Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.6% → 42.2%

Time: 10.7s

Alternatives: 14

Speedup: 0.9×

• could not determine a ground truth

  1. c0 = -8.459655602570199e+150
  2. w = 2.622228954311364e-273
  3. h = 1.4968564237411681e-273
  4. D = -4.5313331694527364e-299
  5. d = 5.6984509751905695e+299
  6. M = -1.3870376128021037e+287

Specification

Math

\[\begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	return (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
}

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(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    t_0 = (c0 * (d_1 * d_1)) / ((w * h) * (d * d))
    code = (c0 / (2.0d0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (m * m))))
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	return (c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	return (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	return Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Initial Program: 24.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	return (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
}

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(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    t_0 = (c0 * (d_1 * d_1)) / ((w * h) * (d * d))
    code = (c0 / (2.0d0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (m * m))))
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	return (c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	return (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	return Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 42.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \frac{\frac{\frac{c0}{w}}{D} \cdot d}{D} \cdot \frac{d}{h}\\ t_1 := \left(-\left|M\right|\right) \cdot \left|M\right|\\ \mathbf{if}\;\left|M\right| \leq 5.5 \cdot 10^{-163}:\\ \;\;\;\;\frac{\sqrt{t\_1} \cdot c0}{w + w}\\ \mathbf{elif}\;\left|M\right| \leq 1.55 \cdot 10^{+139}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - \left|M\right| \cdot \left|M\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot {t\_1}^{0.5}}{w}\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ (* (/ (/ c0 w) D) d) D) (/ d h)))
        (t_1 (* (- (fabs M)) (fabs M))))
   (if (<= (fabs M) 5.5e-163)
     (/ (* (sqrt t_1) c0) (+ w w))
     (if (<= (fabs M) 1.55e+139)
       (*
        (/ c0 (* 2.0 w))
        (+ t_0 (sqrt (- (* t_0 t_0) (* (fabs M) (fabs M))))))
       (* 0.5 (/ (* c0 (pow t_1 0.5)) w))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((((c0 / w) / D) * d) / D) * (d / h);
	double t_1 = -fabs(M) * fabs(M);
	double tmp;
	if (fabs(M) <= 5.5e-163) {
		tmp = (sqrt(t_1) * c0) / (w + w);
	} else if (fabs(M) <= 1.55e+139) {
		tmp = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (fabs(M) * fabs(M)))));
	} else {
		tmp = 0.5 * ((c0 * pow(t_1, 0.5)) / w);
	}
	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(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((((c0 / w) / d) * d_1) / d) * (d_1 / h)
    t_1 = -abs(m) * abs(m)
    if (abs(m) <= 5.5d-163) then
        tmp = (sqrt(t_1) * c0) / (w + w)
    else if (abs(m) <= 1.55d+139) then
        tmp = (c0 / (2.0d0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (abs(m) * abs(m)))))
    else
        tmp = 0.5d0 * ((c0 * (t_1 ** 0.5d0)) / w)
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((((c0 / w) / D) * d) / D) * (d / h);
	double t_1 = -Math.abs(M) * Math.abs(M);
	double tmp;
	if (Math.abs(M) <= 5.5e-163) {
		tmp = (Math.sqrt(t_1) * c0) / (w + w);
	} else if (Math.abs(M) <= 1.55e+139) {
		tmp = (c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (Math.abs(M) * Math.abs(M)))));
	} else {
		tmp = 0.5 * ((c0 * Math.pow(t_1, 0.5)) / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = ((((c0 / w) / D) * d) / D) * (d / h)
	t_1 = -math.fabs(M) * math.fabs(M)
	tmp = 0
	if math.fabs(M) <= 5.5e-163:
		tmp = (math.sqrt(t_1) * c0) / (w + w)
	elif math.fabs(M) <= 1.55e+139:
		tmp = (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (math.fabs(M) * math.fabs(M)))))
	else:
		tmp = 0.5 * ((c0 * math.pow(t_1, 0.5)) / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(Float64(Float64(Float64(c0 / w) / D) * d) / D) * Float64(d / h))
	t_1 = Float64(Float64(-abs(M)) * abs(M))
	tmp = 0.0
	if (abs(M) <= 5.5e-163)
		tmp = Float64(Float64(sqrt(t_1) * c0) / Float64(w + w));
	elseif (abs(M) <= 1.55e+139)
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(abs(M) * abs(M))))));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * (t_1 ^ 0.5)) / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = ((((c0 / w) / D) * d) / D) * (d / h);
	t_1 = -abs(M) * abs(M);
	tmp = 0.0;
	if (abs(M) <= 5.5e-163)
		tmp = (sqrt(t_1) * c0) / (w + w);
	elseif (abs(M) <= 1.55e+139)
		tmp = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (abs(M) * abs(M)))));
	else
		tmp = 0.5 * ((c0 * (t_1 ^ 0.5)) / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(N[(N[(N[(c0 / w), $MachinePrecision] / D), $MachinePrecision] * d), $MachinePrecision] / D), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-N[Abs[M], $MachinePrecision]) * N[Abs[M], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[M], $MachinePrecision], 5.5e-163], N[(N[(N[Sqrt[t$95$1], $MachinePrecision] * c0), $MachinePrecision] / N[(w + w), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[M], $MachinePrecision], 1.55e+139], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(N[Abs[M], $MachinePrecision] * N[Abs[M], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Power[t$95$1, 0.5], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if M < 5.4999999999999998e-163

   1. Initial program 24.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. lower-pow.f64 14.0%

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   4. Applied rewrites 14.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{\color{blue}{w}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(c0 \cdot \sqrt{-{M}^{2}}\right)}{\color{blue}{w}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(c0 \cdot \sqrt{-{M}^{2}}\right)}{\color{blue}{w}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(c0 \cdot \sqrt{-{M}^{2}}\right)}{w} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{-{M}^{2}}}{w} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{-{M}^{2}}}{w} \]

      8. lower-*.f64 14.0%

         \[\leadsto \frac{\left(0.5 \cdot c0\right) \cdot \sqrt{-{M}^{2}}}{w} \]

      9. lift-neg.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      11. pow2 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}{w} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}{w} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}{w} \]

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

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\left(\mathsf{neg}\left(M\right)\right) \cdot M}}{w} \]

      15. lift-neg.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

      16. lower-*.f64 14.0%

          \[\leadsto \frac{\left(0.5 \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

   6. Applied rewrites 14.0%

      \[\leadsto \frac{\left(0.5 \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{\color{blue}{w}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot \color{blue}{c0}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

      2. lift-*.f64 14.0%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{c0}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot \color{blue}{c0}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

      4. lift-*.f64 14.0%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{c0}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{\color{blue}{w}} \]

   8. Applied rewrites 12.1%

      \[\leadsto \color{blue}{\sqrt{\left(-M\right) \cdot M} \cdot \frac{c0}{w + w}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \color{blue}{\frac{c0}{w + w}} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \frac{c0}{\color{blue}{w + w}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\sqrt{\left(-M\right) \cdot M} \cdot c0}{\color{blue}{w + w}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\left(-M\right) \cdot M} \cdot c0}{\color{blue}{w + w}} \]

      5. lower-*.f64 14.0%

         \[\leadsto \frac{\sqrt{\left(-M\right) \cdot M} \cdot c0}{\color{blue}{w} + w} \]

   10. Applied rewrites 14.0%

       \[\leadsto \frac{\sqrt{\left(-M\right) \cdot M} \cdot c0}{\color{blue}{w + w}} \]

   if 5.4999999999999998e-163 < M < 1.55e139

   1. Initial program 24.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{c0 \cdot \left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      7. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      8. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(w \cdot h\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      18. lower-/.f64 24.1%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \color{blue}{\frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Applied rewrites 24.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\color{blue}{c0 \cdot \left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      7. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      8. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(w \cdot h\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      18. lower-/.f64 24.2%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \color{blue}{\frac{d}{D}}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   5. Applied rewrites 24.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\color{blue}{c0 \cdot \left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} - M \cdot M}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} - M \cdot M}\right) \]

      7. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} - M \cdot M}\right) \]

      8. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} - M \cdot M}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} - M \cdot M}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(w \cdot h\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      18. lower-/.f64 34.2%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \color{blue}{\frac{d}{D}}\right) - M \cdot M}\right) \]

   7. Applied rewrites 34.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)} - M \cdot M}\right) \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(h \cdot w\right)}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      3. associate-/r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{d \cdot c0}{D}}{h \cdot w}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot c0}{D}}{\color{blue}{h \cdot w}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      5. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot c0}{D}}{\color{blue}{w \cdot h}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      6. associate-/r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{\frac{d \cdot c0}{D}}{w}}{h}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{\frac{d \cdot c0}{D}}{w}}{h}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\frac{\frac{d \cdot c0}{D}}{w}}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{\frac{\color{blue}{d \cdot c0}}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      10. associate-/l* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \color{blue}{\frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      12. lower-*.f64 32.0%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

   9. Applied rewrites 32.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(h \cdot w\right)}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       3. associate-/r* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\color{blue}{\frac{\frac{d \cdot c0}{D}}{h \cdot w}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot c0}{D}}{\color{blue}{h \cdot w}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       5. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot c0}{D}}{\color{blue}{w \cdot h}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       6. associate-/r* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\color{blue}{\frac{\frac{\frac{d \cdot c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       7. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\color{blue}{\frac{\frac{\frac{d \cdot c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\color{blue}{\frac{\frac{d \cdot c0}{D}}{w}}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{\frac{\color{blue}{d \cdot c0}}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       10. associate-/l* N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       11. lift-/.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \color{blue}{\frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       12. lower-*.f64 32.0%

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

   11. Applied rewrites 32.0%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]
   12. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(h \cdot w\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       3. associate-/r* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\color{blue}{\frac{\frac{d \cdot c0}{D}}{h \cdot w}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot c0}{D}}{\color{blue}{h \cdot w}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       5. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot c0}{D}}{\color{blue}{w \cdot h}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       6. associate-/r* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\color{blue}{\frac{\frac{\frac{d \cdot c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       7. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\color{blue}{\frac{\frac{\frac{d \cdot c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\color{blue}{\frac{\frac{d \cdot c0}{D}}{w}}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{\frac{\color{blue}{d \cdot c0}}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       10. associate-/l* N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       11. lift-/.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \color{blue}{\frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       12. lower-*.f64 36.3%

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

   13. Applied rewrites 36.3%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]
   14. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h}} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       3. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \color{blue}{\frac{d}{D}} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       4. frac-times N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w} \cdot d}{h \cdot D}} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       5. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w} \cdot d}{\color{blue}{D \cdot h}} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       6. times-frac N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{D} \cdot \frac{d}{h}} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{D} \cdot \frac{d}{h}} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{D}} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       9. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\frac{d \cdot \frac{c0}{D}}{w}}}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       11. associate-/l* N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{d \cdot \frac{\frac{c0}{D}}{w}}}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       12. *-commutative N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\frac{\frac{c0}{D}}{w} \cdot d}}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       13. lower-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\frac{\frac{c0}{D}}{w} \cdot d}}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       14. lift-/.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{\color{blue}{\frac{c0}{D}}}{w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       15. associate-/l/ N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\frac{c0}{D \cdot w}} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       16. lift-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{\color{blue}{D \cdot w}} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       17. lower-/.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\frac{c0}{D \cdot w}} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       18. lower-/.f64 32.4%

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \color{blue}{\frac{d}{h}} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

   15. Applied rewrites 32.4%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]
   16. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\color{blue}{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right)} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       3. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \color{blue}{\frac{d}{D}}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       4. frac-times N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w} \cdot d}{h \cdot D}} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       5. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\frac{\frac{d \cdot \frac{c0}{D}}{w} \cdot d}{\color{blue}{D \cdot h}} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       6. times-frac N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\color{blue}{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{D} \cdot \frac{d}{h}\right)} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\color{blue}{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{D} \cdot \frac{d}{h}\right)} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{D}} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       9. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\color{blue}{\frac{d \cdot \frac{c0}{D}}{w}}}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       11. associate-/l* N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\color{blue}{d \cdot \frac{\frac{c0}{D}}{w}}}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       12. *-commutative N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\color{blue}{\frac{\frac{c0}{D}}{w} \cdot d}}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       13. lower-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\color{blue}{\frac{\frac{c0}{D}}{w} \cdot d}}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       14. lift-/.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{\color{blue}{\frac{c0}{D}}}{w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       15. associate-/l/ N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\color{blue}{\frac{c0}{D \cdot w}} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       16. lift-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{\color{blue}{D \cdot w}} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       17. lower-/.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\color{blue}{\frac{c0}{D \cdot w}} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       18. lower-/.f64 32.6%

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \color{blue}{\frac{d}{h}}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

   17. Applied rewrites 32.6%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\color{blue}{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right)} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]
   18. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \color{blue}{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right)} - M \cdot M}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       3. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \color{blue}{\frac{d}{D}}\right) - M \cdot M}\right) \]

       4. frac-times N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w} \cdot d}{h \cdot D}} - M \cdot M}\right) \]

       5. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \frac{\frac{d \cdot \frac{c0}{D}}{w} \cdot d}{\color{blue}{D \cdot h}} - M \cdot M}\right) \]

       6. times-frac N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \color{blue}{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{D} \cdot \frac{d}{h}\right)} - M \cdot M}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \color{blue}{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{D} \cdot \frac{d}{h}\right)} - M \cdot M}\right) \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{D}} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       9. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\color{blue}{\frac{d \cdot \frac{c0}{D}}{w}}}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       11. associate-/l* N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\color{blue}{d \cdot \frac{\frac{c0}{D}}{w}}}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       12. *-commutative N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\color{blue}{\frac{\frac{c0}{D}}{w} \cdot d}}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       13. lower-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\color{blue}{\frac{\frac{c0}{D}}{w} \cdot d}}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       14. lift-/.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{\color{blue}{\frac{c0}{D}}}{w} \cdot d}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       15. associate-/l/ N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\color{blue}{\frac{c0}{D \cdot w}} \cdot d}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       16. lift-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{c0}{\color{blue}{D \cdot w}} \cdot d}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       17. lower-/.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\color{blue}{\frac{c0}{D \cdot w}} \cdot d}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       18. lower-/.f64 34.8%

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \color{blue}{\frac{d}{h}}\right) - M \cdot M}\right) \]

   19. Applied rewrites 34.8%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \color{blue}{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right)} - M \cdot M}\right) \]
   20. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\frac{c0}{D \cdot w}} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{\color{blue}{D \cdot w}} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       3. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{\color{blue}{w \cdot D}} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       4. associate-/r* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\frac{\frac{c0}{w}}{D}} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       5. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\frac{\frac{c0}{w}}{D}} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       6. lower-/.f64 34.2%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{\color{blue}{\frac{c0}{w}}}{D} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

   21. Applied rewrites 34.2%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\frac{\frac{c0}{w}}{D}} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]
   22. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{\frac{c0}{w}}{D} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\color{blue}{\frac{c0}{D \cdot w}} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{\frac{c0}{w}}{D} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{\color{blue}{D \cdot w}} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       3. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{\frac{c0}{w}}{D} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{\color{blue}{w \cdot D}} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       4. associate-/r* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{\frac{c0}{w}}{D} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\color{blue}{\frac{\frac{c0}{w}}{D}} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       5. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{\frac{c0}{w}}{D} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\color{blue}{\frac{\frac{c0}{w}}{D}} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       6. lower-/.f64 34.1%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{\frac{c0}{w}}{D} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{\color{blue}{\frac{c0}{w}}}{D} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

   23. Applied rewrites 34.1%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{\frac{c0}{w}}{D} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\color{blue}{\frac{\frac{c0}{w}}{D}} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]
   24. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{\frac{c0}{w}}{D} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{\frac{c0}{w}}{D} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\color{blue}{\frac{c0}{D \cdot w}} \cdot d}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{\frac{c0}{w}}{D} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{\frac{c0}{w}}{D} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{c0}{\color{blue}{D \cdot w}} \cdot d}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       3. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{\frac{c0}{w}}{D} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{\frac{c0}{w}}{D} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{c0}{\color{blue}{w \cdot D}} \cdot d}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       4. associate-/r* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{\frac{c0}{w}}{D} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{\frac{c0}{w}}{D} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\color{blue}{\frac{\frac{c0}{w}}{D}} \cdot d}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       5. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{\frac{c0}{w}}{D} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{\frac{c0}{w}}{D} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\color{blue}{\frac{\frac{c0}{w}}{D}} \cdot d}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       6. lower-/.f64 35.5%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{\frac{c0}{w}}{D} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{\frac{c0}{w}}{D} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{\color{blue}{\frac{c0}{w}}}{D} \cdot d}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

