Henrywood and Agarwal, Equation (13)

Percentage Accurate: 25.4% → 45.6%

Time: 10.5s

Alternatives: 16

Speedup: 1.2×

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: 25.4% 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: 45.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left(c0 \cdot \frac{d}{D}\right) \cdot d}{\left(h \cdot D\right) \cdot w}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_1 \cdot \left(t\_2 + \sqrt{t\_2 \cdot t\_2 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_1 \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w}\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
  :precision binary64
  (let* ((t_0 (/ (* (* c0 (/ d D)) d) (* (* h D) w)))
       (t_1 (/ c0 (* 2.0 w)))
       (t_2 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
  (if (<= (* t_1 (+ t_2 (sqrt (- (* t_2 t_2) (* M M))))) INFINITY)
    (* t_1 (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
    (* 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 * (d / D)) * d) / ((h * D) * w);
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_1 * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	} 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 * (d / D)) * d) / ((h * D) * w);
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_1 * (t_2 + Math.sqrt(((t_2 * t_2) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_1 * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
	} 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 * (d / D)) * d) / ((h * D) * w)
	t_1 = c0 / (2.0 * w)
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if (t_1 * (t_2 + math.sqrt(((t_2 * t_2) - (M * M))))) <= math.inf:
		tmp = t_1 * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))
	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(Float64(Float64(c0 * Float64(d / D)) * d) / Float64(Float64(h * D) * w))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_1 * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_1 * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))));
	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 * (d / D)) * d) / ((h * D) * w);
	t_1 = c0 / (2.0 * w);
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= Inf)
		tmp = t_1 * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	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[(N[(N[(c0 * N[(d / D), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision] / N[(N[(h * D), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 * N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$1 * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $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 25.4%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 24.7%

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

   3. Applied rewrites 24.7%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 25.0%

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

   5. Applied rewrites 25.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 35.5%

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

   7. Applied rewrites 35.5%

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

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. associate-/l* N/A

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

      20. lift-/.f64 N/A

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

      21. lower-*.f64 33.7%

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

      22. lift-*.f64 N/A

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

      23. lift-*.f64 N/A

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

      24. associate-*r* N/A

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

      25. lower-*.f64 N/A

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

      26. *-commutative N/A

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

      27. lower-*.f64 32.3%

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

   9. Applied rewrites 32.3%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-/r* N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lift-*.f64 N/A

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

       8. lift-/.f64 N/A

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

       9. frac-times N/A

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

       10. *-commutative N/A

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

       11. lift-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lift-*.f64 N/A

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

       14. lift-*.f64 N/A

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

       15. lower-/.f64 N/A

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

       16. lower-*.f64 N/A

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

       17. lift-*.f64 N/A

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

       18. *-commutative N/A

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

       19. associate-/l* N/A

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

       20. lift-/.f64 N/A

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

       21. lower-*.f64 32.5%

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

       22. lift-*.f64 N/A

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

       23. lift-*.f64 N/A

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

       24. associate-*r* N/A

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

       25. lower-*.f64 N/A

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

       26. *-commutative N/A

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

       27. lower-*.f64 32.5%

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

   11. Applied rewrites 32.5%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-/r* N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lift-*.f64 N/A

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

       8. lift-/.f64 N/A

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

       9. frac-times N/A

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

       10. *-commutative N/A

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

       11. lift-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lift-*.f64 N/A

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

       14. lift-*.f64 N/A

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

       15. lower-/.f64 N/A

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

       16. lower-*.f64 N/A

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

       17. lift-*.f64 N/A

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

       18. *-commutative N/A

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

       19. associate-/l* N/A

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

       20. lift-/.f64 N/A

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

       21. lower-*.f64 33.0%

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

       22. lift-*.f64 N/A

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

       23. lift-*.f64 N/A

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

       24. associate-*r* N/A

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

       25. lower-*.f64 N/A

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

       26. *-commutative N/A

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

       27. lower-*.f64 34.9%

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

   13. Applied rewrites 34.9%

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

   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 25.4%

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

   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 14.6%

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

   4. Applied rewrites 14.6%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   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. lower-pow.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. lift-*.f64 N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 22.2%

