N[(w0 * N[(N[(N[(N[Sqrt[N[(-0.25 * N[(h / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[d], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e-18], N[(N[Sqrt[N[(N[(N[(t$95$3 * t$95$3), $MachinePrecision] * -0.25), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(0.25 * N[(t$95$3 * N[(N[(N[(h * t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision] / N[Abs[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $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)) < -1e8

   1. Initial program 80.6%

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

   2. Taylor expanded in D around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 9.5%

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

   4. Applied rewrites 9.5%

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

   5. Taylor expanded in M around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 11.7%

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

   7. Applied rewrites 11.7%

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

   8. Taylor expanded in d around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 13.5%

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

   10. Applied rewrites 13.5%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lower-*.f64 N/A

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

   12. Applied rewrites 13.6%

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

   if -1e8 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < 2.0000000000000001e-18

   1. Initial program 80.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. frac-times N/A

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

      12. metadata-eval N/A

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

      13. cosh-0-rev N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 80.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 80.5%

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

   5. Applied rewrites 65.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l/ N/A

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

      14. lift-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-*.f64 80.5%

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 80.5%

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 80.5%

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

   7. Applied rewrites 80.5%

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

   if 2.0000000000000001e-18 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 80.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. frac-times N/A

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

      12. metadata-eval N/A

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

      13. cosh-0-rev N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 80.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. mult-flip N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 88.8%

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

   5. Applied rewrites 88.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 83.8%

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

   7. Applied rewrites 83.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 83.2%

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 83.2%

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 83.2%

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

   9. Applied rewrites 83.2%

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

Alternative 3: 91.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_1 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_2 := t\_1 \cdot \frac{t\_0}{\left|d\right|}\\ t_3 := t\_1 \cdot t\_0\\ t_4 := 1 - {\left(\frac{t\_3}{2 \cdot \left|d\right|}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t\_4 \leq 1000000:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(t\_2 \cdot t\_2\right) \cdot -0.25, \frac{h}{\ell}, 1\right)} \cdot w0\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;w0 \cdot \left(\left(\frac{\sqrt{-0.25 \cdot \frac{h}{\ell}}}{\left|d\right|} \cdot t\_1\right) \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{\left(t\_3 \cdot t\_0\right) \cdot t\_1}{\left|d\right|} \cdot h}{\left(4 \cdot \left|d\right|\right) \cdot \ell}}\\ \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (fmax (fabs M) (fabs D)))
        (t_1 (fmin (fabs M) (fabs D)))
        (t_2 (* t_1 (/ t_0 (fabs d))))
        (t_3 (* t_1 t_0))
        (t_4 (- 1.0 (* (pow (/ t_3 (* 2.0 (fabs d))) 2.0) (/ h l)))))
   (if (<= t_4 1000000.0)
     (* (sqrt (fma (* (* t_2 t_2) -0.25) (/ h l) 1.0)) w0)
     (if (<= t_4 INFINITY)
       (* w0 (* (* (/ (sqrt (* -0.25 (/ h l))) (fabs d)) t_1) t_0))
       (*
        w0
        (sqrt
         (-
          1.0
          (/
           (* (/ (* (* t_3 t_0) t_1) (fabs d)) h)
           (* (* 4.0 (fabs d)) l)))))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = fmax(fabs(M), fabs(D));
	double t_1 = fmin(fabs(M), fabs(D));
	double t_2 = t_1 * (t_0 / fabs(d));
	double t_3 = t_1 * t_0;
	double t_4 = 1.0 - (pow((t_3 / (2.0 * fabs(d))), 2.0) * (h / l));
	double tmp;
	if (t_4 <= 1000000.0) {
		tmp = sqrt(fma(((t_2 * t_2) * -0.25), (h / l), 1.0)) * w0;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = w0 * (((sqrt((-0.25 * (h / l))) / fabs(d)) * t_1) * t_0);
	} else {
		tmp = w0 * sqrt((1.0 - (((((t_3 * t_0) * t_1) / fabs(d)) * h) / ((4.0 * fabs(d)) * l))));
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	t_0 = fmax(abs(M), abs(D))
	t_1 = fmin(abs(M), abs(D))
	t_2 = Float64(t_1 * Float64(t_0 / abs(d)))
	t_3 = Float64(t_1 * t_0)
	t_4 = Float64(1.0 - Float64((Float64(t_3 / Float64(2.0 * abs(d))) ^ 2.0) * Float64(h / l)))
	tmp = 0.0
	if (t_4 <= 1000000.0)
		tmp = Float64(sqrt(fma(Float64(Float64(t_2 * t_2) * -0.25), Float64(h / l), 1.0)) * w0);
	elseif (t_4 <= Inf)
		tmp = Float64(w0 * Float64(Float64(Float64(sqrt(Float64(-0.25 * Float64(h / l))) / abs(d)) * t_1) * t_0));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(t_3 * t_0) * t_1) / abs(d)) * h) / Float64(Float64(4.0 * abs(d)) * l)))));
	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$1 * N[(t$95$0 / N[Abs[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[(1.0 - N[(N[Power[N[(t$95$3 / N[(2.0 * N[Abs[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 1000000.0], N[(N[Sqrt[N[(N[(N[(t$95$2 * t$95$2), $MachinePrecision] * -0.25), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(w0 * N[(N[(N[(N[Sqrt[N[(-0.25 * N[(h / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[d], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(N[(t$95$3 * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision] / N[Abs[d], $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / N[(N[(4.0 * N[Abs[d], $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.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))) < 1e6