   25. Applied rewrites 35.5%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{\frac{c0}{w}}{D} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{\frac{c0}{w}}{D} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\color{blue}{\frac{\frac{c0}{w}}{D}} \cdot d}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

   if 1.55e139 < M

   1. Initial program 24.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. lower-pow.f64 14.0%

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   4. Applied rewrites 14.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      2. pow1/2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(-{M}^{2}\right)}^{\frac{1}{2}}}{w} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      5. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

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

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      9. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      11. lower-*.f64 21.9%

          \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]

   6. Applied rewrites 21.9%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 41.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \frac{\frac{c0}{D} \cdot \frac{d}{w}}{h} \cdot \frac{d}{D}\\ t_1 := \left(-\left|M\right|\right) \cdot \left|M\right|\\ \mathbf{if}\;\left|M\right| \leq 4.2 \cdot 10^{-163}:\\ \;\;\;\;\frac{\sqrt{t\_1} \cdot c0}{w + w}\\ \mathbf{elif}\;\left|M\right| \leq 1.55 \cdot 10^{+139}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - \left|M\right| \cdot \left|M\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot {t\_1}^{0.5}}{w}\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ (* (/ c0 D) (/ d w)) h) (/ d D)))
        (t_1 (* (- (fabs M)) (fabs M))))
   (if (<= (fabs M) 4.2e-163)
     (/ (* (sqrt t_1) c0) (+ w w))
     (if (<= (fabs M) 1.55e+139)
       (*
        (/ c0 (* 2.0 w))
        (+ t_0 (sqrt (- (* t_0 t_0) (* (fabs M) (fabs M))))))
       (* 0.5 (/ (* c0 (pow t_1 0.5)) w))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (((c0 / D) * (d / w)) / h) * (d / D);
	double t_1 = -fabs(M) * fabs(M);
	double tmp;
	if (fabs(M) <= 4.2e-163) {
		tmp = (sqrt(t_1) * c0) / (w + w);
	} else if (fabs(M) <= 1.55e+139) {
		tmp = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (fabs(M) * fabs(M)))));
	} else {
		tmp = 0.5 * ((c0 * pow(t_1, 0.5)) / w);
	}
	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(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((c0 / d) * (d_1 / w)) / h) * (d_1 / d)
    t_1 = -abs(m) * abs(m)
    if (abs(m) <= 4.2d-163) then
        tmp = (sqrt(t_1) * c0) / (w + w)
    else if (abs(m) <= 1.55d+139) then
        tmp = (c0 / (2.0d0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (abs(m) * abs(m)))))
    else
        tmp = 0.5d0 * ((c0 * (t_1 ** 0.5d0)) / w)
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (((c0 / D) * (d / w)) / h) * (d / D);
	double t_1 = -Math.abs(M) * Math.abs(M);
	double tmp;
	if (Math.abs(M) <= 4.2e-163) {
		tmp = (Math.sqrt(t_1) * c0) / (w + w);
	} else if (Math.abs(M) <= 1.55e+139) {
		tmp = (c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (Math.abs(M) * Math.abs(M)))));
	} else {
		tmp = 0.5 * ((c0 * Math.pow(t_1, 0.5)) / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (((c0 / D) * (d / w)) / h) * (d / D)
	t_1 = -math.fabs(M) * math.fabs(M)
	tmp = 0
	if math.fabs(M) <= 4.2e-163:
		tmp = (math.sqrt(t_1) * c0) / (w + w)
	elif math.fabs(M) <= 1.55e+139:
		tmp = (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (math.fabs(M) * math.fabs(M)))))
	else:
		tmp = 0.5 * ((c0 * math.pow(t_1, 0.5)) / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(Float64(Float64(c0 / D) * Float64(d / w)) / h) * Float64(d / D))
	t_1 = Float64(Float64(-abs(M)) * abs(M))
	tmp = 0.0
	if (abs(M) <= 4.2e-163)
		tmp = Float64(Float64(sqrt(t_1) * c0) / Float64(w + w));
	elseif (abs(M) <= 1.55e+139)
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(abs(M) * abs(M))))));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * (t_1 ^ 0.5)) / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (((c0 / D) * (d / w)) / h) * (d / D);
	t_1 = -abs(M) * abs(M);
	tmp = 0.0;
	if (abs(M) <= 4.2e-163)
		tmp = (sqrt(t_1) * c0) / (w + w);
	elseif (abs(M) <= 1.55e+139)
		tmp = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (abs(M) * abs(M)))));
	else
		tmp = 0.5 * ((c0 * (t_1 ^ 0.5)) / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(N[(N[(c0 / D), $MachinePrecision] * N[(d / w), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-N[Abs[M], $MachinePrecision]) * N[Abs[M], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[M], $MachinePrecision], 4.2e-163], N[(N[(N[Sqrt[t$95$1], $MachinePrecision] * c0), $MachinePrecision] / N[(w + w), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[M], $MachinePrecision], 1.55e+139], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(N[Abs[M], $MachinePrecision] * N[Abs[M], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Power[t$95$1, 0.5], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if M < 4.19999999999999996e-163

   1. Initial program 24.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. lower-pow.f64 14.0%

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   4. Applied rewrites 14.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{\color{blue}{w}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(c0 \cdot \sqrt{-{M}^{2}}\right)}{\color{blue}{w}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(c0 \cdot \sqrt{-{M}^{2}}\right)}{\color{blue}{w}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(c0 \cdot \sqrt{-{M}^{2}}\right)}{w} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{-{M}^{2}}}{w} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{-{M}^{2}}}{w} \]

      8. lower-*.f64 14.0%

         \[\leadsto \frac{\left(0.5 \cdot c0\right) \cdot \sqrt{-{M}^{2}}}{w} \]

      9. lift-neg.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      11. pow2 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}{w} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}{w} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}{w} \]

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

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\left(\mathsf{neg}\left(M\right)\right) \cdot M}}{w} \]

      15. lift-neg.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

      16. lower-*.f64 14.0%

          \[\leadsto \frac{\left(0.5 \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

   6. Applied rewrites 14.0%

      \[\leadsto \frac{\left(0.5 \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{\color{blue}{w}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot \color{blue}{c0}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

      2. lift-*.f64 14.0%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{c0}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot \color{blue}{c0}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

      4. lift-*.f64 14.0%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{c0}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{\color{blue}{w}} \]

   8. Applied rewrites 12.1%

      \[\leadsto \color{blue}{\sqrt{\left(-M\right) \cdot M} \cdot \frac{c0}{w + w}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \color{blue}{\frac{c0}{w + w}} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \frac{c0}{\color{blue}{w + w}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\sqrt{\left(-M\right) \cdot M} \cdot c0}{\color{blue}{w + w}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\left(-M\right) \cdot M} \cdot c0}{\color{blue}{w + w}} \]

      5. lower-*.f64 14.0%

         \[\leadsto \frac{\sqrt{\left(-M\right) \cdot M} \cdot c0}{\color{blue}{w} + w} \]

   10. Applied rewrites 14.0%

       \[\leadsto \frac{\sqrt{\left(-M\right) \cdot M} \cdot c0}{\color{blue}{w + w}} \]

   if 4.19999999999999996e-163 < M < 1.55e139

   1. Initial program 24.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{c0 \cdot \left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      7. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      8. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(w \cdot h\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      18. lower-/.f64 24.1%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \color{blue}{\frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Applied rewrites 24.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\color{blue}{c0 \cdot \left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      7. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      8. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(w \cdot h\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      18. lower-/.f64 24.2%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \color{blue}{\frac{d}{D}}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   5. Applied rewrites 24.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\color{blue}{c0 \cdot \left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} - M \cdot M}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} - M \cdot M}\right) \]

      7. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} - M \cdot M}\right) \]

      8. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} - M \cdot M}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} - M \cdot M}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(w \cdot h\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      18. lower-/.f64 34.2%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \color{blue}{\frac{d}{D}}\right) - M \cdot M}\right) \]

   7. Applied rewrites 34.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)} - M \cdot M}\right) \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(h \cdot w\right)}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      3. associate-/r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{d \cdot c0}{D}}{h \cdot w}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot c0}{D}}{\color{blue}{h \cdot w}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      5. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot c0}{D}}{\color{blue}{w \cdot h}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      6. associate-/r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{\frac{d \cdot c0}{D}}{w}}{h}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{\frac{d \cdot c0}{D}}{w}}{h}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\frac{\frac{d \cdot c0}{D}}{w}}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{\frac{\color{blue}{d \cdot c0}}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      10. associate-/l* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \color{blue}{\frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      12. lower-*.f64 32.0%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

   9. Applied rewrites 32.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(h \cdot w\right)}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       3. associate-/r* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\color{blue}{\frac{\frac{d \cdot c0}{D}}{h \cdot w}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot c0}{D}}{\color{blue}{h \cdot w}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       5. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot c0}{D}}{\color{blue}{w \cdot h}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       6. associate-/r* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\color{blue}{\frac{\frac{\frac{d \cdot c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       7. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\color{blue}{\frac{\frac{\frac{d \cdot c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\color{blue}{\frac{\frac{d \cdot c0}{D}}{w}}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{\frac{\color{blue}{d \cdot c0}}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       10. associate-/l* N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       11. lift-/.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \color{blue}{\frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       12. lower-*.f64 32.0%

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

   11. Applied rewrites 32.0%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]
   12. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(h \cdot w\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       3. associate-/r* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\color{blue}{\frac{\frac{d \cdot c0}{D}}{h \cdot w}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot c0}{D}}{\color{blue}{h \cdot w}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       5. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot c0}{D}}{\color{blue}{w \cdot h}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       6. associate-/r* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\color{blue}{\frac{\frac{\frac{d \cdot c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       7. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\color{blue}{\frac{\frac{\frac{d \cdot c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\color{blue}{\frac{\frac{d \cdot c0}{D}}{w}}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{\frac{\color{blue}{d \cdot c0}}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       10. associate-/l* N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       11. lift-/.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \color{blue}{\frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       12. lower-*.f64 36.3%

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

   13. Applied rewrites 36.3%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]
   14. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\frac{d \cdot \frac{c0}{D}}{w}}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       3. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{\color{blue}{\frac{c0}{D} \cdot d}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       4. associate-/l* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\frac{c0}{D} \cdot \frac{d}{w}}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\frac{c0}{D} \cdot \frac{d}{w}}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       6. lower-/.f64 34.6%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D} \cdot \color{blue}{\frac{d}{w}}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

   15. Applied rewrites 34.6%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\frac{c0}{D} \cdot \frac{d}{w}}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]
   16. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D} \cdot \frac{d}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\color{blue}{\frac{d \cdot \frac{c0}{D}}{w}}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D} \cdot \frac{d}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       3. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D} \cdot \frac{d}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{\color{blue}{\frac{c0}{D} \cdot d}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       4. associate-/l* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D} \cdot \frac{d}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\color{blue}{\frac{c0}{D} \cdot \frac{d}{w}}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D} \cdot \frac{d}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\color{blue}{\frac{c0}{D} \cdot \frac{d}{w}}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       6. lower-/.f64 34.8%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D} \cdot \frac{d}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{c0}{D} \cdot \color{blue}{\frac{d}{w}}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

   17. Applied rewrites 34.8%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D} \cdot \frac{d}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\color{blue}{\frac{c0}{D} \cdot \frac{d}{w}}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]
   18. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D} \cdot \frac{d}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{c0}{D} \cdot \frac{d}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\color{blue}{\frac{d \cdot \frac{c0}{D}}{w}}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D} \cdot \frac{d}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{c0}{D} \cdot \frac{d}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       3. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D} \cdot \frac{d}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{c0}{D} \cdot \frac{d}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{\color{blue}{\frac{c0}{D} \cdot d}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       4. associate-/l* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D} \cdot \frac{d}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{c0}{D} \cdot \frac{d}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\color{blue}{\frac{c0}{D} \cdot \frac{d}{w}}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D} \cdot \frac{d}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{c0}{D} \cdot \frac{d}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\color{blue}{\frac{c0}{D} \cdot \frac{d}{w}}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       6. lower-/.f64 35.5%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D} \cdot \frac{d}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{c0}{D} \cdot \frac{d}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{c0}{D} \cdot \color{blue}{\frac{d}{w}}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

   19. Applied rewrites 35.5%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D} \cdot \frac{d}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{c0}{D} \cdot \frac{d}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\color{blue}{\frac{c0}{D} \cdot \frac{d}{w}}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

   if 1.55e139 < M

   1. Initial program 24.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. lower-pow.f64 14.0%

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   4. Applied rewrites 14.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      2. pow1/2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(-{M}^{2}\right)}^{\frac{1}{2}}}{w} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      5. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

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

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      9. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      11. lower-*.f64 21.9%

          \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]

   6. Applied rewrites 21.9%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 41.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\\ t_1 := \left(-\left|M\right|\right) \cdot \left|M\right|\\ \mathbf{if}\;\left|M\right| \leq 5.5 \cdot 10^{-163}:\\ \;\;\;\;\frac{\sqrt{t\_1} \cdot c0}{w + w}\\ \mathbf{elif}\;\left|M\right| \leq 5.8 \cdot 10^{+138}:\\ \;\;\;\;\frac{c0}{w + w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - \left|M\right| \cdot \left|M\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot {t\_1}^{0.5}}{w}\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ (* (/ c0 (* D w)) d) D) (/ d h)))
        (t_1 (* (- (fabs M)) (fabs M))))
   (if (<= (fabs M) 5.5e-163)
     (/ (* (sqrt t_1) c0) (+ w w))
     (if (<= (fabs M) 5.8e+138)
       (* (/ c0 (+ w w)) (+ t_0 (sqrt (- (* t_0 t_0) (* (fabs M) (fabs M))))))
       (* 0.5 (/ (* c0 (pow t_1 0.5)) w))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (((c0 / (D * w)) * d) / D) * (d / h);
	double t_1 = -fabs(M) * fabs(M);
	double tmp;
	if (fabs(M) <= 5.5e-163) {
		tmp = (sqrt(t_1) * c0) / (w + w);
	} else if (fabs(M) <= 5.8e+138) {
		tmp = (c0 / (w + w)) * (t_0 + sqrt(((t_0 * t_0) - (fabs(M) * fabs(M)))));
	} else {
		tmp = 0.5 * ((c0 * pow(t_1, 0.5)) / w);
	}
	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(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((c0 / (d * w)) * d_1) / d) * (d_1 / h)
    t_1 = -abs(m) * abs(m)
    if (abs(m) <= 5.5d-163) then
        tmp = (sqrt(t_1) * c0) / (w + w)
    else if (abs(m) <= 5.8d+138) then
        tmp = (c0 / (w + w)) * (t_0 + sqrt(((t_0 * t_0) - (abs(m) * abs(m)))))
    else
        tmp = 0.5d0 * ((c0 * (t_1 ** 0.5d0)) / w)
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (((c0 / (D * w)) * d) / D) * (d / h);
	double t_1 = -Math.abs(M) * Math.abs(M);
	double tmp;
	if (Math.abs(M) <= 5.5e-163) {
		tmp = (Math.sqrt(t_1) * c0) / (w + w);
	} else if (Math.abs(M) <= 5.8e+138) {
		tmp = (c0 / (w + w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (Math.abs(M) * Math.abs(M)))));
	} else {
		tmp = 0.5 * ((c0 * Math.pow(t_1, 0.5)) / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (((c0 / (D * w)) * d) / D) * (d / h)
	t_1 = -math.fabs(M) * math.fabs(M)
	tmp = 0
	if math.fabs(M) <= 5.5e-163:
		tmp = (math.sqrt(t_1) * c0) / (w + w)
	elif math.fabs(M) <= 5.8e+138:
		tmp = (c0 / (w + w)) * (t_0 + math.sqrt(((t_0 * t_0) - (math.fabs(M) * math.fabs(M)))))
	else:
		tmp = 0.5 * ((c0 * math.pow(t_1, 0.5)) / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(Float64(Float64(c0 / Float64(D * w)) * d) / D) * Float64(d / h))
	t_1 = Float64(Float64(-abs(M)) * abs(M))
	tmp = 0.0
	if (abs(M) <= 5.5e-163)
		tmp = Float64(Float64(sqrt(t_1) * c0) / Float64(w + w));
	elseif (abs(M) <= 5.8e+138)
		tmp = Float64(Float64(c0 / Float64(w + w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(abs(M) * abs(M))))));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * (t_1 ^ 0.5)) / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (((c0 / (D * w)) * d) / D) * (d / h);
	t_1 = -abs(M) * abs(M);
	tmp = 0.0;
	if (abs(M) <= 5.5e-163)
		tmp = (sqrt(t_1) * c0) / (w + w);
	elseif (abs(M) <= 5.8e+138)
		tmp = (c0 / (w + w)) * (t_0 + sqrt(((t_0 * t_0) - (abs(M) * abs(M)))));
	else
		tmp = 0.5 * ((c0 * (t_1 ^ 0.5)) / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(N[(N[(c0 / N[(D * w), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision] / D), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-N[Abs[M], $MachinePrecision]) * N[Abs[M], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[M], $MachinePrecision], 5.5e-163], N[(N[(N[Sqrt[t$95$1], $MachinePrecision] * c0), $MachinePrecision] / N[(w + w), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[M], $MachinePrecision], 5.8e+138], N[(N[(c0 / N[(w + w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(N[Abs[M], $MachinePrecision] * N[Abs[M], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Power[t$95$1, 0.5], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if M < 5.4999999999999998e-163