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

   6. Applied rewrites 22.2%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 45.2% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (c0 w h D d M)
  :precision binary64
  (let* ((t_0 (* (/ (* d c0) (* D (* h w))) (/ d D)))
       (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
  (if (<=
       (* (/ c0 (* 2.0 w)) (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))
       INFINITY)
    (* (/ c0 (+ w w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
    (* 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 = ((d * c0) / (D * (h * w))) * (d / D);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = (c0 / (w + w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	} 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 = ((d * c0) / (D * (h * w))) * (d / D);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = (c0 / (w + w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
	} 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 = ((d * c0) / (D * (h * w))) * (d / D)
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if ((c0 / (2.0 * w)) * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))) <= math.inf:
		tmp = (c0 / (w + w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))
	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(Float64(Float64(d * c0) / Float64(D * Float64(h * w))) * Float64(d / D))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))))) <= Inf)
		tmp = Float64(Float64(c0 / Float64(w + w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))));
	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 = ((d * c0) / (D * (h * w))) * (d / D);
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= Inf)
		tmp = (c0 / (w + w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	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[(N[(N[(d * c0), $MachinePrecision] / N[(D * N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / D), $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[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * 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[(N[(c0 / N[(w + 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], 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 25.4%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 24.7%

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

   3. Applied rewrites 24.7%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 25.0%

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

   5. Applied rewrites 25.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 35.5%

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

   7. Applied rewrites 35.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 35.5%

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

   9. Applied rewrites 35.5%

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

   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 25.4%

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

   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 14.6%

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

   4. Applied rewrites 14.6%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   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. lower-pow.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. lift-*.f64 N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 22.2%

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

   6. Applied rewrites 22.2%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 45.1% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (c0 w h D d M)
  :precision binary64
  (let* ((t_0 (* (/ d (* (* h D) w)) (* c0 (/ d D))))
       (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
  (if (<=
       (* (/ c0 (* 2.0 w)) (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))
       INFINITY)
    (* (/ c0 (+ w w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
    (* 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 = (d / ((h * D) * w)) * (c0 * (d / D));
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = (c0 / (w + w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	} 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 = (d / ((h * D) * w)) * (c0 * (d / D));
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = (c0 / (w + w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
	} 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 = (d / ((h * D) * w)) * (c0 * (d / D))
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if ((c0 / (2.0 * w)) * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))) <= math.inf:
		tmp = (c0 / (w + w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))
	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(Float64(d / Float64(Float64(h * D) * w)) * Float64(c0 * Float64(d / D)))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))))) <= Inf)
		tmp = Float64(Float64(c0 / Float64(w + w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))));
	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 = (d / ((h * D) * w)) * (c0 * (d / D));
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= Inf)
		tmp = (c0 / (w + w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	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[(N[(d / N[(N[(h * D), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] * N[(c0 * N[(d / D), $MachinePrecision]), $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[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * 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[(N[(c0 / N[(w + 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], 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 25.4%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 24.7%

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

   3. Applied rewrites 24.7%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 25.0%

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

   5. Applied rewrites 25.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 35.5%

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

   7. Applied rewrites 35.5%

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

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lift-/.f64 N/A

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

      19. lower-*.f64 33.0%

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

   9. Applied rewrites 33.0%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. frac-times N/A

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

       5. *-commutative N/A

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

       6. times-frac N/A

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

       7. lower-*.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. associate-*r* N/A

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

       12. lower-*.f64 N/A

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

       13. *-commutative N/A

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

       14. lower-*.f64 N/A

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

       15. lift-*.f64 N/A

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

       16. *-commutative N/A

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

       17. associate-/l* N/A

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

       18. lift-/.f64 N/A

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

       19. lower-*.f64 33.0%

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

   11. Applied rewrites 33.0%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. frac-times N/A

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

       5. *-commutative N/A

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

       6. times-frac N/A

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

       7. lower-*.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. associate-*r* N/A

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

       12. lower-*.f64 N/A

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

       13. *-commutative N/A

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

       14. lower-*.f64 N/A

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

       15. lift-*.f64 N/A

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

       16. *-commutative N/A

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

       17. associate-/l* N/A

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

       18. lift-/.f64 N/A

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

       19. lower-*.f64 35.9%

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

   13. Applied rewrites 35.9%

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

       1. lift-*.f64 N/A

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

       2. count-2-rev N/A

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

       3. lower-+.f64 35.9%

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

   15. Applied rewrites 35.9%

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

   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 25.4%

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

   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 14.6%

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

   4. Applied rewrites 14.6%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   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. lower-pow.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. lift-*.f64 N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 22.2%