   1. Initial program 80.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. frac-times N/A

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

      12. metadata-eval N/A

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

      13. cosh-0-rev N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 80.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 80.5%

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

   5. Applied rewrites 65.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l/ N/A

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

      14. lift-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-*.f64 80.5%

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 80.5%

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 80.5%

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

   7. Applied rewrites 80.5%

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

   if 1e6 < (-.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 80.6%

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

   2. Taylor expanded in D around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 9.5%

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

   4. Applied rewrites 9.5%

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

   5. Taylor expanded in M around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 11.7%

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

   7. Applied rewrites 11.7%

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

   8. Taylor expanded in d around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 13.5%

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

   10. Applied rewrites 13.5%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lower-*.f64 N/A

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

   12. Applied rewrites 13.6%

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

   if +inf.0 < (-.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 80.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. frac-times N/A

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

      12. metadata-eval N/A

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

      13. cosh-0-rev N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 80.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. mult-flip N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 88.8%

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

   5. Applied rewrites 88.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. mult-flip N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. associate-*l/ N/A

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

      14. lift-*.f64 N/A

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

      15. times-frac N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

   7. Applied rewrites 78.0%

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

Alternative 4: 91.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_1 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_2 := t\_1 \cdot \frac{t\_0}{\left|d\right|}\\ t_3 := 1 - {\left(\frac{t\_1 \cdot t\_0}{2 \cdot \left|d\right|}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t\_3 \leq 1000000:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(t\_2 \cdot t\_2\right) \cdot -0.25, \frac{h}{\ell}, 1\right)} \cdot w0\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;w0 \cdot \left(\left(\frac{\sqrt{-0.25 \cdot \frac{h}{\ell}}}{\left|d\right|} \cdot t\_1\right) \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(\left(\left(t\_0 \cdot t\_1\right) \cdot t\_0\right) \cdot t\_1\right) \cdot h}{\left|d\right|} \cdot \frac{0.25}{\left|d\right| \cdot \ell}}\\ \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (fmax (fabs M) (fabs D)))
        (t_1 (fmin (fabs M) (fabs D)))
        (t_2 (* t_1 (/ t_0 (fabs d))))
        (t_3 (- 1.0 (* (pow (/ (* t_1 t_0) (* 2.0 (fabs d))) 2.0) (/ h l)))))
   (if (<= t_3 1000000.0)
     (* (sqrt (fma (* (* t_2 t_2) -0.25) (/ h l) 1.0)) w0)
     (if (<= t_3 INFINITY)
       (* w0 (* (* (/ (sqrt (* -0.25 (/ h l))) (fabs d)) t_1) t_0))
       (*
        w0
        (sqrt
         (-
          1.0
          (*
           (/ (* (* (* (* t_0 t_1) t_0) t_1) h) (fabs d))
           (/ 0.25 (* (fabs d) l))))))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = fmax(fabs(M), fabs(D));
	double t_1 = fmin(fabs(M), fabs(D));
	double t_2 = t_1 * (t_0 / fabs(d));
	double t_3 = 1.0 - (pow(((t_1 * t_0) / (2.0 * fabs(d))), 2.0) * (h / l));
	double tmp;
	if (t_3 <= 1000000.0) {
		tmp = sqrt(fma(((t_2 * t_2) * -0.25), (h / l), 1.0)) * w0;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = w0 * (((sqrt((-0.25 * (h / l))) / fabs(d)) * t_1) * t_0);
	} else {
		tmp = w0 * sqrt((1.0 - ((((((t_0 * t_1) * t_0) * t_1) * h) / fabs(d)) * (0.25 / (fabs(d) * l)))));
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	t_0 = fmax(abs(M), abs(D))
	t_1 = fmin(abs(M), abs(D))
	t_2 = Float64(t_1 * Float64(t_0 / abs(d)))
	t_3 = Float64(1.0 - Float64((Float64(Float64(t_1 * t_0) / Float64(2.0 * abs(d))) ^ 2.0) * Float64(h / l)))
	tmp = 0.0
	if (t_3 <= 1000000.0)
		tmp = Float64(sqrt(fma(Float64(Float64(t_2 * t_2) * -0.25), Float64(h / l), 1.0)) * w0);
	elseif (t_3 <= Inf)
		tmp = Float64(w0 * Float64(Float64(Float64(sqrt(Float64(-0.25 * Float64(h / l))) / abs(d)) * t_1) * t_0));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(t_0 * t_1) * t_0) * t_1) * h) / abs(d)) * Float64(0.25 / Float64(abs(d) * l))))));
	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$1 * N[(t$95$0 / N[Abs[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 - N[(N[Power[N[(N[(t$95$1 * t$95$0), $MachinePrecision] / N[(2.0 * N[Abs[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 1000000.0], N[(N[Sqrt[N[(N[(N[(t$95$2 * t$95$2), $MachinePrecision] * -0.25), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(w0 * N[(N[(N[(N[Sqrt[N[(-0.25 * N[(h / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[d], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 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] * h), $MachinePrecision] / N[Abs[d], $MachinePrecision]), $MachinePrecision] * N[(0.25 / N[(N[Abs[d], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.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))) < 1e6