   1. Initial program 24.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. lower-pow.f64 14.0%

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   4. Applied rewrites 14.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{\color{blue}{w}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(c0 \cdot \sqrt{-{M}^{2}}\right)}{\color{blue}{w}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(c0 \cdot \sqrt{-{M}^{2}}\right)}{\color{blue}{w}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(c0 \cdot \sqrt{-{M}^{2}}\right)}{w} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{-{M}^{2}}}{w} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{-{M}^{2}}}{w} \]

      8. lower-*.f64 14.0%

         \[\leadsto \frac{\left(0.5 \cdot c0\right) \cdot \sqrt{-{M}^{2}}}{w} \]

      9. lift-neg.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      11. pow2 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}{w} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}{w} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}{w} \]

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

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\left(\mathsf{neg}\left(M\right)\right) \cdot M}}{w} \]

      15. lift-neg.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

      16. lower-*.f64 14.0%

          \[\leadsto \frac{\left(0.5 \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

   6. Applied rewrites 14.0%

      \[\leadsto \frac{\left(0.5 \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{\color{blue}{w}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot \color{blue}{c0}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

      2. lift-*.f64 14.0%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{c0}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot \color{blue}{c0}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

      4. lift-*.f64 14.0%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{c0}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{\color{blue}{w}} \]

   8. Applied rewrites 12.1%

      \[\leadsto \color{blue}{\sqrt{\left(-M\right) \cdot M} \cdot \frac{c0}{w + w}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \color{blue}{\frac{c0}{w + w}} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \frac{c0}{\color{blue}{w + w}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\sqrt{\left(-M\right) \cdot M} \cdot c0}{\color{blue}{w + w}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\left(-M\right) \cdot M} \cdot c0}{\color{blue}{w + w}} \]

      5. lower-*.f64 14.0%

         \[\leadsto \frac{\sqrt{\left(-M\right) \cdot M} \cdot c0}{\color{blue}{w} + w} \]

   10. Applied rewrites 14.0%

       \[\leadsto \frac{\sqrt{\left(-M\right) \cdot M} \cdot c0}{\color{blue}{w + w}} \]

   if 5.4999999999999998e-163 < M < 5.80000000000000019e138

   1. Initial program 24.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{c0 \cdot \left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      7. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      8. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(w \cdot h\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      18. lower-/.f64 24.1%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \color{blue}{\frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Applied rewrites 24.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\color{blue}{c0 \cdot \left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      7. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      8. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(w \cdot h\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      18. lower-/.f64 24.2%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \color{blue}{\frac{d}{D}}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   5. Applied rewrites 24.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\color{blue}{c0 \cdot \left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} - M \cdot M}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} - M \cdot M}\right) \]

      7. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} - M \cdot M}\right) \]

      8. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} - M \cdot M}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} - M \cdot M}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(w \cdot h\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      18. lower-/.f64 34.2%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \color{blue}{\frac{d}{D}}\right) - M \cdot M}\right) \]

   7. Applied rewrites 34.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)} - M \cdot M}\right) \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(h \cdot w\right)}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      3. associate-/r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{d \cdot c0}{D}}{h \cdot w}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot c0}{D}}{\color{blue}{h \cdot w}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      5. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot c0}{D}}{\color{blue}{w \cdot h}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      6. associate-/r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{\frac{d \cdot c0}{D}}{w}}{h}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{\frac{d \cdot c0}{D}}{w}}{h}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\frac{\frac{d \cdot c0}{D}}{w}}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{\frac{\color{blue}{d \cdot c0}}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      10. associate-/l* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \color{blue}{\frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      12. lower-*.f64 32.0%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

   9. Applied rewrites 32.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(h \cdot w\right)}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       3. associate-/r* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\color{blue}{\frac{\frac{d \cdot c0}{D}}{h \cdot w}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot c0}{D}}{\color{blue}{h \cdot w}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       5. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot c0}{D}}{\color{blue}{w \cdot h}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       6. associate-/r* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\color{blue}{\frac{\frac{\frac{d \cdot c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       7. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\color{blue}{\frac{\frac{\frac{d \cdot c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\color{blue}{\frac{\frac{d \cdot c0}{D}}{w}}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{\frac{\color{blue}{d \cdot c0}}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       10. associate-/l* N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       11. lift-/.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \color{blue}{\frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       12. lower-*.f64 32.0%

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

   11. Applied rewrites 32.0%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]
   12. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(h \cdot w\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       3. associate-/r* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\color{blue}{\frac{\frac{d \cdot c0}{D}}{h \cdot w}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot c0}{D}}{\color{blue}{h \cdot w}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       5. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot c0}{D}}{\color{blue}{w \cdot h}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       6. associate-/r* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\color{blue}{\frac{\frac{\frac{d \cdot c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       7. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\color{blue}{\frac{\frac{\frac{d \cdot c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\color{blue}{\frac{\frac{d \cdot c0}{D}}{w}}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{\frac{\color{blue}{d \cdot c0}}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       10. associate-/l* N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       11. lift-/.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \color{blue}{\frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       12. lower-*.f64 36.3%

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

   13. Applied rewrites 36.3%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]
   14. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h}} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       3. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \color{blue}{\frac{d}{D}} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       4. frac-times N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w} \cdot d}{h \cdot D}} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       5. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w} \cdot d}{\color{blue}{D \cdot h}} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       6. times-frac N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{D} \cdot \frac{d}{h}} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{D} \cdot \frac{d}{h}} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{D}} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       9. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\frac{d \cdot \frac{c0}{D}}{w}}}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       11. associate-/l* N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{d \cdot \frac{\frac{c0}{D}}{w}}}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       12. *-commutative N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\frac{\frac{c0}{D}}{w} \cdot d}}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       13. lower-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\frac{\frac{c0}{D}}{w} \cdot d}}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       14. lift-/.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{\color{blue}{\frac{c0}{D}}}{w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       15. associate-/l/ N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\frac{c0}{D \cdot w}} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       16. lift-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{\color{blue}{D \cdot w}} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       17. lower-/.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\frac{c0}{D \cdot w}} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       18. lower-/.f64 32.4%

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \color{blue}{\frac{d}{h}} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

   15. Applied rewrites 32.4%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]
   16. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\color{blue}{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right)} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       3. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \color{blue}{\frac{d}{D}}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       4. frac-times N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w} \cdot d}{h \cdot D}} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       5. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\frac{\frac{d \cdot \frac{c0}{D}}{w} \cdot d}{\color{blue}{D \cdot h}} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       6. times-frac N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\color{blue}{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{D} \cdot \frac{d}{h}\right)} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\color{blue}{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{D} \cdot \frac{d}{h}\right)} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{D}} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       9. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\color{blue}{\frac{d \cdot \frac{c0}{D}}{w}}}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       11. associate-/l* N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\color{blue}{d \cdot \frac{\frac{c0}{D}}{w}}}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       12. *-commutative N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\color{blue}{\frac{\frac{c0}{D}}{w} \cdot d}}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       13. lower-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\color{blue}{\frac{\frac{c0}{D}}{w} \cdot d}}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       14. lift-/.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{\color{blue}{\frac{c0}{D}}}{w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       15. associate-/l/ N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\color{blue}{\frac{c0}{D \cdot w}} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       16. lift-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{\color{blue}{D \cdot w}} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       17. lower-/.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\color{blue}{\frac{c0}{D \cdot w}} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       18. lower-/.f64 32.6%

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \color{blue}{\frac{d}{h}}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

   17. Applied rewrites 32.6%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\color{blue}{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right)} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]
   18. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \color{blue}{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right)} - M \cdot M}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       3. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \color{blue}{\frac{d}{D}}\right) - M \cdot M}\right) \]

       4. frac-times N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w} \cdot d}{h \cdot D}} - M \cdot M}\right) \]

       5. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \frac{\frac{d \cdot \frac{c0}{D}}{w} \cdot d}{\color{blue}{D \cdot h}} - M \cdot M}\right) \]

       6. times-frac N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \color{blue}{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{D} \cdot \frac{d}{h}\right)} - M \cdot M}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \color{blue}{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{D} \cdot \frac{d}{h}\right)} - M \cdot M}\right) \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{D}} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       9. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\color{blue}{\frac{d \cdot \frac{c0}{D}}{w}}}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       11. associate-/l* N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\color{blue}{d \cdot \frac{\frac{c0}{D}}{w}}}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       12. *-commutative N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\color{blue}{\frac{\frac{c0}{D}}{w} \cdot d}}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       13. lower-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\color{blue}{\frac{\frac{c0}{D}}{w} \cdot d}}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       14. lift-/.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{\color{blue}{\frac{c0}{D}}}{w} \cdot d}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       15. associate-/l/ N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\color{blue}{\frac{c0}{D \cdot w}} \cdot d}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       16. lift-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{c0}{\color{blue}{D \cdot w}} \cdot d}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       17. lower-/.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\color{blue}{\frac{c0}{D \cdot w}} \cdot d}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       18. lower-/.f64 34.8%

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \color{blue}{\frac{d}{h}}\right) - M \cdot M}\right) \]

   19. Applied rewrites 34.8%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \color{blue}{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right)} - M \cdot M}\right) \]
   20. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{c0}{2 \cdot w}} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       2. mult-flip N/A

          \[\leadsto \color{blue}{\left(c0 \cdot \frac{1}{2 \cdot w}\right)} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(c0 \cdot \frac{1}{\color{blue}{2 \cdot w}}\right) \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       4. count-2-rev N/A

          \[\leadsto \left(c0 \cdot \frac{1}{\color{blue}{w + w}}\right) \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       5. lift-+.f64 N/A

          \[\leadsto \left(c0 \cdot \frac{1}{\color{blue}{w + w}}\right) \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       6. mult-flip N/A

          \[\leadsto \color{blue}{\frac{c0}{w + w}} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       7. lift-/.f64 34.8%

          \[\leadsto \color{blue}{\frac{c0}{w + w}} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

   21. Applied rewrites 34.8%

       \[\leadsto \color{blue}{\frac{c0}{w + w}} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

   if 5.80000000000000019e138 < M

   1. Initial program 24.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. lower-pow.f64 14.0%

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   4. Applied rewrites 14.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      2. pow1/2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(-{M}^{2}\right)}^{\frac{1}{2}}}{w} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      5. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

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

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      9. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      11. lower-*.f64 21.9%

          \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]

   6. Applied rewrites 21.9%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 41.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right)\\ t_1 := \left(-\left|M\right|\right) \cdot \left|M\right|\\ \mathbf{if}\;\left|M\right| \leq 6.4 \cdot 10^{-163}:\\ \;\;\;\;\frac{\sqrt{t\_1} \cdot c0}{w + w}\\ \mathbf{elif}\;\left|M\right| \leq 5.8 \cdot 10^{+138}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - \left|M\right| \cdot \left|M\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot {t\_1}^{0.5}}{w}\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ d (* h D)) (* (/ d (* D w)) c0)))
        (t_1 (* (- (fabs M)) (fabs M))))
   (if (<= (fabs M) 6.4e-163)
     (/ (* (sqrt t_1) c0) (+ w w))
     (if (<= (fabs M) 5.8e+138)
       (*
        (/ c0 (* 2.0 w))
        (+ t_0 (sqrt (- (* t_0 t_0) (* (fabs M) (fabs M))))))
       (* 0.5 (/ (* c0 (pow t_1 0.5)) w))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (d / (h * D)) * ((d / (D * w)) * c0);
	double t_1 = -fabs(M) * fabs(M);
	double tmp;
	if (fabs(M) <= 6.4e-163) {
		tmp = (sqrt(t_1) * c0) / (w + w);
	} else if (fabs(M) <= 5.8e+138) {
		tmp = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (fabs(M) * fabs(M)))));
	} else {
		tmp = 0.5 * ((c0 * pow(t_1, 0.5)) / w);
	}
	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(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (d_1 / (h * d)) * ((d_1 / (d * w)) * c0)
    t_1 = -abs(m) * abs(m)
    if (abs(m) <= 6.4d-163) then
        tmp = (sqrt(t_1) * c0) / (w + w)
    else if (abs(m) <= 5.8d+138) then
        tmp = (c0 / (2.0d0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (abs(m) * abs(m)))))
    else
        tmp = 0.5d0 * ((c0 * (t_1 ** 0.5d0)) / w)
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (d / (h * D)) * ((d / (D * w)) * c0);
	double t_1 = -Math.abs(M) * Math.abs(M);
	double tmp;
	if (Math.abs(M) <= 6.4e-163) {
		tmp = (Math.sqrt(t_1) * c0) / (w + w);
	} else if (Math.abs(M) <= 5.8e+138) {
		tmp = (c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (Math.abs(M) * Math.abs(M)))));
	} else {
		tmp = 0.5 * ((c0 * Math.pow(t_1, 0.5)) / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (d / (h * D)) * ((d / (D * w)) * c0)
	t_1 = -math.fabs(M) * math.fabs(M)
	tmp = 0
	if math.fabs(M) <= 6.4e-163:
		tmp = (math.sqrt(t_1) * c0) / (w + w)
	elif math.fabs(M) <= 5.8e+138:
		tmp = (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (math.fabs(M) * math.fabs(M)))))
	else:
		tmp = 0.5 * ((c0 * math.pow(t_1, 0.5)) / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(d / Float64(h * D)) * Float64(Float64(d / Float64(D * w)) * c0))
	t_1 = Float64(Float64(-abs(M)) * abs(M))
	tmp = 0.0
	if (abs(M) <= 6.4e-163)
		tmp = Float64(Float64(sqrt(t_1) * c0) / Float64(w + w));
	elseif (abs(M) <= 5.8e+138)
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(abs(M) * abs(M))))));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * (t_1 ^ 0.5)) / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (d / (h * D)) * ((d / (D * w)) * c0);
	t_1 = -abs(M) * abs(M);
	tmp = 0.0;
	if (abs(M) <= 6.4e-163)
		tmp = (sqrt(t_1) * c0) / (w + w);
	elseif (abs(M) <= 5.8e+138)
		tmp = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (abs(M) * abs(M)))));
	else
		tmp = 0.5 * ((c0 * (t_1 ^ 0.5)) / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(d / N[(h * D), $MachinePrecision]), $MachinePrecision] * N[(N[(d / N[(D * w), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-N[Abs[M], $MachinePrecision]) * N[Abs[M], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[M], $MachinePrecision], 6.4e-163], N[(N[(N[Sqrt[t$95$1], $MachinePrecision] * c0), $MachinePrecision] / N[(w + w), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[M], $MachinePrecision], 5.8e+138], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(N[Abs[M], $MachinePrecision] * N[Abs[M], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Power[t$95$1, 0.5], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if M < 6.39999999999999976e-163

   1. Initial program 24.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. lower-pow.f64 14.0%

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   4. Applied rewrites 14.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{\color{blue}{w}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(c0 \cdot \sqrt{-{M}^{2}}\right)}{\color{blue}{w}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(c0 \cdot \sqrt{-{M}^{2}}\right)}{\color{blue}{w}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(c0 \cdot \sqrt{-{M}^{2}}\right)}{w} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{-{M}^{2}}}{w} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{-{M}^{2}}}{w} \]