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

   6. Applied rewrites 22.2%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 45.0% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (c0 w h D d M)
  :precision binary64
  (let* ((t_0 (* (/ d (* h D)) (* c0 (/ d (* D w)))))
       (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
  (if (<=
       (* (/ c0 (* 2.0 w)) (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))
       INFINITY)
    (* (/ c0 (+ w w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
    (* 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 = (d / (h * D)) * (c0 * (d / (D * w)));
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = (c0 / (w + w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	} 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 = (d / (h * D)) * (c0 * (d / (D * w)));
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = (c0 / (w + w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
	} 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 = (d / (h * D)) * (c0 * (d / (D * w)))
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if ((c0 / (2.0 * w)) * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))) <= math.inf:
		tmp = (c0 / (w + w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))
	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(Float64(d / Float64(h * D)) * Float64(c0 * Float64(d / Float64(D * w))))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))))) <= Inf)
		tmp = Float64(Float64(c0 / Float64(w + w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))));
	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 = (d / (h * D)) * (c0 * (d / (D * w)));
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= Inf)
		tmp = (c0 / (w + w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	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[(N[(d / N[(h * D), $MachinePrecision]), $MachinePrecision] * N[(c0 * N[(d / N[(D * w), $MachinePrecision]), $MachinePrecision]), $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[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * 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[(N[(c0 / N[(w + 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], 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 25.4%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 24.7%

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

   3. Applied rewrites 24.7%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 25.0%

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

   5. Applied rewrites 25.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 35.5%

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

   7. Applied rewrites 35.5%

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

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lift-/.f64 N/A

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

      16. lower-*.f64 33.1%

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 33.1%

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

   9. Applied rewrites 33.1%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-/r* N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lift-*.f64 N/A

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

       9. associate-*r/ N/A

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

       10. lower-/.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. *-commutative N/A

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

       14. associate-/l* N/A

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

       15. lift-/.f64 N/A

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

       16. lower-*.f64 33.6%

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

       17. lift-*.f64 N/A

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

       18. *-commutative N/A

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

       19. lift-*.f64 33.6%

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

   11. Applied rewrites 33.6%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-/r* N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lift-*.f64 N/A

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

       9. associate-*r/ N/A

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

       10. lower-/.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. *-commutative N/A

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

       14. associate-/l* N/A

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

       15. lift-/.f64 N/A

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

       16. lower-*.f64 36.0%

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

       17. lift-*.f64 N/A

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

       18. *-commutative N/A

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

       19. lift-*.f64 36.0%

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

   13. Applied rewrites 36.0%

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

       1. lift-*.f64 N/A

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

       2. count-2-rev N/A

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

       3. lower-+.f64 36.0%

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

   15. Applied rewrites 36.0%

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

       1. lift-/.f64 N/A

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

       2. mult-flip N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. mult-flip N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. associate-/l* N/A

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

       9. lift-/.f64 N/A

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

       10. times-frac N/A

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

       11. lift-*.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. *-commutative N/A

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

       14. associate-*l* N/A

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

       15. lift-*.f64 N/A

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

       16. lift-*.f64 N/A

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

       17. lift-/.f64 N/A

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

       18. lift-/.f64 N/A

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

       19. lift-*.f64 N/A

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

       20. associate-/r* N/A

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

       21. associate-*r/ N/A

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

   17. Applied rewrites 31.4%

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

       1. lift-/.f64 N/A

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

       2. mult-flip N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. mult-flip N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. associate-/l* N/A

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

       9. lift-/.f64 N/A

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

       10. times-frac N/A

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

       11. lift-*.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. *-commutative N/A

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

       14. associate-*l* N/A

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

       15. lift-*.f64 N/A

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

       16. lift-*.f64 N/A

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

       17. lift-/.f64 N/A

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

       18. lift-/.f64 N/A

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

       19. lift-*.f64 N/A

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

       20. associate-/r* N/A

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

       21. associate-*r/ N/A

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

   19. Applied rewrites 31.5%

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

       1. lift-/.f64 N/A

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

       2. mult-flip N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. mult-flip N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. associate-/l* N/A

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

       9. lift-/.f64 N/A

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

       10. times-frac N/A

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

       11. lift-*.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. *-commutative N/A

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

       14. associate-*l* N/A

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

       15. lift-*.f64 N/A

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

       16. lift-*.f64 N/A

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

       17. lift-/.f64 N/A

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

       18. lift-/.f64 N/A

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

       19. lift-*.f64 N/A

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

       20. associate-/r* N/A

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

       21. associate-*r/ N/A

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

   21. Applied rewrites 36.2%

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

   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 25.4%

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

   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 14.6%

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

   4. Applied rewrites 14.6%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   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. lower-pow.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. lift-*.f64 N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 22.2%