   1. Initial program 80.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. frac-times N/A

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

      12. metadata-eval N/A

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

      13. cosh-0-rev N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 80.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 80.5%

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

   5. Applied rewrites 65.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l/ N/A

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

      14. lift-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-*.f64 80.5%

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 80.5%

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 80.5%

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

   7. Applied rewrites 80.5%

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

   if 1e6 < (-.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 80.6%

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

   2. Taylor expanded in D around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 9.5%

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

   4. Applied rewrites 9.5%

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

   5. Taylor expanded in M around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 11.7%

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

   7. Applied rewrites 11.7%

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

   8. Taylor expanded in d around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 13.5%

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

   10. Applied rewrites 13.5%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lower-*.f64 N/A

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

   12. Applied rewrites 13.6%

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

   if +inf.0 < (-.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 80.6%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. swap-sqr N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   3. Applied rewrites 64.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 76.2%

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

   5. Applied rewrites 76.2%

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

Alternative 5: 90.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_1 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_2 := t\_1 \cdot \frac{t\_0}{\left|d\right|}\\ t_3 := t\_1 \cdot t\_0\\ t_4 := 1 - {\left(\frac{t\_3}{2 \cdot \left|d\right|}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t\_4 \leq 1000000:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(t\_2 \cdot t\_2\right) \cdot -0.25, \frac{h}{\ell}, 1\right)} \cdot w0\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;w0 \cdot \left(\left(\frac{\sqrt{-0.25 \cdot \frac{h}{\ell}}}{\left|d\right|} \cdot t\_1\right) \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-0.25, \frac{\left(\left(t\_3 \cdot t\_0\right) \cdot t\_1\right) \cdot h}{\left(\ell \cdot \left|d\right|\right) \cdot \left|d\right|}, 1\right)} \cdot w0\\ \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (fmax (fabs M) (fabs D)))
        (t_1 (fmin (fabs M) (fabs D)))
        (t_2 (* t_1 (/ t_0 (fabs d))))
        (t_3 (* t_1 t_0))
        (t_4 (- 1.0 (* (pow (/ t_3 (* 2.0 (fabs d))) 2.0) (/ h l)))))
   (if (<= t_4 1000000.0)
     (* (sqrt (fma (* (* t_2 t_2) -0.25) (/ h l) 1.0)) w0)
     (if (<= t_4 INFINITY)
       (* w0 (* (* (/ (sqrt (* -0.25 (/ h l))) (fabs d)) t_1) t_0))
       (*
        (sqrt
         (fma
          -0.25
          (/ (* (* (* t_3 t_0) t_1) h) (* (* l (fabs d)) (fabs d)))
          1.0))
        w0)))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = fmax(fabs(M), fabs(D));
	double t_1 = fmin(fabs(M), fabs(D));
	double t_2 = t_1 * (t_0 / fabs(d));
	double t_3 = t_1 * t_0;
	double t_4 = 1.0 - (pow((t_3 / (2.0 * fabs(d))), 2.0) * (h / l));
	double tmp;
	if (t_4 <= 1000000.0) {
		tmp = sqrt(fma(((t_2 * t_2) * -0.25), (h / l), 1.0)) * w0;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = w0 * (((sqrt((-0.25 * (h / l))) / fabs(d)) * t_1) * t_0);
	} else {
		tmp = sqrt(fma(-0.25, ((((t_3 * t_0) * t_1) * h) / ((l * fabs(d)) * fabs(d))), 1.0)) * w0;
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	t_0 = fmax(abs(M), abs(D))