      8. lower-*.f64 14.0%

         \[\leadsto \frac{\left(0.5 \cdot c0\right) \cdot \sqrt{-{M}^{2}}}{w} \]

      9. lift-neg.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      11. pow2 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}{w} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}{w} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}{w} \]

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

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\left(\mathsf{neg}\left(M\right)\right) \cdot M}}{w} \]

      15. lift-neg.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

      16. lower-*.f64 14.0%

          \[\leadsto \frac{\left(0.5 \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

   6. Applied rewrites 14.0%

      \[\leadsto \frac{\left(0.5 \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{\color{blue}{w}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot \color{blue}{c0}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

      2. lift-*.f64 14.0%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{c0}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot \color{blue}{c0}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

      4. lift-*.f64 14.0%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{c0}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{\color{blue}{w}} \]

   8. Applied rewrites 12.1%

      \[\leadsto \color{blue}{\sqrt{\left(-M\right) \cdot M} \cdot \frac{c0}{w + w}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \color{blue}{\frac{c0}{w + w}} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \frac{c0}{\color{blue}{w + w}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\sqrt{\left(-M\right) \cdot M} \cdot c0}{\color{blue}{w + w}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\left(-M\right) \cdot M} \cdot c0}{\color{blue}{w + w}} \]

      5. lower-*.f64 14.0%

         \[\leadsto \frac{\sqrt{\left(-M\right) \cdot M} \cdot c0}{\color{blue}{w} + w} \]

   10. Applied rewrites 14.0%

       \[\leadsto \frac{\sqrt{\left(-M\right) \cdot M} \cdot c0}{\color{blue}{w + w}} \]

   if 6.39999999999999976e-163 < M < 5.80000000000000019e138

   1. Initial program 24.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{c0 \cdot \left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      7. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      8. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(w \cdot h\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      18. lower-/.f64 24.1%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \color{blue}{\frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Applied rewrites 24.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\color{blue}{c0 \cdot \left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      7. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      8. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(w \cdot h\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      18. lower-/.f64 24.2%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \color{blue}{\frac{d}{D}}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   5. Applied rewrites 24.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\color{blue}{c0 \cdot \left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} - M \cdot M}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} - M \cdot M}\right) \]

      7. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} - M \cdot M}\right) \]

      8. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} - M \cdot M}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} - M \cdot M}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(w \cdot h\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      18. lower-/.f64 34.2%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \color{blue}{\frac{d}{D}}\right) - M \cdot M}\right) \]

   7. Applied rewrites 34.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)} - M \cdot M}\right) \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(h \cdot w\right)}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      3. associate-/r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{d \cdot c0}{D}}{h \cdot w}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot c0}{D}}{\color{blue}{h \cdot w}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      5. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot c0}{D}}{\color{blue}{w \cdot h}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      6. associate-/r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{\frac{d \cdot c0}{D}}{w}}{h}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{\frac{d \cdot c0}{D}}{w}}{h}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\frac{\frac{d \cdot c0}{D}}{w}}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{\frac{\color{blue}{d \cdot c0}}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      10. associate-/l* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \color{blue}{\frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      12. lower-*.f64 32.0%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

   9. Applied rewrites 32.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(h \cdot w\right)}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       3. associate-/r* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\color{blue}{\frac{\frac{d \cdot c0}{D}}{h \cdot w}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot c0}{D}}{\color{blue}{h \cdot w}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       5. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot c0}{D}}{\color{blue}{w \cdot h}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       6. associate-/r* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\color{blue}{\frac{\frac{\frac{d \cdot c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       7. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\color{blue}{\frac{\frac{\frac{d \cdot c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\color{blue}{\frac{\frac{d \cdot c0}{D}}{w}}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{\frac{\color{blue}{d \cdot c0}}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       10. associate-/l* N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       11. lift-/.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \color{blue}{\frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       12. lower-*.f64 32.0%

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

   11. Applied rewrites 32.0%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]
   12. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(h \cdot w\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       3. associate-/r* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\color{blue}{\frac{\frac{d \cdot c0}{D}}{h \cdot w}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot c0}{D}}{\color{blue}{h \cdot w}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       5. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot c0}{D}}{\color{blue}{w \cdot h}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       6. associate-/r* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\color{blue}{\frac{\frac{\frac{d \cdot c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       7. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\color{blue}{\frac{\frac{\frac{d \cdot c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\color{blue}{\frac{\frac{d \cdot c0}{D}}{w}}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{\frac{\color{blue}{d \cdot c0}}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       10. associate-/l* N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       11. lift-/.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \color{blue}{\frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       12. lower-*.f64 36.3%

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

   13. Applied rewrites 36.3%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]
   14. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h}} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       3. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \color{blue}{\frac{d}{D}} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       4. frac-times N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w} \cdot d}{h \cdot D}} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w} \cdot d}{\color{blue}{h \cdot D}} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       6. associate-/l* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{d \cdot \frac{c0}{D}}{w} \cdot \frac{d}{h \cdot D}} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       7. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{d \cdot \frac{c0}{D}}{w}} \cdot \frac{d}{h \cdot D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       8. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w} \cdot \frac{d}{h \cdot D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       9. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot \color{blue}{\frac{c0}{D}}}{w} \cdot \frac{d}{h \cdot D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       10. associate-*r/ N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\frac{d \cdot c0}{D}}}{w} \cdot \frac{d}{h \cdot D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       11. lift-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{\color{blue}{d \cdot c0}}{D}}{w} \cdot \frac{d}{h \cdot D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       12. associate-/l/ N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{d \cdot c0}{D \cdot w}} \cdot \frac{d}{h \cdot D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       13. lift-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot w}} \cdot \frac{d}{h \cdot D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       14. times-frac N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\left(d \cdot c0\right) \cdot d}{\left(D \cdot w\right) \cdot \left(h \cdot D\right)}} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       15. *-commutative N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left(d \cdot c0\right) \cdot d}{\color{blue}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)}} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       16. *-commutative N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{d \cdot \left(d \cdot c0\right)}}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       17. lift-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot \color{blue}{\left(d \cdot c0\right)}}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       18. associate-*l* N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\left(d \cdot d\right) \cdot c0}}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       19. lift-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\left(d \cdot d\right)} \cdot c0}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       20. lift-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left(d \cdot d\right) \cdot c0}{\color{blue}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)}} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       21. associate-*l/ N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

   15. Applied rewrites 31.8%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right)} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]
   16. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right) + \sqrt{\color{blue}{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right)} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right) + \sqrt{\left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       3. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right) + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \color{blue}{\frac{d}{D}}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       4. frac-times N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right) + \sqrt{\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w} \cdot d}{h \cdot D}} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right) + \sqrt{\frac{\frac{d \cdot \frac{c0}{D}}{w} \cdot d}{\color{blue}{h \cdot D}} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       6. associate-/l* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right) + \sqrt{\color{blue}{\left(\frac{d \cdot \frac{c0}{D}}{w} \cdot \frac{d}{h \cdot D}\right)} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       7. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right) + \sqrt{\left(\color{blue}{\frac{d \cdot \frac{c0}{D}}{w}} \cdot \frac{d}{h \cdot D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       8. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right) + \sqrt{\left(\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w} \cdot \frac{d}{h \cdot D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       9. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right) + \sqrt{\left(\frac{d \cdot \color{blue}{\frac{c0}{D}}}{w} \cdot \frac{d}{h \cdot D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       10. associate-*r/ N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right) + \sqrt{\left(\frac{\color{blue}{\frac{d \cdot c0}{D}}}{w} \cdot \frac{d}{h \cdot D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       11. lift-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right) + \sqrt{\left(\frac{\frac{\color{blue}{d \cdot c0}}{D}}{w} \cdot \frac{d}{h \cdot D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       12. associate-/l/ N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right) + \sqrt{\left(\color{blue}{\frac{d \cdot c0}{D \cdot w}} \cdot \frac{d}{h \cdot D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       13. lift-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right) + \sqrt{\left(\frac{d \cdot c0}{\color{blue}{D \cdot w}} \cdot \frac{d}{h \cdot D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       14. times-frac N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right) + \sqrt{\color{blue}{\frac{\left(d \cdot c0\right) \cdot d}{\left(D \cdot w\right) \cdot \left(h \cdot D\right)}} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       15. *-commutative N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right) + \sqrt{\frac{\left(d \cdot c0\right) \cdot d}{\color{blue}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)}} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       16. *-commutative N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right) + \sqrt{\frac{\color{blue}{d \cdot \left(d \cdot c0\right)}}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       17. lift-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right) + \sqrt{\frac{d \cdot \color{blue}{\left(d \cdot c0\right)}}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       18. associate-*l* N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right) + \sqrt{\frac{\color{blue}{\left(d \cdot d\right) \cdot c0}}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       19. lift-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right) + \sqrt{\frac{\color{blue}{\left(d \cdot d\right)} \cdot c0}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       20. lift-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right) + \sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\color{blue}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)}} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       21. associate-*l/ N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right) + \sqrt{\color{blue}{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

   17. Applied rewrites 31.9%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right) + \sqrt{\color{blue}{\left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right)\right)} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]
   18. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right) + \sqrt{\left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right)\right) \cdot \color{blue}{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right)} - M \cdot M}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right) + \sqrt{\left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right)\right) \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       3. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right) + \sqrt{\left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right)\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \color{blue}{\frac{d}{D}}\right) - M \cdot M}\right) \]

       4. frac-times N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right) + \sqrt{\left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right)\right) \cdot \color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w} \cdot d}{h \cdot D}} - M \cdot M}\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right) + \sqrt{\left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right)\right) \cdot \frac{\frac{d \cdot \frac{c0}{D}}{w} \cdot d}{\color{blue}{h \cdot D}} - M \cdot M}\right) \]

       6. associate-/l* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right) + \sqrt{\left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right)\right) \cdot \color{blue}{\left(\frac{d \cdot \frac{c0}{D}}{w} \cdot \frac{d}{h \cdot D}\right)} - M \cdot M}\right) \]

       7. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right) + \sqrt{\left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right)\right) \cdot \left(\color{blue}{\frac{d \cdot \frac{c0}{D}}{w}} \cdot \frac{d}{h \cdot D}\right) - M \cdot M}\right) \]

       8. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right) + \sqrt{\left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right)\right) \cdot \left(\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w} \cdot \frac{d}{h \cdot D}\right) - M \cdot M}\right) \]

       9. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right) + \sqrt{\left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right)\right) \cdot \left(\frac{d \cdot \color{blue}{\frac{c0}{D}}}{w} \cdot \frac{d}{h \cdot D}\right) - M \cdot M}\right) \]

       10. associate-*r/ N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right) + \sqrt{\left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right)\right) \cdot \left(\frac{\color{blue}{\frac{d \cdot c0}{D}}}{w} \cdot \frac{d}{h \cdot D}\right) - M \cdot M}\right) \]

       11. lift-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right) + \sqrt{\left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right)\right) \cdot \left(\frac{\frac{\color{blue}{d \cdot c0}}{D}}{w} \cdot \frac{d}{h \cdot D}\right) - M \cdot M}\right) \]

       12. associate-/l/ N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right) + \sqrt{\left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right)\right) \cdot \left(\color{blue}{\frac{d \cdot c0}{D \cdot w}} \cdot \frac{d}{h \cdot D}\right) - M \cdot M}\right) \]

       13. lift-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right) + \sqrt{\left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right)\right) \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot w}} \cdot \frac{d}{h \cdot D}\right) - M \cdot M}\right) \]

       14. times-frac N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right) + \sqrt{\left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right)\right) \cdot \color{blue}{\frac{\left(d \cdot c0\right) \cdot d}{\left(D \cdot w\right) \cdot \left(h \cdot D\right)}} - M \cdot M}\right) \]

       15. *-commutative N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right) + \sqrt{\left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right)\right) \cdot \frac{\left(d \cdot c0\right) \cdot d}{\color{blue}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)}} - M \cdot M}\right) \]

       16. *-commutative N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right) + \sqrt{\left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right)\right) \cdot \frac{\color{blue}{d \cdot \left(d \cdot c0\right)}}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} - M \cdot M}\right) \]

       17. lift-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right) + \sqrt{\left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right)\right) \cdot \frac{d \cdot \color{blue}{\left(d \cdot c0\right)}}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} - M \cdot M}\right) \]

       18. associate-*l* N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right) + \sqrt{\left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right)\right) \cdot \frac{\color{blue}{\left(d \cdot d\right) \cdot c0}}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} - M \cdot M}\right) \]

       19. lift-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right) + \sqrt{\left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right)\right) \cdot \frac{\color{blue}{\left(d \cdot d\right)} \cdot c0}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} - M \cdot M}\right) \]

       20. lift-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right) + \sqrt{\left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right)\right) \cdot \frac{\left(d \cdot d\right) \cdot c0}{\color{blue}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)}} - M \cdot M}\right) \]

       21. associate-*l/ N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right) + \sqrt{\left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right)\right) \cdot \color{blue}{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)} - M \cdot M}\right) \]

   19. Applied rewrites 34.5%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right) + \sqrt{\left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right)\right) \cdot \color{blue}{\left(\frac{d}{h \cdot D} \cdot \left(\frac{d}{D \cdot w} \cdot c0\right)\right)} - M \cdot M}\right) \]

   if 5.80000000000000019e138 < M

   1. Initial program 24.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. lower-pow.f64 14.0%

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   4. Applied rewrites 14.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      2. pow1/2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(-{M}^{2}\right)}^{\frac{1}{2}}}{w} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      5. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

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

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      9. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      11. lower-*.f64 21.9%

          \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]

   6. Applied rewrites 21.9%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 41.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}\\ t_1 := \left(-\left|M\right|\right) \cdot \left|M\right|\\ \mathbf{if}\;\left|M\right| \leq 6.4 \cdot 10^{-163}:\\ \;\;\;\;\frac{\sqrt{t\_1} \cdot c0}{w + w}\\ \mathbf{elif}\;\left|M\right| \leq 5.8 \cdot 10^{+138}:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(t\_0, c0, \sqrt{{\left(t\_0 \cdot c0\right)}^{2} - \left|M\right| \cdot \left|M\right|}\right)}{w + w}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot {t\_1}^{0.5}}{w}\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ d (* (* h w) D)) (/ d D))) (t_1 (* (- (fabs M)) (fabs M))))
   (if (<= (fabs M) 6.4e-163)
     (/ (* (sqrt t_1) c0) (+ w w))
     (if (<= (fabs M) 5.8e+138)
       (*
        c0
        (/
         (fma t_0 c0 (sqrt (- (pow (* t_0 c0) 2.0) (* (fabs M) (fabs M)))))
         (+ w w)))
       (* 0.5 (/ (* c0 (pow t_1 0.5)) w))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (d / ((h * w) * D)) * (d / D);
	double t_1 = -fabs(M) * fabs(M);
	double tmp;
	if (fabs(M) <= 6.4e-163) {
		tmp = (sqrt(t_1) * c0) / (w + w);
	} else if (fabs(M) <= 5.8e+138) {
		tmp = c0 * (fma(t_0, c0, sqrt((pow((t_0 * c0), 2.0) - (fabs(M) * fabs(M))))) / (w + w));
	} else {
		tmp = 0.5 * ((c0 * pow(t_1, 0.5)) / w);
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(d / Float64(Float64(h * w) * D)) * Float64(d / D))
	t_1 = Float64(Float64(-abs(M)) * abs(M))
	tmp = 0.0
	if (abs(M) <= 6.4e-163)
		tmp = Float64(Float64(sqrt(t_1) * c0) / Float64(w + w));
	elseif (abs(M) <= 5.8e+138)
		tmp = Float64(c0 * Float64(fma(t_0, c0, sqrt(Float64((Float64(t_0 * c0) ^ 2.0) - Float64(abs(M) * abs(M))))) / Float64(w + w)));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * (t_1 ^ 0.5)) / w));
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(d / N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-N[Abs[M], $MachinePrecision]) * N[Abs[M], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[M], $MachinePrecision], 6.4e-163], N[(N[(N[Sqrt[t$95$1], $MachinePrecision] * c0), $MachinePrecision] / N[(w + w), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[M], $MachinePrecision], 5.8e+138], N[(c0 * N[(N[(t$95$0 * c0 + N[Sqrt[N[(N[Power[N[(t$95$0 * c0), $MachinePrecision], 2.0], $MachinePrecision] - N[(N[Abs[M], $MachinePrecision] * N[Abs[M], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(w + w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Power[t$95$1, 0.5], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if M < 6.39999999999999976e-163