	t_1 = fmin(abs(M), abs(D))
	t_2 = Float64(t_1 * Float64(t_0 / abs(d)))
	t_3 = Float64(t_1 * t_0)
	t_4 = Float64(1.0 - Float64((Float64(t_3 / Float64(2.0 * abs(d))) ^ 2.0) * Float64(h / l)))
	tmp = 0.0
	if (t_4 <= 1000000.0)
		tmp = Float64(sqrt(fma(Float64(Float64(t_2 * t_2) * -0.25), Float64(h / l), 1.0)) * w0);
	elseif (t_4 <= Inf)
		tmp = Float64(w0 * Float64(Float64(Float64(sqrt(Float64(-0.25 * Float64(h / l))) / abs(d)) * t_1) * t_0));
	else
		tmp = Float64(sqrt(fma(-0.25, Float64(Float64(Float64(Float64(t_3 * t_0) * t_1) * h) / Float64(Float64(l * abs(d)) * abs(d))), 1.0)) * w0);
	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$1 * N[(t$95$0 / N[Abs[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[(1.0 - N[(N[Power[N[(t$95$3 / N[(2.0 * N[Abs[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 1000000.0], N[(N[Sqrt[N[(N[(N[(t$95$2 * t$95$2), $MachinePrecision] * -0.25), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(w0 * N[(N[(N[(N[Sqrt[N[(-0.25 * N[(h / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[d], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(-0.25 * N[(N[(N[(N[(t$95$3 * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision] * h), $MachinePrecision] / N[(N[(l * N[Abs[d], $MachinePrecision]), $MachinePrecision] * N[Abs[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.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))) < 1e6

   1. Initial program 80.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. frac-times N/A

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

      12. metadata-eval N/A

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

      13. cosh-0-rev N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 80.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 80.5%

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

   5. Applied rewrites 65.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l/ N/A

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

      14. lift-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-*.f64 80.5%

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 80.5%

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 80.5%

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

   7. Applied rewrites 80.5%

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

   if 1e6 < (-.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 80.6%

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

   2. Taylor expanded in D around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 9.5%

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

   4. Applied rewrites 9.5%

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

   5. Taylor expanded in M around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 11.7%

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

   7. Applied rewrites 11.7%

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

   8. Taylor expanded in d around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 13.5%

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

   10. Applied rewrites 13.5%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lower-*.f64 N/A

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

   12. Applied rewrites 13.6%

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

   if +inf.0 < (-.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 80.6%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. swap-sqr N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   3. Applied rewrites 64.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 64.4%

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

   5. Applied rewrites 68.0%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

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

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 71.9%

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

Alternative 6: 90.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_1 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ \mathbf{if}\;{\left(\frac{t\_1 \cdot t\_0}{2 \cdot \left|d\right|}\right)}^{2} \cdot \frac{h}{\ell} \leq -400000:\\ \;\;\;\;w0 \cdot \left(\left(\frac{\sqrt{-0.25 \cdot \frac{h}{\ell}}}{\left|d\right|} \cdot t\_1\right) \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]