   1. Initial program 24.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. lower-pow.f64 14.0%

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   4. Applied rewrites 14.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{\color{blue}{w}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(c0 \cdot \sqrt{-{M}^{2}}\right)}{\color{blue}{w}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(c0 \cdot \sqrt{-{M}^{2}}\right)}{\color{blue}{w}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(c0 \cdot \sqrt{-{M}^{2}}\right)}{w} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{-{M}^{2}}}{w} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{-{M}^{2}}}{w} \]

      8. lower-*.f64 14.0%

         \[\leadsto \frac{\left(0.5 \cdot c0\right) \cdot \sqrt{-{M}^{2}}}{w} \]

      9. lift-neg.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      11. pow2 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}{w} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}{w} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}{w} \]

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

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\left(\mathsf{neg}\left(M\right)\right) \cdot M}}{w} \]

      15. lift-neg.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

      16. lower-*.f64 14.0%

          \[\leadsto \frac{\left(0.5 \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

   6. Applied rewrites 14.0%

      \[\leadsto \frac{\left(0.5 \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{\color{blue}{w}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot \color{blue}{c0}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

      2. lift-*.f64 14.0%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{c0}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot \color{blue}{c0}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

      4. lift-*.f64 14.0%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{c0}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{\color{blue}{w}} \]

   8. Applied rewrites 12.1%

      \[\leadsto \color{blue}{\sqrt{\left(-M\right) \cdot M} \cdot \frac{c0}{w + w}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \color{blue}{\frac{c0}{w + w}} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \frac{c0}{\color{blue}{w + w}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\sqrt{\left(-M\right) \cdot M} \cdot c0}{\color{blue}{w + w}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\left(-M\right) \cdot M} \cdot c0}{\color{blue}{w + w}} \]

      5. lower-*.f64 14.0%

         \[\leadsto \frac{\sqrt{\left(-M\right) \cdot M} \cdot c0}{\color{blue}{w} + w} \]

   10. Applied rewrites 14.0%

       \[\leadsto \frac{\sqrt{\left(-M\right) \cdot M} \cdot c0}{\color{blue}{w + w}} \]

   if 6.39999999999999976e-163 < M < 5.80000000000000019e138

   1. Initial program 24.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{c0 \cdot \left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      7. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      8. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(w \cdot h\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      18. lower-/.f64 24.1%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \color{blue}{\frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Applied rewrites 24.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\color{blue}{c0 \cdot \left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      7. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      8. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(w \cdot h\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      18. lower-/.f64 24.2%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \color{blue}{\frac{d}{D}}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   5. Applied rewrites 24.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\color{blue}{c0 \cdot \left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} - M \cdot M}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} - M \cdot M}\right) \]

      7. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} - M \cdot M}\right) \]

      8. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} - M \cdot M}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} - M \cdot M}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(w \cdot h\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      18. lower-/.f64 34.2%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \color{blue}{\frac{d}{D}}\right) - M \cdot M}\right) \]

   7. Applied rewrites 34.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)} - M \cdot M}\right) \]

   9. Applied rewrites 27.8%

      \[\leadsto \color{blue}{c0 \cdot \frac{\mathsf{fma}\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d \cdot d}{\color{blue}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)}}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       2. lift-*.f64 N/A

          \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \color{blue}{\left(D \cdot w\right)}}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       3. associate-*l* N/A

          \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d \cdot d}{\color{blue}{\left(\left(h \cdot D\right) \cdot D\right) \cdot w}}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       4. lift-*.f64 N/A

          \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d \cdot d}{\left(\color{blue}{\left(h \cdot D\right)} \cdot D\right) \cdot w}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       5. associate-*l* N/A

          \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d \cdot d}{\color{blue}{\left(h \cdot \left(D \cdot D\right)\right)} \cdot w}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       6. lift-*.f64 N/A

          \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d \cdot d}{\left(h \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot w}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       7. *-commutative N/A

          \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d \cdot d}{\color{blue}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)}}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       8. associate-*l* N/A

          \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       9. lift-*.f64 N/A

          \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       10. lift-*.f64 N/A

           \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d \cdot d}{\color{blue}{\left(w \cdot h\right)} \cdot \left(D \cdot D\right)}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       11. lift-*.f64 N/A

           \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       12. lift-*.f64 24.3%

           \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       13. lower-/.f64 N/A

           \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\color{blue}{\frac{d \cdot d}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       14. lift-*.f64 N/A

           \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{\color{blue}{d \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       15. lift-*.f64 N/A

           \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       16. lift-*.f64 N/A

           \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d \cdot d}{\color{blue}{\left(w \cdot h\right)} \cdot \left(D \cdot D\right)}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       17. lift-*.f64 N/A

           \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       18. lift-*.f64 N/A

           \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d \cdot d}{\color{blue}{\left(w \cdot h\right)} \cdot \left(D \cdot D\right)}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       19. associate-*r* N/A

           \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       20. times-frac N/A

           \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\color{blue}{\frac{d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       21. lift-/.f64 N/A

           \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d}{\left(w \cdot h\right) \cdot D} \cdot \color{blue}{\frac{d}{D}}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       22. lower-*.f64 N/A

           \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\color{blue}{\frac{d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

   11. Applied rewrites 26.6%

       \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\color{blue}{\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}, c0, \sqrt{{\left(\frac{d \cdot d}{\color{blue}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)}} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       2. lift-*.f64 N/A

          \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \color{blue}{\left(D \cdot w\right)}} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       3. associate-*l* N/A

          \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}, c0, \sqrt{{\left(\frac{d \cdot d}{\color{blue}{\left(\left(h \cdot D\right) \cdot D\right) \cdot w}} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       4. lift-*.f64 N/A

          \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(\color{blue}{\left(h \cdot D\right)} \cdot D\right) \cdot w} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       5. associate-*l* N/A

          \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}, c0, \sqrt{{\left(\frac{d \cdot d}{\color{blue}{\left(h \cdot \left(D \cdot D\right)\right)} \cdot w} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       6. lift-*.f64 N/A

          \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot w} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       7. *-commutative N/A

          \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}, c0, \sqrt{{\left(\frac{d \cdot d}{\color{blue}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)}} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       8. associate-*l* N/A

          \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}, c0, \sqrt{{\left(\frac{d \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       9. lift-*.f64 N/A

          \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       10. lift-*.f64 N/A

           \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}, c0, \sqrt{{\left(\frac{d \cdot d}{\color{blue}{\left(w \cdot h\right)} \cdot \left(D \cdot D\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       11. lift-*.f64 N/A

           \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       12. lift-*.f64 24.5%

           \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}, c0, \sqrt{{\left(\frac{d \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       13. lower-/.f64 N/A

           \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}, c0, \sqrt{{\left(\color{blue}{\frac{d \cdot d}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       14. lift-*.f64 N/A

           \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}, c0, \sqrt{{\left(\frac{\color{blue}{d \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       15. lift-*.f64 N/A

           \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}, c0, \sqrt{{\left(\frac{d \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       16. lift-*.f64 N/A

           \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}, c0, \sqrt{{\left(\frac{d \cdot d}{\color{blue}{\left(w \cdot h\right)} \cdot \left(D \cdot D\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       17. lift-*.f64 N/A

           \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       18. lift-*.f64 N/A

           \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}, c0, \sqrt{{\left(\frac{d \cdot d}{\color{blue}{\left(w \cdot h\right)} \cdot \left(D \cdot D\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       19. associate-*r* N/A

           \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}, c0, \sqrt{{\left(\frac{d \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       20. times-frac N/A

           \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}, c0, \sqrt{{\left(\color{blue}{\left(\frac{d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       21. lift-/.f64 N/A

           \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}, c0, \sqrt{{\left(\left(\frac{d}{\left(w \cdot h\right) \cdot D} \cdot \color{blue}{\frac{d}{D}}\right) \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       22. lower-*.f64 N/A

           \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}, c0, \sqrt{{\left(\color{blue}{\left(\frac{d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

   13. Applied rewrites 32.3%

       \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}, c0, \sqrt{{\left(\color{blue}{\left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

   if 5.80000000000000019e138 < M

   1. Initial program 24.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. lower-pow.f64 14.0%

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   4. Applied rewrites 14.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      2. pow1/2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(-{M}^{2}\right)}^{\frac{1}{2}}}{w} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      5. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

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

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      9. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      11. lower-*.f64 21.9%

          \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]

   6. Applied rewrites 21.9%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 40.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \frac{d}{\left(\left(h \cdot D\right) \cdot w\right) \cdot D}\\ t_1 := \left(-\left|M\right|\right) \cdot \left|M\right|\\ \mathbf{if}\;\left|M\right| \leq 6.8 \cdot 10^{-163}:\\ \;\;\;\;\frac{\sqrt{t\_1} \cdot c0}{w + w}\\ \mathbf{elif}\;\left|M\right| \leq 4 \cdot 10^{+139}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d, t\_0 \cdot c0, \sqrt{{\left(\left(t\_0 \cdot d\right) \cdot c0\right)}^{2} - \left|M\right| \cdot \left|M\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot {t\_1}^{0.5}}{w}\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ d (* (* (* h D) w) D))) (t_1 (* (- (fabs M)) (fabs M))))
   (if (<= (fabs M) 6.8e-163)
     (/ (* (sqrt t_1) c0) (+ w w))
     (if (<= (fabs M) 4e+139)
       (*
        (/ c0 (* 2.0 w))
        (fma
         d
         (* t_0 c0)
         (sqrt (- (pow (* (* t_0 d) c0) 2.0) (* (fabs M) (fabs M))))))
       (* 0.5 (/ (* c0 (pow t_1 0.5)) w))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = d / (((h * D) * w) * D);
	double t_1 = -fabs(M) * fabs(M);
	double tmp;
	if (fabs(M) <= 6.8e-163) {
		tmp = (sqrt(t_1) * c0) / (w + w);
	} else if (fabs(M) <= 4e+139) {
		tmp = (c0 / (2.0 * w)) * fma(d, (t_0 * c0), sqrt((pow(((t_0 * d) * c0), 2.0) - (fabs(M) * fabs(M)))));
	} else {
		tmp = 0.5 * ((c0 * pow(t_1, 0.5)) / w);
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(d / Float64(Float64(Float64(h * D) * w) * D))
	t_1 = Float64(Float64(-abs(M)) * abs(M))
	tmp = 0.0
	if (abs(M) <= 6.8e-163)
		tmp = Float64(Float64(sqrt(t_1) * c0) / Float64(w + w));
	elseif (abs(M) <= 4e+139)
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * fma(d, Float64(t_0 * c0), sqrt(Float64((Float64(Float64(t_0 * d) * c0) ^ 2.0) - Float64(abs(M) * abs(M))))));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * (t_1 ^ 0.5)) / w));
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(d / N[(N[(N[(h * D), $MachinePrecision] * w), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-N[Abs[M], $MachinePrecision]) * N[Abs[M], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[M], $MachinePrecision], 6.8e-163], N[(N[(N[Sqrt[t$95$1], $MachinePrecision] * c0), $MachinePrecision] / N[(w + w), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[M], $MachinePrecision], 4e+139], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(d * N[(t$95$0 * c0), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(N[(t$95$0 * d), $MachinePrecision] * c0), $MachinePrecision], 2.0], $MachinePrecision] - N[(N[Abs[M], $MachinePrecision] * N[Abs[M], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Power[t$95$1, 0.5], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if M < 6.80000000000000028e-163

   1. Initial program 24.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. lower-pow.f64 14.0%

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   4. Applied rewrites 14.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{\color{blue}{w}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(c0 \cdot \sqrt{-{M}^{2}}\right)}{\color{blue}{w}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(c0 \cdot \sqrt{-{M}^{2}}\right)}{\color{blue}{w}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(c0 \cdot \sqrt{-{M}^{2}}\right)}{w} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{-{M}^{2}}}{w} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{-{M}^{2}}}{w} \]

      8. lower-*.f64 14.0%

         \[\leadsto \frac{\left(0.5 \cdot c0\right) \cdot \sqrt{-{M}^{2}}}{w} \]

      9. lift-neg.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      11. pow2 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}{w} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}{w} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}{w} \]

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

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\left(\mathsf{neg}\left(M\right)\right) \cdot M}}{w} \]

      15. lift-neg.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

      16. lower-*.f64 14.0%

          \[\leadsto \frac{\left(0.5 \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

   6. Applied rewrites 14.0%

      \[\leadsto \frac{\left(0.5 \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{\color{blue}{w}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot \color{blue}{c0}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

      2. lift-*.f64 14.0%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{c0}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot \color{blue}{c0}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

      4. lift-*.f64 14.0%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{c0}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{\color{blue}{w}} \]

   8. Applied rewrites 12.1%

      \[\leadsto \color{blue}{\sqrt{\left(-M\right) \cdot M} \cdot \frac{c0}{w + w}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \color{blue}{\frac{c0}{w + w}} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \frac{c0}{\color{blue}{w + w}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\sqrt{\left(-M\right) \cdot M} \cdot c0}{\color{blue}{w + w}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\left(-M\right) \cdot M} \cdot c0}{\color{blue}{w + w}} \]

      5. lower-*.f64 14.0%

         \[\leadsto \frac{\sqrt{\left(-M\right) \cdot M} \cdot c0}{\color{blue}{w} + w} \]

   10. Applied rewrites 14.0%

       \[\leadsto \frac{\sqrt{\left(-M\right) \cdot M} \cdot c0}{\color{blue}{w + w}} \]

   if 6.80000000000000028e-163 < M < 4.00000000000000013e139

   1. Initial program 24.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{c0 \cdot \left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      7. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      8. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(w \cdot h\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      18. lower-/.f64 24.1%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \color{blue}{\frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Applied rewrites 24.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\color{blue}{c0 \cdot \left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      7. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      8. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(w \cdot h\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      18. lower-/.f64 24.2%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \color{blue}{\frac{d}{D}}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   5. Applied rewrites 24.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\color{blue}{c0 \cdot \left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} - M \cdot M}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} - M \cdot M}\right) \]

      7. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} - M \cdot M}\right) \]

      8. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} - M \cdot M}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} - M \cdot M}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(w \cdot h\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      18. lower-/.f64 34.2%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \color{blue}{\frac{d}{D}}\right) - M \cdot M}\right) \]

   7. Applied rewrites 34.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)} - M \cdot M}\right) \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(h \cdot w\right)}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      3. associate-/r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{d \cdot c0}{D}}{h \cdot w}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot c0}{D}}{\color{blue}{h \cdot w}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      5. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot c0}{D}}{\color{blue}{w \cdot h}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      6. associate-/r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{\frac{d \cdot c0}{D}}{w}}{h}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{\frac{d \cdot c0}{D}}{w}}{h}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\frac{\frac{d \cdot c0}{D}}{w}}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{\frac{\color{blue}{d \cdot c0}}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      10. associate-/l* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \color{blue}{\frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      12. lower-*.f64 32.0%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

   9. Applied rewrites 32.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(h \cdot w\right)}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       3. associate-/r* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\color{blue}{\frac{\frac{d \cdot c0}{D}}{h \cdot w}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot c0}{D}}{\color{blue}{h \cdot w}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       5. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot c0}{D}}{\color{blue}{w \cdot h}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       6. associate-/r* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\color{blue}{\frac{\frac{\frac{d \cdot c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       7. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\color{blue}{\frac{\frac{\frac{d \cdot c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\color{blue}{\frac{\frac{d \cdot c0}{D}}{w}}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{\frac{\color{blue}{d \cdot c0}}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       10. associate-/l* N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       11. lift-/.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \color{blue}{\frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       12. lower-*.f64 32.0%

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

   11. Applied rewrites 32.0%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]
   12. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(h \cdot w\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       3. associate-/r* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\color{blue}{\frac{\frac{d \cdot c0}{D}}{h \cdot w}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot c0}{D}}{\color{blue}{h \cdot w}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       5. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot c0}{D}}{\color{blue}{w \cdot h}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       6. associate-/r* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\color{blue}{\frac{\frac{\frac{d \cdot c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       7. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\color{blue}{\frac{\frac{\frac{d \cdot c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\color{blue}{\frac{\frac{d \cdot c0}{D}}{w}}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{\frac{\color{blue}{d \cdot c0}}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       10. associate-/l* N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       11. lift-/.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \color{blue}{\frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       12. lower-*.f64 36.3%

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

   13. Applied rewrites 36.3%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]
   14. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h}} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       3. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \color{blue}{\frac{d}{D}} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       4. frac-times N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w} \cdot d}{h \cdot D}} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       5. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w} \cdot d}{\color{blue}{D \cdot h}} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       6. times-frac N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{D} \cdot \frac{d}{h}} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{D} \cdot \frac{d}{h}} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{D}} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       9. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\frac{d \cdot \frac{c0}{D}}{w}}}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       11. associate-/l* N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{d \cdot \frac{\frac{c0}{D}}{w}}}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       12. *-commutative N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\frac{\frac{c0}{D}}{w} \cdot d}}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       13. lower-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\frac{\frac{c0}{D}}{w} \cdot d}}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       14. lift-/.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{\color{blue}{\frac{c0}{D}}}{w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       15. associate-/l/ N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\frac{c0}{D \cdot w}} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       16. lift-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{\color{blue}{D \cdot w}} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       17. lower-/.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\frac{c0}{D \cdot w}} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       18. lower-/.f64 32.4%

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \color{blue}{\frac{d}{h}} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

   15. Applied rewrites 32.4%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]
   16. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\color{blue}{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right)} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       3. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \color{blue}{\frac{d}{D}}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       4. frac-times N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w} \cdot d}{h \cdot D}} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       5. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\frac{\frac{d \cdot \frac{c0}{D}}{w} \cdot d}{\color{blue}{D \cdot h}} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       6. times-frac N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\color{blue}{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{D} \cdot \frac{d}{h}\right)} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\color{blue}{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{D} \cdot \frac{d}{h}\right)} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{D}} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       9. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\color{blue}{\frac{d \cdot \frac{c0}{D}}{w}}}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       11. associate-/l* N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\color{blue}{d \cdot \frac{\frac{c0}{D}}{w}}}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       12. *-commutative N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\color{blue}{\frac{\frac{c0}{D}}{w} \cdot d}}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       13. lower-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\color{blue}{\frac{\frac{c0}{D}}{w} \cdot d}}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       14. lift-/.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{\color{blue}{\frac{c0}{D}}}{w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       15. associate-/l/ N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\color{blue}{\frac{c0}{D \cdot w}} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       16. lift-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{\color{blue}{D \cdot w}} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       17. lower-/.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\color{blue}{\frac{c0}{D \cdot w}} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       18. lower-/.f64 32.6%

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \color{blue}{\frac{d}{h}}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

   17. Applied rewrites 32.6%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\color{blue}{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right)} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]
   18. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \color{blue}{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right)} - M \cdot M}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       3. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \color{blue}{\frac{d}{D}}\right) - M \cdot M}\right) \]

       4. frac-times N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w} \cdot d}{h \cdot D}} - M \cdot M}\right) \]

       5. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \frac{\frac{d \cdot \frac{c0}{D}}{w} \cdot d}{\color{blue}{D \cdot h}} - M \cdot M}\right) \]

       6. times-frac N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \color{blue}{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{D} \cdot \frac{d}{h}\right)} - M \cdot M}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \color{blue}{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{D} \cdot \frac{d}{h}\right)} - M \cdot M}\right) \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{D}} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       9. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\color{blue}{\frac{d \cdot \frac{c0}{D}}{w}}}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       11. associate-/l* N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\color{blue}{d \cdot \frac{\frac{c0}{D}}{w}}}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       12. *-commutative N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\color{blue}{\frac{\frac{c0}{D}}{w} \cdot d}}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       13. lower-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\color{blue}{\frac{\frac{c0}{D}}{w} \cdot d}}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       14. lift-/.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{\color{blue}{\frac{c0}{D}}}{w} \cdot d}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       15. associate-/l/ N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\color{blue}{\frac{c0}{D \cdot w}} \cdot d}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       16. lift-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{c0}{\color{blue}{D \cdot w}} \cdot d}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       17. lower-/.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\color{blue}{\frac{c0}{D \cdot w}} \cdot d}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       18. lower-/.f64 34.8%

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \color{blue}{\frac{d}{h}}\right) - M \cdot M}\right) \]

   19. Applied rewrites 34.8%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \color{blue}{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right)} - M \cdot M}\right) \]

   21. Applied rewrites 31.3%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(d, \frac{d}{\left(\left(h \cdot D\right) \cdot w\right) \cdot D} \cdot c0, \sqrt{{\left(\left(\frac{d}{\left(\left(h \cdot D\right) \cdot w\right) \cdot D} \cdot d\right) \cdot c0\right)}^{2} - M \cdot M}\right)} \]

   if 4.00000000000000013e139 < M

   1. Initial program 24.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. lower-pow.f64 14.0%

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   4. Applied rewrites 14.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      2. pow1/2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(-{M}^{2}\right)}^{\frac{1}{2}}}{w} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      5. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

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

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      9. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      11. lower-*.f64 21.9%

          \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]

   6. Applied rewrites 21.9%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 40.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \frac{d}{\left(\left(h \cdot D\right) \cdot w\right) \cdot D} \cdot d\\ t_1 := \left(-\left|M\right|\right) \cdot \left|M\right|\\ \mathbf{if}\;\left|M\right| \leq 6.8 \cdot 10^{-163}:\\ \;\;\;\;\frac{\sqrt{t\_1} \cdot c0}{w + w}\\ \mathbf{elif}\;\left|M\right| \leq 5.8 \cdot 10^{+138}:\\ \;\;\;\;\frac{c0 \cdot \mathsf{fma}\left(t\_0, c0, \sqrt{{\left(t\_0 \cdot c0\right)}^{2} - \left|M\right| \cdot \left|M\right|}\right)}{w + w}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot {t\_1}^{0.5}}{w}\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ d (* (* (* h D) w) D)) d)) (t_1 (* (- (fabs M)) (fabs M))))
   (if (<= (fabs M) 6.8e-163)
     (/ (* (sqrt t_1) c0) (+ w w))
     (if (<= (fabs M) 5.8e+138)
       (/
        (*
         c0
         (fma t_0 c0 (sqrt (- (pow (* t_0 c0) 2.0) (* (fabs M) (fabs M))))))
        (+ w w))
       (* 0.5 (/ (* c0 (pow t_1 0.5)) w))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (d / (((h * D) * w) * D)) * d;
	double t_1 = -fabs(M) * fabs(M);
	double tmp;
	if (fabs(M) <= 6.8e-163) {
		tmp = (sqrt(t_1) * c0) / (w + w);
	} else if (fabs(M) <= 5.8e+138) {
		tmp = (c0 * fma(t_0, c0, sqrt((pow((t_0 * c0), 2.0) - (fabs(M) * fabs(M)))))) / (w + w);
	} else {
		tmp = 0.5 * ((c0 * pow(t_1, 0.5)) / w);
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(d / Float64(Float64(Float64(h * D) * w) * D)) * d)
	t_1 = Float64(Float64(-abs(M)) * abs(M))
	tmp = 0.0
	if (abs(M) <= 6.8e-163)
		tmp = Float64(Float64(sqrt(t_1) * c0) / Float64(w + w));
	elseif (abs(M) <= 5.8e+138)
		tmp = Float64(Float64(c0 * fma(t_0, c0, sqrt(Float64((Float64(t_0 * c0) ^ 2.0) - Float64(abs(M) * abs(M)))))) / Float64(w + w));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * (t_1 ^ 0.5)) / w));
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(d / N[(N[(N[(h * D), $MachinePrecision] * w), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]}, Block[{t$95$1 = N[((-N[Abs[M], $MachinePrecision]) * N[Abs[M], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[M], $MachinePrecision], 6.8e-163], N[(N[(N[Sqrt[t$95$1], $MachinePrecision] * c0), $MachinePrecision] / N[(w + w), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[M], $MachinePrecision], 5.8e+138], N[(N[(c0 * N[(t$95$0 * c0 + N[Sqrt[N[(N[Power[N[(t$95$0 * c0), $MachinePrecision], 2.0], $MachinePrecision] - N[(N[Abs[M], $MachinePrecision] * N[Abs[M], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(w + w), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Power[t$95$1, 0.5], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if M < 6.80000000000000028e-163

   1. Initial program 24.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. lower-pow.f64 14.0%

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   4. Applied rewrites 14.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{\color{blue}{w}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(c0 \cdot \sqrt{-{M}^{2}}\right)}{\color{blue}{w}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(c0 \cdot \sqrt{-{M}^{2}}\right)}{\color{blue}{w}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(c0 \cdot \sqrt{-{M}^{2}}\right)}{w} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{-{M}^{2}}}{w} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{-{M}^{2}}}{w} \]

      8. lower-*.f64 14.0%

         \[\leadsto \frac{\left(0.5 \cdot c0\right) \cdot \sqrt{-{M}^{2}}}{w} \]

      9. lift-neg.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      11. pow2 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}{w} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}{w} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}{w} \]

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

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\left(\mathsf{neg}\left(M\right)\right) \cdot M}}{w} \]

      15. lift-neg.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

      16. lower-*.f64 14.0%

          \[\leadsto \frac{\left(0.5 \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

   6. Applied rewrites 14.0%

      \[\leadsto \frac{\left(0.5 \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{\color{blue}{w}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot \color{blue}{c0}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

      2. lift-*.f64 14.0%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{c0}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot \color{blue}{c0}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

      4. lift-*.f64 14.0%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{c0}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{\color{blue}{w}} \]

   8. Applied rewrites 12.1%

      \[\leadsto \color{blue}{\sqrt{\left(-M\right) \cdot M} \cdot \frac{c0}{w + w}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \color{blue}{\frac{c0}{w + w}} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \frac{c0}{\color{blue}{w + w}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\sqrt{\left(-M\right) \cdot M} \cdot c0}{\color{blue}{w + w}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\left(-M\right) \cdot M} \cdot c0}{\color{blue}{w + w}} \]

      5. lower-*.f64 14.0%

         \[\leadsto \frac{\sqrt{\left(-M\right) \cdot M} \cdot c0}{\color{blue}{w} + w} \]

   10. Applied rewrites 14.0%

       \[\leadsto \frac{\sqrt{\left(-M\right) \cdot M} \cdot c0}{\color{blue}{w + w}} \]

   if 6.80000000000000028e-163 < M < 5.80000000000000019e138

   1. Initial program 24.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{c0 \cdot \left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      7. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      8. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(w \cdot h\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      18. lower-/.f64 24.1%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \color{blue}{\frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Applied rewrites 24.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\color{blue}{c0 \cdot \left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      7. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      8. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(w \cdot h\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      18. lower-/.f64 24.2%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \color{blue}{\frac{d}{D}}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   5. Applied rewrites 24.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\color{blue}{c0 \cdot \left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} - M \cdot M}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} - M \cdot M}\right) \]

      7. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} - M \cdot M}\right) \]

      8. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} - M \cdot M}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} - M \cdot M}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(w \cdot h\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      18. lower-/.f64 34.2%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \color{blue}{\frac{d}{D}}\right) - M \cdot M}\right) \]

   7. Applied rewrites 34.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)} - M \cdot M}\right) \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(h \cdot w\right)}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      3. associate-/r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{d \cdot c0}{D}}{h \cdot w}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot c0}{D}}{\color{blue}{h \cdot w}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      5. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot c0}{D}}{\color{blue}{w \cdot h}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      6. associate-/r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{\frac{d \cdot c0}{D}}{w}}{h}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{\frac{d \cdot c0}{D}}{w}}{h}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\frac{\frac{d \cdot c0}{D}}{w}}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{\frac{\color{blue}{d \cdot c0}}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      10. associate-/l* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \color{blue}{\frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      12. lower-*.f64 32.0%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

   9. Applied rewrites 32.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h}} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(h \cdot w\right)}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       3. associate-/r* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\color{blue}{\frac{\frac{d \cdot c0}{D}}{h \cdot w}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot c0}{D}}{\color{blue}{h \cdot w}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       5. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot c0}{D}}{\color{blue}{w \cdot h}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       6. associate-/r* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\color{blue}{\frac{\frac{\frac{d \cdot c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       7. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\color{blue}{\frac{\frac{\frac{d \cdot c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\color{blue}{\frac{\frac{d \cdot c0}{D}}{w}}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{\frac{\color{blue}{d \cdot c0}}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       10. associate-/l* N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       11. lift-/.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \color{blue}{\frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       12. lower-*.f64 32.0%

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

   11. Applied rewrites 32.0%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]
   12. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(h \cdot w\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       3. associate-/r* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\color{blue}{\frac{\frac{d \cdot c0}{D}}{h \cdot w}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot c0}{D}}{\color{blue}{h \cdot w}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       5. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot c0}{D}}{\color{blue}{w \cdot h}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       6. associate-/r* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\color{blue}{\frac{\frac{\frac{d \cdot c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       7. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\color{blue}{\frac{\frac{\frac{d \cdot c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\color{blue}{\frac{\frac{d \cdot c0}{D}}{w}}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{\frac{\color{blue}{d \cdot c0}}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       10. associate-/l* N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       11. lift-/.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \color{blue}{\frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       12. lower-*.f64 36.3%

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

   13. Applied rewrites 36.3%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]
   14. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h}} \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       3. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \color{blue}{\frac{d}{D}} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       4. frac-times N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w} \cdot d}{h \cdot D}} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       5. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w} \cdot d}{\color{blue}{D \cdot h}} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       6. times-frac N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{D} \cdot \frac{d}{h}} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{D} \cdot \frac{d}{h}} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{D}} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       9. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\frac{d \cdot \frac{c0}{D}}{w}}}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       11. associate-/l* N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{d \cdot \frac{\frac{c0}{D}}{w}}}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       12. *-commutative N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\frac{\frac{c0}{D}}{w} \cdot d}}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       13. lower-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\frac{\frac{c0}{D}}{w} \cdot d}}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       14. lift-/.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{\color{blue}{\frac{c0}{D}}}{w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       15. associate-/l/ N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\frac{c0}{D \cdot w}} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       16. lift-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{\color{blue}{D \cdot w}} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       17. lower-/.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\frac{c0}{D \cdot w}} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       18. lower-/.f64 32.4%

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \color{blue}{\frac{d}{h}} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

   15. Applied rewrites 32.4%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]
   16. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\color{blue}{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right)} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       3. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \color{blue}{\frac{d}{D}}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       4. frac-times N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w} \cdot d}{h \cdot D}} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       5. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\frac{\frac{d \cdot \frac{c0}{D}}{w} \cdot d}{\color{blue}{D \cdot h}} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       6. times-frac N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\color{blue}{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{D} \cdot \frac{d}{h}\right)} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\color{blue}{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{D} \cdot \frac{d}{h}\right)} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{D}} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       9. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\color{blue}{\frac{d \cdot \frac{c0}{D}}{w}}}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       11. associate-/l* N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\color{blue}{d \cdot \frac{\frac{c0}{D}}{w}}}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       12. *-commutative N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\color{blue}{\frac{\frac{c0}{D}}{w} \cdot d}}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       13. lower-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\color{blue}{\frac{\frac{c0}{D}}{w} \cdot d}}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       14. lift-/.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{\color{blue}{\frac{c0}{D}}}{w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       15. associate-/l/ N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\color{blue}{\frac{c0}{D \cdot w}} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       16. lift-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{\color{blue}{D \cdot w}} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       17. lower-/.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\color{blue}{\frac{c0}{D \cdot w}} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       18. lower-/.f64 32.6%

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \color{blue}{\frac{d}{h}}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

   17. Applied rewrites 32.6%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\color{blue}{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right)} \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]
   18. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \color{blue}{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \frac{d}{D}\right)} - M \cdot M}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

       3. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{h} \cdot \color{blue}{\frac{d}{D}}\right) - M \cdot M}\right) \]

       4. frac-times N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w} \cdot d}{h \cdot D}} - M \cdot M}\right) \]

       5. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \frac{\frac{d \cdot \frac{c0}{D}}{w} \cdot d}{\color{blue}{D \cdot h}} - M \cdot M}\right) \]