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 (fabs d))) 2.0) (/ h l)) -400000.0)
     (* w0 (* (* (/ (sqrt (* -0.25 (/ h l))) (fabs d)) t_1) t_0))
     (* 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 * fabs(d))), 2.0) * (h / l)) <= -400000.0) {
		tmp = w0 * (((sqrt((-0.25 * (h / l))) / fabs(d)) * t_1) * t_0);
	} 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 * abs(d_1))) ** 2.0d0) * (h / l)) <= (-400000.0d0)) then
        tmp = w0 * (((sqrt(((-0.25d0) * (h / l))) / abs(d_1)) * t_1) * t_0)
    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 * Math.abs(d))), 2.0) * (h / l)) <= -400000.0) {
		tmp = w0 * (((Math.sqrt((-0.25 * (h / l))) / Math.abs(d)) * t_1) * t_0);
	} 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 * math.fabs(d))), 2.0) * (h / l)) <= -400000.0:
		tmp = w0 * (((math.sqrt((-0.25 * (h / l))) / math.fabs(d)) * t_1) * t_0)
	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 * abs(d))) ^ 2.0) * Float64(h / l)) <= -400000.0)
		tmp = Float64(w0 * Float64(Float64(Float64(sqrt(Float64(-0.25 * Float64(h / l))) / abs(d)) * t_1) * t_0));
	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 * abs(d))) ^ 2.0) * (h / l)) <= -400000.0)
		tmp = w0 * (((sqrt((-0.25 * (h / l))) / abs(d)) * t_1) * t_0);
	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 * N[Abs[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -400000.0], N[(w0 * N[(N[(N[(N[Sqrt[N[(-0.25 * N[(h / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[d], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$0), $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)) < -4e5

   1. Initial program 80.6%

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

   2. Taylor expanded in D around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 9.5%

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

   4. Applied rewrites 9.5%

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

   5. Taylor expanded in M around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 11.7%

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

   7. Applied rewrites 11.7%

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

   8. Taylor expanded in d around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 13.5%

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

   10. Applied rewrites 13.5%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lower-*.f64 N/A

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

   12. Applied rewrites 13.6%

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

   if -4e5 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 80.6%

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

   2. Taylor expanded in M around 0

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

   4. Applied rewrites 68.0%

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

Alternative 7: 90.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_1 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ \mathbf{if}\;{\left(\frac{t\_1 \cdot t\_0}{2 \cdot \left|d\right|}\right)}^{2} \cdot \frac{h}{\ell} \leq -500000:\\ \;\;\;\;\left(\left(\frac{\sqrt{-0.25 \cdot \frac{h}{\ell}}}{\left|d\right|} \cdot t\_1\right) \cdot w0\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]

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 (fabs d))) 2.0) (/ h l)) -500000.0)
     (* (* (* (/ (sqrt (* -0.25 (/ h l))) (fabs d)) t_1) w0) t_0)
     (* 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 * fabs(d))), 2.0) * (h / l)) <= -500000.0) {
		tmp = (((sqrt((-0.25 * (h / l))) / fabs(d)) * t_1) * w0) * t_0;
	} 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 * abs(d_1))) ** 2.0d0) * (h / l)) <= (-500000.0d0)) then
        tmp = (((sqrt(((-0.25d0) * (h / l))) / abs(d_1)) * t_1) * w0) * t_0
    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 * Math.abs(d))), 2.0) * (h / l)) <= -500000.0) {
		tmp = (((Math.sqrt((-0.25 * (h / l))) / Math.abs(d)) * t_1) * w0) * t_0;
	} 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 * math.fabs(d))), 2.0) * (h / l)) <= -500000.0:
		tmp = (((math.sqrt((-0.25 * (h / l))) / math.fabs(d)) * t_1) * w0) * t_0
	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 * abs(d))) ^ 2.0) * Float64(h / l)) <= -500000.0)
		tmp = Float64(Float64(Float64(Float64(sqrt(Float64(-0.25 * Float64(h / l))) / abs(d)) * t_1) * w0) * t_0);
	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 * abs(d))) ^ 2.0) * (h / l)) <= -500000.0)
		tmp = (((sqrt((-0.25 * (h / l))) / abs(d)) * t_1) * w0) * t_0;
	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 * N[Abs[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -500000.0], N[(N[(N[(N[(N[Sqrt[N[(-0.25 * N[(h / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[d], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * w0), $MachinePrecision] * t$95$0), $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)) < -5e5

   1. Initial program 80.6%

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

   2. Taylor expanded in D around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 9.5%

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

   4. Applied rewrites 9.5%

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

   5. Taylor expanded in M around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 11.7%

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

   7. Applied rewrites 11.7%

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

   8. Taylor expanded in d around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 13.5%

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

   10. Applied rewrites 13.5%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 13.5%

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

   12. Applied rewrites 13.5%

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

   if -5e5 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 80.6%

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

   2. Taylor expanded in M around 0

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

   4. Applied rewrites 68.0%

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

Alternative 8: 68.0% accurate, 10.1× 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 80.6%

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

2. Taylor expanded in M around 0

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

4. Applied rewrites 68.0%

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

Reproduce

herbie shell --seed 2025187 
(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))))))