       6. times-frac N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \color{blue}{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{D} \cdot \frac{d}{h}\right)} - M \cdot M}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \color{blue}{\left(\frac{\frac{d \cdot \frac{c0}{D}}{w}}{D} \cdot \frac{d}{h}\right)} - M \cdot M}\right) \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\color{blue}{\frac{\frac{d \cdot \frac{c0}{D}}{w}}{D}} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       9. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\color{blue}{\frac{d \cdot \frac{c0}{D}}{w}}}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{\color{blue}{d \cdot \frac{c0}{D}}}{w}}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       11. associate-/l* N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\color{blue}{d \cdot \frac{\frac{c0}{D}}{w}}}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       12. *-commutative N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\color{blue}{\frac{\frac{c0}{D}}{w} \cdot d}}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       13. lower-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\color{blue}{\frac{\frac{c0}{D}}{w} \cdot d}}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       14. lift-/.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{\color{blue}{\frac{c0}{D}}}{w} \cdot d}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       15. associate-/l/ N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\color{blue}{\frac{c0}{D \cdot w}} \cdot d}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       16. lift-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{c0}{\color{blue}{D \cdot w}} \cdot d}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       17. lower-/.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\color{blue}{\frac{c0}{D \cdot w}} \cdot d}{D} \cdot \frac{d}{h}\right) - M \cdot M}\right) \]

       18. lower-/.f64 34.8%

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \color{blue}{\frac{d}{h}}\right) - M \cdot M}\right) \]

   19. Applied rewrites 34.8%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h} + \sqrt{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right) \cdot \color{blue}{\left(\frac{\frac{c0}{D \cdot w} \cdot d}{D} \cdot \frac{d}{h}\right)} - M \cdot M}\right) \]

   21. Applied rewrites 31.4%

       \[\leadsto \color{blue}{\frac{c0 \cdot \mathsf{fma}\left(\frac{d}{\left(\left(h \cdot D\right) \cdot w\right) \cdot D} \cdot d, c0, \sqrt{{\left(\left(\frac{d}{\left(\left(h \cdot D\right) \cdot w\right) \cdot D} \cdot d\right) \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w}} \]

   if 5.80000000000000019e138 < M

   1. Initial program 24.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. lower-pow.f64 14.0%

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   4. Applied rewrites 14.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      2. pow1/2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(-{M}^{2}\right)}^{\frac{1}{2}}}{w} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      5. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

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

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      9. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      11. lower-*.f64 21.9%

          \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]

   6. Applied rewrites 21.9%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 40.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \frac{d}{\left(\left(D \cdot w\right) \cdot h\right) \cdot D} \cdot d\\ t_1 := \left(-\left|M\right|\right) \cdot \left|M\right|\\ \mathbf{if}\;\left|M\right| \leq 6.8 \cdot 10^{-163}:\\ \;\;\;\;\frac{\sqrt{t\_1} \cdot c0}{w + w}\\ \mathbf{elif}\;\left|M\right| \leq 5.8 \cdot 10^{+138}:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(t\_0, c0, \sqrt{{\left(t\_0 \cdot c0\right)}^{2} - \left|M\right| \cdot \left|M\right|}\right)}{w + w}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot {t\_1}^{0.5}}{w}\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ d (* (* (* D w) h) D)) d)) (t_1 (* (- (fabs M)) (fabs M))))
   (if (<= (fabs M) 6.8e-163)
     (/ (* (sqrt t_1) c0) (+ w w))
     (if (<= (fabs M) 5.8e+138)
       (*
        c0
        (/
         (fma t_0 c0 (sqrt (- (pow (* t_0 c0) 2.0) (* (fabs M) (fabs M)))))
         (+ w w)))
       (* 0.5 (/ (* c0 (pow t_1 0.5)) w))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (d / (((D * w) * h) * D)) * d;
	double t_1 = -fabs(M) * fabs(M);
	double tmp;
	if (fabs(M) <= 6.8e-163) {
		tmp = (sqrt(t_1) * c0) / (w + w);
	} else if (fabs(M) <= 5.8e+138) {
		tmp = c0 * (fma(t_0, c0, sqrt((pow((t_0 * c0), 2.0) - (fabs(M) * fabs(M))))) / (w + w));
	} else {
		tmp = 0.5 * ((c0 * pow(t_1, 0.5)) / w);
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(d / Float64(Float64(Float64(D * w) * h) * D)) * d)
	t_1 = Float64(Float64(-abs(M)) * abs(M))
	tmp = 0.0
	if (abs(M) <= 6.8e-163)
		tmp = Float64(Float64(sqrt(t_1) * c0) / Float64(w + w));
	elseif (abs(M) <= 5.8e+138)
		tmp = Float64(c0 * Float64(fma(t_0, c0, sqrt(Float64((Float64(t_0 * c0) ^ 2.0) - Float64(abs(M) * abs(M))))) / Float64(w + w)));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * (t_1 ^ 0.5)) / w));
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(d / N[(N[(N[(D * w), $MachinePrecision] * h), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]}, Block[{t$95$1 = N[((-N[Abs[M], $MachinePrecision]) * N[Abs[M], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[M], $MachinePrecision], 6.8e-163], N[(N[(N[Sqrt[t$95$1], $MachinePrecision] * c0), $MachinePrecision] / N[(w + w), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[M], $MachinePrecision], 5.8e+138], N[(c0 * N[(N[(t$95$0 * c0 + N[Sqrt[N[(N[Power[N[(t$95$0 * c0), $MachinePrecision], 2.0], $MachinePrecision] - N[(N[Abs[M], $MachinePrecision] * N[Abs[M], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(w + w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Power[t$95$1, 0.5], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if M < 6.80000000000000028e-163

   1. Initial program 24.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. lower-pow.f64 14.0%

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   4. Applied rewrites 14.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{\color{blue}{w}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(c0 \cdot \sqrt{-{M}^{2}}\right)}{\color{blue}{w}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(c0 \cdot \sqrt{-{M}^{2}}\right)}{\color{blue}{w}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(c0 \cdot \sqrt{-{M}^{2}}\right)}{w} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{-{M}^{2}}}{w} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{-{M}^{2}}}{w} \]

      8. lower-*.f64 14.0%

         \[\leadsto \frac{\left(0.5 \cdot c0\right) \cdot \sqrt{-{M}^{2}}}{w} \]

      9. lift-neg.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      11. pow2 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}{w} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}{w} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}{w} \]

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

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\left(\mathsf{neg}\left(M\right)\right) \cdot M}}{w} \]

      15. lift-neg.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

      16. lower-*.f64 14.0%

          \[\leadsto \frac{\left(0.5 \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

   6. Applied rewrites 14.0%

      \[\leadsto \frac{\left(0.5 \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{\color{blue}{w}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot \color{blue}{c0}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

      2. lift-*.f64 14.0%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{c0}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot \color{blue}{c0}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

      4. lift-*.f64 14.0%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{c0}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{\color{blue}{w}} \]

   8. Applied rewrites 12.1%

      \[\leadsto \color{blue}{\sqrt{\left(-M\right) \cdot M} \cdot \frac{c0}{w + w}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \color{blue}{\frac{c0}{w + w}} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \frac{c0}{\color{blue}{w + w}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\sqrt{\left(-M\right) \cdot M} \cdot c0}{\color{blue}{w + w}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\left(-M\right) \cdot M} \cdot c0}{\color{blue}{w + w}} \]

      5. lower-*.f64 14.0%

         \[\leadsto \frac{\sqrt{\left(-M\right) \cdot M} \cdot c0}{\color{blue}{w} + w} \]

   10. Applied rewrites 14.0%

       \[\leadsto \frac{\sqrt{\left(-M\right) \cdot M} \cdot c0}{\color{blue}{w + w}} \]

   if 6.80000000000000028e-163 < M < 5.80000000000000019e138

   1. Initial program 24.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{c0 \cdot \left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      7. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      8. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(w \cdot h\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      18. lower-/.f64 24.1%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \color{blue}{\frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Applied rewrites 24.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\color{blue}{c0 \cdot \left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      7. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      8. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(w \cdot h\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      18. lower-/.f64 24.2%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \color{blue}{\frac{d}{D}}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   5. Applied rewrites 24.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\color{blue}{c0 \cdot \left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} - M \cdot M}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} - M \cdot M}\right) \]

      7. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} - M \cdot M}\right) \]

      8. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} - M \cdot M}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} - M \cdot M}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(w \cdot h\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      18. lower-/.f64 34.2%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \color{blue}{\frac{d}{D}}\right) - M \cdot M}\right) \]

   7. Applied rewrites 34.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)} - M \cdot M}\right) \]

   9. Applied rewrites 27.8%

      \[\leadsto \color{blue}{c0 \cdot \frac{\mathsf{fma}\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d \cdot d}{\color{blue}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)}}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       2. lift-*.f64 N/A

          \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \color{blue}{\left(D \cdot w\right)}}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       3. associate-*l* N/A

          \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d \cdot d}{\color{blue}{\left(\left(h \cdot D\right) \cdot D\right) \cdot w}}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       4. lift-*.f64 N/A

          \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d \cdot d}{\left(\color{blue}{\left(h \cdot D\right)} \cdot D\right) \cdot w}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       5. associate-*l* N/A

          \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d \cdot d}{\color{blue}{\left(h \cdot \left(D \cdot D\right)\right)} \cdot w}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       6. lift-*.f64 N/A

          \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d \cdot d}{\left(h \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot w}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       7. *-commutative N/A

          \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d \cdot d}{\color{blue}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)}}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       8. associate-*l* N/A

          \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       9. lift-*.f64 N/A

          \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       10. lift-*.f64 N/A

           \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d \cdot d}{\color{blue}{\left(w \cdot h\right)} \cdot \left(D \cdot D\right)}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       11. lift-*.f64 N/A

           \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       12. lift-*.f64 24.3%

           \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       13. lower-/.f64 N/A

           \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\color{blue}{\frac{d \cdot d}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       14. lift-*.f64 N/A

           \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{\color{blue}{d \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       15. associate-/l* N/A

           \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\color{blue}{d \cdot \frac{d}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       16. *-commutative N/A

           \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\color{blue}{\frac{d}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot d}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       17. lower-*.f64 N/A

           \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\color{blue}{\frac{d}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot d}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

   11. Applied rewrites 27.8%

       \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\color{blue}{\frac{d}{\left(\left(D \cdot w\right) \cdot h\right) \cdot D} \cdot d}, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d}{\left(\left(D \cdot w\right) \cdot h\right) \cdot D} \cdot d, c0, \sqrt{{\left(\frac{d \cdot d}{\color{blue}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)}} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       2. lift-*.f64 N/A

          \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d}{\left(\left(D \cdot w\right) \cdot h\right) \cdot D} \cdot d, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \color{blue}{\left(D \cdot w\right)}} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       3. associate-*l* N/A

          \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d}{\left(\left(D \cdot w\right) \cdot h\right) \cdot D} \cdot d, c0, \sqrt{{\left(\frac{d \cdot d}{\color{blue}{\left(\left(h \cdot D\right) \cdot D\right) \cdot w}} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       4. lift-*.f64 N/A

          \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d}{\left(\left(D \cdot w\right) \cdot h\right) \cdot D} \cdot d, c0, \sqrt{{\left(\frac{d \cdot d}{\left(\color{blue}{\left(h \cdot D\right)} \cdot D\right) \cdot w} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       5. associate-*l* N/A

          \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d}{\left(\left(D \cdot w\right) \cdot h\right) \cdot D} \cdot d, c0, \sqrt{{\left(\frac{d \cdot d}{\color{blue}{\left(h \cdot \left(D \cdot D\right)\right)} \cdot w} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       6. lift-*.f64 N/A

          \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d}{\left(\left(D \cdot w\right) \cdot h\right) \cdot D} \cdot d, c0, \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot w} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       7. *-commutative N/A

          \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d}{\left(\left(D \cdot w\right) \cdot h\right) \cdot D} \cdot d, c0, \sqrt{{\left(\frac{d \cdot d}{\color{blue}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)}} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       8. associate-*l* N/A

          \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d}{\left(\left(D \cdot w\right) \cdot h\right) \cdot D} \cdot d, c0, \sqrt{{\left(\frac{d \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       9. lift-*.f64 N/A

          \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d}{\left(\left(D \cdot w\right) \cdot h\right) \cdot D} \cdot d, c0, \sqrt{{\left(\frac{d \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       10. lift-*.f64 N/A

           \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d}{\left(\left(D \cdot w\right) \cdot h\right) \cdot D} \cdot d, c0, \sqrt{{\left(\frac{d \cdot d}{\color{blue}{\left(w \cdot h\right)} \cdot \left(D \cdot D\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       11. lift-*.f64 N/A

           \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d}{\left(\left(D \cdot w\right) \cdot h\right) \cdot D} \cdot d, c0, \sqrt{{\left(\frac{d \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       12. lift-*.f64 24.5%

           \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d}{\left(\left(D \cdot w\right) \cdot h\right) \cdot D} \cdot d, c0, \sqrt{{\left(\frac{d \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       13. lower-/.f64 N/A

           \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d}{\left(\left(D \cdot w\right) \cdot h\right) \cdot D} \cdot d, c0, \sqrt{{\left(\color{blue}{\frac{d \cdot d}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       14. lift-*.f64 N/A

           \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d}{\left(\left(D \cdot w\right) \cdot h\right) \cdot D} \cdot d, c0, \sqrt{{\left(\frac{\color{blue}{d \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       15. associate-/l* N/A

           \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d}{\left(\left(D \cdot w\right) \cdot h\right) \cdot D} \cdot d, c0, \sqrt{{\left(\color{blue}{\left(d \cdot \frac{d}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       16. *-commutative N/A

           \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d}{\left(\left(D \cdot w\right) \cdot h\right) \cdot D} \cdot d, c0, \sqrt{{\left(\color{blue}{\left(\frac{d}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot d\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

       17. lower-*.f64 N/A

           \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d}{\left(\left(D \cdot w\right) \cdot h\right) \cdot D} \cdot d, c0, \sqrt{{\left(\color{blue}{\left(\frac{d}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot d\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

   13. Applied rewrites 31.8%

       \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(\frac{d}{\left(\left(D \cdot w\right) \cdot h\right) \cdot D} \cdot d, c0, \sqrt{{\left(\color{blue}{\left(\frac{d}{\left(\left(D \cdot w\right) \cdot h\right) \cdot D} \cdot d\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w} \]

   if 5.80000000000000019e138 < M

   1. Initial program 24.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. lower-pow.f64 14.0%

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   4. Applied rewrites 14.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      2. pow1/2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(-{M}^{2}\right)}^{\frac{1}{2}}}{w} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      5. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

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

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      9. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      11. lower-*.f64 21.9%

          \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]

   6. Applied rewrites 21.9%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 40.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - \left|M\right| \cdot \left|M\right|}\right) \leq \infty:\\ \;\;\;\;t\_0 \cdot \left(t\_1 + \sqrt{\left|M\right|} \cdot \sqrt{\left|\frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} \cdot c0\right| - \left|M\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot {\left(\left(-\left|M\right|\right) \cdot \left|M\right|\right)}^{0.5}}{w}\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w))) (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<=
        (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* (fabs M) (fabs M))))))
        INFINITY)
     (*
      t_0
      (+
       t_1
       (*
        (sqrt (fabs M))
        (sqrt (- (fabs (* (/ (* d d) (* (* (* D D) w) h)) c0)) (fabs M))))))
     (* 0.5 (/ (* c0 (pow (* (- (fabs M)) (fabs M)) 0.5)) w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (fabs(M) * fabs(M)))))) <= ((double) INFINITY)) {
		tmp = t_0 * (t_1 + (sqrt(fabs(M)) * sqrt((fabs((((d * d) / (((D * D) * w) * h)) * c0)) - fabs(M)))));
	} else {
		tmp = 0.5 * ((c0 * pow((-fabs(M) * fabs(M)), 0.5)) / w);
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (Math.abs(M) * Math.abs(M)))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * (t_1 + (Math.sqrt(Math.abs(M)) * Math.sqrt((Math.abs((((d * d) / (((D * D) * w) * h)) * c0)) - Math.abs(M)))));
	} else {
		tmp = 0.5 * ((c0 * Math.pow((-Math.abs(M) * Math.abs(M)), 0.5)) / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if (t_0 * (t_1 + math.sqrt(((t_1 * t_1) - (math.fabs(M) * math.fabs(M)))))) <= math.inf:
		tmp = t_0 * (t_1 + (math.sqrt(math.fabs(M)) * math.sqrt((math.fabs((((d * d) / (((D * D) * w) * h)) * c0)) - math.fabs(M)))))
	else:
		tmp = 0.5 * ((c0 * math.pow((-math.fabs(M) * math.fabs(M)), 0.5)) / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(abs(M) * abs(M)))))) <= Inf)
		tmp = Float64(t_0 * Float64(t_1 + Float64(sqrt(abs(M)) * sqrt(Float64(abs(Float64(Float64(Float64(d * d) / Float64(Float64(Float64(D * D) * w) * h)) * c0)) - abs(M))))));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * (Float64(Float64(-abs(M)) * abs(M)) ^ 0.5)) / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (abs(M) * abs(M)))))) <= Inf)
		tmp = t_0 * (t_1 + (sqrt(abs(M)) * sqrt((abs((((d * d) / (((D * D) * w) * h)) * c0)) - abs(M)))));
	else
		tmp = 0.5 * ((c0 * ((-abs(M) * abs(M)) ^ 0.5)) / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(N[Abs[M], $MachinePrecision] * N[Abs[M], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 * N[(t$95$1 + N[(N[Sqrt[N[Abs[M], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[Abs[N[(N[(N[(d * d), $MachinePrecision] / N[(N[(N[(D * D), $MachinePrecision] * w), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision]], $MachinePrecision] - N[Abs[M], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Power[N[((-N[Abs[M], $MachinePrecision]) * N[Abs[M], $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 24.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \color{blue}{\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} - M \cdot M}\right) \]

      4. sqr-abs-rev N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\color{blue}{\left|\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right| \cdot \left|\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right|} - M \cdot M}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\left|\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right| \cdot \left|\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right| - \color{blue}{M \cdot M}}\right) \]

      6. difference-of-squares N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\color{blue}{\left(\left|\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right| + M\right) \cdot \left(\left|\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right| - M\right)}}\right) \]

      7. sqrt-prod N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \color{blue}{\sqrt{\left|\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right| + M} \cdot \sqrt{\left|\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right| - M}}\right) \]

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

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \color{blue}{\sqrt{\left|\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right| + M} \cdot \sqrt{\left|\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right| - M}}\right) \]

   3. Applied rewrites 30.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \color{blue}{\sqrt{\mathsf{fma}\left(d \cdot d, \left|\frac{c0}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right|, M\right)} \cdot \sqrt{\left|\frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} \cdot c0\right| - M}}\right) \]

   4. Taylor expanded in d around 0

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\color{blue}{M}} \cdot \sqrt{\left|\frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} \cdot c0\right| - M}\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 14.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\color{blue}{M}} \cdot \sqrt{\left|\frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} \cdot c0\right| - M}\right) \]

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 24.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. lower-pow.f64 14.0%

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   4. Applied rewrites 14.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      2. pow1/2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(-{M}^{2}\right)}^{\frac{1}{2}}}{w} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      5. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

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

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      9. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      11. lower-*.f64 21.9%

          \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]

   6. Applied rewrites 21.9%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 27.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_0 \cdot \left(t\_1 + \sqrt{-1 \cdot {M}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w}\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w))) (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M))))) INFINITY)
     (* t_0 (+ t_1 (sqrt (* -1.0 (pow M 2.0)))))
     (* 0.5 (/ (* c0 (pow (* (- M) M) 0.5)) w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_0 * (t_1 + sqrt((-1.0 * pow(M, 2.0))));
	} else {
		tmp = 0.5 * ((c0 * pow((-M * M), 0.5)) / w);
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * (t_1 + Math.sqrt((-1.0 * Math.pow(M, 2.0))));
	} else {
		tmp = 0.5 * ((c0 * Math.pow((-M * M), 0.5)) / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if (t_0 * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))) <= math.inf:
		tmp = t_0 * (t_1 + math.sqrt((-1.0 * math.pow(M, 2.0))))
	else:
		tmp = 0.5 * ((c0 * math.pow((-M * M), 0.5)) / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_0 * Float64(t_1 + sqrt(Float64(-1.0 * (M ^ 2.0)))));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * (Float64(Float64(-M) * M) ^ 0.5)) / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= Inf)
		tmp = t_0 * (t_1 + sqrt((-1.0 * (M ^ 2.0))));
	else
		tmp = 0.5 * ((c0 * ((-M * M) ^ 0.5)) / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(-1.0 * N[Power[M, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Power[N[((-M) * M), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 24.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\color{blue}{-1 \cdot {M}^{2}}}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{-1 \cdot \color{blue}{{M}^{2}}}\right) \]

      2. lower-pow.f64 7.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{-1 \cdot {M}^{\color{blue}{2}}}\right) \]

   4. Applied rewrites 7.9%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\color{blue}{-1 \cdot {M}^{2}}}\right) \]

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 24.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. lower-pow.f64 14.0%

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   4. Applied rewrites 14.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      2. pow1/2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(-{M}^{2}\right)}^{\frac{1}{2}}}{w} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      5. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

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

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      9. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      11. lower-*.f64 21.9%

          \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]

   6. Applied rewrites 21.9%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 21.9% accurate, 2.6× speedup

Math

\[0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (* 0.5 (/ (* c0 (pow (* (- M) M) 0.5)) w)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return 0.5 * ((c0 * pow((-M * M), 0.5)) / w);
}

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(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = 0.5d0 * ((c0 * ((-m * m) ** 0.5d0)) / w)
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return 0.5 * ((c0 * Math.pow((-M * M), 0.5)) / w);
}

Python

def code(c0, w, h, D, d, M):
	return 0.5 * ((c0 * math.pow((-M * M), 0.5)) / w)

Julia

function code(c0, w, h, D, d, M)
	return Float64(0.5 * Float64(Float64(c0 * (Float64(Float64(-M) * M) ^ 0.5)) / w))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = 0.5 * ((c0 * ((-M * M) ^ 0.5)) / w);
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(0.5 * N[(N[(c0 * N[Power[N[((-M) * M), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 24.6%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

2. Taylor expanded in c0 around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   5. lower-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   6. lower-pow.f64 14.0%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

4. Applied rewrites 14.0%

   \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
5. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   2. pow1/2 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(-{M}^{2}\right)}^{\frac{1}{2}}}{w} \]

   3. lift-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

   4. lift-pow.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

   5. pow2 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

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

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

   9. lift-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

   10. lower-pow.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

   11. lower-*.f64 21.9%

       \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]

6. Applied rewrites 21.9%

   \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]
7. Add Preprocessing

Alternative 12: 20.0% accurate, 2.6× speedup

Math

\[{\left(\left(-M\right) \cdot M\right)}^{0.5} \cdot \frac{c0}{w + w} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (* (pow (* (- M) M) 0.5) (/ c0 (+ w w))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return pow((-M * M), 0.5) * (c0 / (w + w));
}

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(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = ((-m * m) ** 0.5d0) * (c0 / (w + w))
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return Math.pow((-M * M), 0.5) * (c0 / (w + w));
}

Python

def code(c0, w, h, D, d, M):
	return math.pow((-M * M), 0.5) * (c0 / (w + w))

Julia

function code(c0, w, h, D, d, M)
	return Float64((Float64(Float64(-M) * M) ^ 0.5) * Float64(c0 / Float64(w + w)))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = ((-M * M) ^ 0.5) * (c0 / (w + w));
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(N[Power[N[((-M) * M), $MachinePrecision], 0.5], $MachinePrecision] * N[(c0 / N[(w + w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 24.6%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

2. Taylor expanded in c0 around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   5. lower-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   6. lower-pow.f64 14.0%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

4. Applied rewrites 14.0%

   \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{\color{blue}{w}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \left(c0 \cdot \sqrt{-{M}^{2}}\right)}{\color{blue}{w}} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \left(c0 \cdot \sqrt{-{M}^{2}}\right)}{\color{blue}{w}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \left(c0 \cdot \sqrt{-{M}^{2}}\right)}{w} \]

   6. associate-*r* N/A

      \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{-{M}^{2}}}{w} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{-{M}^{2}}}{w} \]

   8. lower-*.f64 14.0%

      \[\leadsto \frac{\left(0.5 \cdot c0\right) \cdot \sqrt{-{M}^{2}}}{w} \]

   9. lift-neg.f64 N/A

      \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   10. lift-pow.f64 N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   11. pow2 N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}{w} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}{w} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}{w} \]

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

       \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\left(\mathsf{neg}\left(M\right)\right) \cdot M}}{w} \]

   15. lift-neg.f64 N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

   16. lower-*.f64 14.0%

       \[\leadsto \frac{\left(0.5 \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

6. Applied rewrites 14.0%

   \[\leadsto \frac{\left(0.5 \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{\color{blue}{w}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\left(\frac{1}{2} \cdot \color{blue}{c0}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

   2. lift-*.f64 14.0%

      \[\leadsto \frac{\left(0.5 \cdot \color{blue}{c0}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\left(\frac{1}{2} \cdot \color{blue}{c0}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

   4. lift-*.f64 14.0%

      \[\leadsto \frac{\left(0.5 \cdot \color{blue}{c0}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

   5. lift-/.f64 N/A

      \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{\color{blue}{w}} \]

8. Applied rewrites 12.1%

   \[\leadsto \color{blue}{\sqrt{\left(-M\right) \cdot M} \cdot \frac{c0}{w + w}} \]
9. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \frac{\color{blue}{c0}}{w + w} \]

   2. pow1/2 N/A

      \[\leadsto {\left(\left(-M\right) \cdot M\right)}^{\frac{1}{2}} \cdot \frac{\color{blue}{c0}}{w + w} \]

   3. lower-pow.f64 20.0%

      \[\leadsto {\left(\left(-M\right) \cdot M\right)}^{0.5} \cdot \frac{\color{blue}{c0}}{w + w} \]

10. Applied rewrites 20.0%

    \[\leadsto {\left(\left(-M\right) \cdot M\right)}^{0.5} \cdot \frac{\color{blue}{c0}}{w + w} \]
11. Add Preprocessing

Alternative 13: 14.0% accurate, 4.9× speedup

Math

\[\frac{\sqrt{\left(-M\right) \cdot M} \cdot c0}{w + w} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (/ (* (sqrt (* (- M) M)) c0) (+ w w)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return (sqrt((-M * M)) * c0) / (w + w);
}

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(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = (sqrt((-m * m)) * c0) / (w + w)
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return (Math.sqrt((-M * M)) * c0) / (w + w);
}

Python

def code(c0, w, h, D, d, M):
	return (math.sqrt((-M * M)) * c0) / (w + w)

Julia

function code(c0, w, h, D, d, M)
	return Float64(Float64(sqrt(Float64(Float64(-M) * M)) * c0) / Float64(w + w))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = (sqrt((-M * M)) * c0) / (w + w);
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(N[(N[Sqrt[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision] / N[(w + w), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 24.6%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

2. Taylor expanded in c0 around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   5. lower-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   6. lower-pow.f64 14.0%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

4. Applied rewrites 14.0%

   \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{\color{blue}{w}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \left(c0 \cdot \sqrt{-{M}^{2}}\right)}{\color{blue}{w}} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \left(c0 \cdot \sqrt{-{M}^{2}}\right)}{\color{blue}{w}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \left(c0 \cdot \sqrt{-{M}^{2}}\right)}{w} \]

   6. associate-*r* N/A

      \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{-{M}^{2}}}{w} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{-{M}^{2}}}{w} \]

   8. lower-*.f64 14.0%

      \[\leadsto \frac{\left(0.5 \cdot c0\right) \cdot \sqrt{-{M}^{2}}}{w} \]

   9. lift-neg.f64 N/A

      \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   10. lift-pow.f64 N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   11. pow2 N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}{w} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}{w} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}{w} \]

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

       \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\left(\mathsf{neg}\left(M\right)\right) \cdot M}}{w} \]

   15. lift-neg.f64 N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

   16. lower-*.f64 14.0%

       \[\leadsto \frac{\left(0.5 \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

6. Applied rewrites 14.0%

   \[\leadsto \frac{\left(0.5 \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{\color{blue}{w}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\left(\frac{1}{2} \cdot \color{blue}{c0}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

   2. lift-*.f64 14.0%

      \[\leadsto \frac{\left(0.5 \cdot \color{blue}{c0}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\left(\frac{1}{2} \cdot \color{blue}{c0}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

   4. lift-*.f64 14.0%

      \[\leadsto \frac{\left(0.5 \cdot \color{blue}{c0}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

   5. lift-/.f64 N/A

      \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{\color{blue}{w}} \]

8. Applied rewrites 12.1%

   \[\leadsto \color{blue}{\sqrt{\left(-M\right) \cdot M} \cdot \frac{c0}{w + w}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \color{blue}{\frac{c0}{w + w}} \]

   2. lift-/.f64 N/A

      \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \frac{c0}{\color{blue}{w + w}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\sqrt{\left(-M\right) \cdot M} \cdot c0}{\color{blue}{w + w}} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{\sqrt{\left(-M\right) \cdot M} \cdot c0}{\color{blue}{w + w}} \]

   5. lower-*.f64 14.0%

      \[\leadsto \frac{\sqrt{\left(-M\right) \cdot M} \cdot c0}{\color{blue}{w} + w} \]

10. Applied rewrites 14.0%

    \[\leadsto \frac{\sqrt{\left(-M\right) \cdot M} \cdot c0}{\color{blue}{w + w}} \]
11. Add Preprocessing

Alternative 14: 12.1% accurate, 4.9× speedup

Math

\[\sqrt{\left(-M\right) \cdot M} \cdot \frac{c0}{w + w} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (* (sqrt (* (- M) M)) (/ c0 (+ w w))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return sqrt((-M * M)) * (c0 / (w + w));
}

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(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = sqrt((-m * m)) * (c0 / (w + w))
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return Math.sqrt((-M * M)) * (c0 / (w + w));
}

Python

def code(c0, w, h, D, d, M):
	return math.sqrt((-M * M)) * (c0 / (w + w))

Julia

function code(c0, w, h, D, d, M)
	return Float64(sqrt(Float64(Float64(-M) * M)) * Float64(c0 / Float64(w + w)))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = sqrt((-M * M)) * (c0 / (w + w));
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(N[Sqrt[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * N[(c0 / N[(w + w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 24.6%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

2. Taylor expanded in c0 around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   5. lower-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   6. lower-pow.f64 14.0%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

4. Applied rewrites 14.0%

   \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{\color{blue}{w}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \left(c0 \cdot \sqrt{-{M}^{2}}\right)}{\color{blue}{w}} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \left(c0 \cdot \sqrt{-{M}^{2}}\right)}{\color{blue}{w}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \left(c0 \cdot \sqrt{-{M}^{2}}\right)}{w} \]

   6. associate-*r* N/A

      \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{-{M}^{2}}}{w} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{-{M}^{2}}}{w} \]

   8. lower-*.f64 14.0%

      \[\leadsto \frac{\left(0.5 \cdot c0\right) \cdot \sqrt{-{M}^{2}}}{w} \]

   9. lift-neg.f64 N/A

      \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   10. lift-pow.f64 N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   11. pow2 N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}{w} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}{w} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}{w} \]

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

       \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\left(\mathsf{neg}\left(M\right)\right) \cdot M}}{w} \]

   15. lift-neg.f64 N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

   16. lower-*.f64 14.0%

       \[\leadsto \frac{\left(0.5 \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

6. Applied rewrites 14.0%

   \[\leadsto \frac{\left(0.5 \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{\color{blue}{w}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\left(\frac{1}{2} \cdot \color{blue}{c0}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

   2. lift-*.f64 14.0%

      \[\leadsto \frac{\left(0.5 \cdot \color{blue}{c0}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\left(\frac{1}{2} \cdot \color{blue}{c0}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

   4. lift-*.f64 14.0%

      \[\leadsto \frac{\left(0.5 \cdot \color{blue}{c0}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

   5. lift-/.f64 N/A

      \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{\color{blue}{w}} \]

8. Applied rewrites 12.1%

   \[\leadsto \color{blue}{\sqrt{\left(-M\right) \cdot M} \cdot \frac{c0}{w + w}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025183 
(FPCore (c0 w h D d M)
  :name "Henrywood and Agarwal, Equation (13)"
  :precision binary64
  (* (/ c0 (* 2.0 w)) (+ (/ (* c0 (* d d)) (* (* w h) (* D D))) (sqrt (- (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* M M))))))