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

   6. Applied rewrites 22.2%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 45.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := c0 \cdot \left(d \cdot d\right)\\ t_2 := \frac{t\_1}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_3 := \frac{t\_1}{\left(D \cdot \left(h \cdot w\right)\right) \cdot D}\\ \mathbf{if}\;t\_0 \cdot \left(t\_2 + \sqrt{t\_2 \cdot t\_2 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_0 \cdot \left(t\_3 + \sqrt{t\_3 \cdot t\_3 - M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w}\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
  :precision binary64
  (let* ((t_0 (/ c0 (* 2.0 w)))
       (t_1 (* c0 (* d d)))
       (t_2 (/ t_1 (* (* w h) (* D D))))
       (t_3 (/ t_1 (* (* D (* h w)) D))))
  (if (<= (* t_0 (+ t_2 (sqrt (- (* t_2 t_2) (* M M))))) INFINITY)
    (* t_0 (+ t_3 (sqrt (- (* t_3 t_3) (* M M)))))
    (* 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);
	double t_2 = t_1 / ((w * h) * (D * D));
	double t_3 = t_1 / ((D * (h * w)) * D);
	double tmp;
	if ((t_0 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_0 * (t_3 + sqrt(((t_3 * t_3) - (M * M))));
	} 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);
	double t_2 = t_1 / ((w * h) * (D * D));
	double t_3 = t_1 / ((D * (h * w)) * D);
	double tmp;
	if ((t_0 * (t_2 + Math.sqrt(((t_2 * t_2) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * (t_3 + Math.sqrt(((t_3 * t_3) - (M * M))));
	} 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)
	t_2 = t_1 / ((w * h) * (D * D))
	t_3 = t_1 / ((D * (h * w)) * D)
	tmp = 0
	if (t_0 * (t_2 + math.sqrt(((t_2 * t_2) - (M * M))))) <= math.inf:
		tmp = t_0 * (t_3 + math.sqrt(((t_3 * t_3) - (M * M))))
	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(c0 * Float64(d * d))
	t_2 = Float64(t_1 / Float64(Float64(w * h) * Float64(D * D)))
	t_3 = Float64(t_1 / Float64(Float64(D * Float64(h * w)) * D))
	tmp = 0.0
	if (Float64(t_0 * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_0 * Float64(t_3 + sqrt(Float64(Float64(t_3 * t_3) - Float64(M * M)))));
	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);
	t_2 = t_1 / ((w * h) * (D * D));
	t_3 = t_1 / ((D * (h * w)) * D);
	tmp = 0.0;
	if ((t_0 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= Inf)
		tmp = t_0 * (t_3 + sqrt(((t_3 * t_3) - (M * M))));
	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[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / N[(N[(D * N[(h * w), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 * N[(t$95$3 + N[Sqrt[N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(M * M), $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 25.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 25.0%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 25.0%

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

   3. Applied rewrites 25.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 25.2%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 25.2%

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

   5. Applied rewrites 25.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 28.2%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 28.2%

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

   7. Applied rewrites 28.2%

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

   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 25.4%

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

   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 14.6%

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

   4. Applied rewrites 14.6%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   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. lower-pow.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. lift-*.f64 N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 22.2%

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

   6. Applied rewrites 22.2%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 44.8% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (c0 w h D d M)
  :precision binary64
  (let* ((t_0 (* (/ (* d d) (* (* (* h D) w) D)) c0))
       (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
  (if (<=
       (* (/ c0 (* 2.0 w)) (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))
       INFINITY)
    (* (/ c0 (+ w w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
    (* 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 = ((d * d) / (((h * D) * w) * D)) * c0;
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = (c0 / (w + w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	} 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 = ((d * d) / (((h * D) * w) * D)) * c0;
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = (c0 / (w + w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
	} 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 = ((d * d) / (((h * D) * w) * D)) * c0
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if ((c0 / (2.0 * w)) * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))) <= math.inf:
		tmp = (c0 / (w + w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))
	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(Float64(Float64(d * d) / Float64(Float64(Float64(h * D) * w) * D)) * c0)
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))))) <= Inf)
		tmp = Float64(Float64(c0 / Float64(w + w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))));
	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 = ((d * d) / (((h * D) * w) * D)) * c0;
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= Inf)
		tmp = (c0 / (w + w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	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[(N[(N[(d * d), $MachinePrecision] / N[(N[(N[(h * D), $MachinePrecision] * w), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * c0), $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[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * 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[(N[(c0 / N[(w + 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], 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 25.4%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 24.7%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 24.2%

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

   3. Applied rewrites 24.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 24.3%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 24.4%

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

   5. Applied rewrites 24.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 24.9%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 26.3%

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

   7. Applied rewrites 26.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 25.1%

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 25.6%

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

   9. Applied rewrites 25.6%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. associate-*l* N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. lift-*.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 24.5%

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

       12. lift-*.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. associate-*r* N/A

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

       15. lower-*.f64 N/A

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

       16. *-commutative N/A

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

       17. lower-*.f64 25.6%

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

   11. Applied rewrites 25.6%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. associate-*l* N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. lift-*.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 27.6%

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

       12. lift-*.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. associate-*r* N/A

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

       15. lower-*.f64 N/A

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

       16. *-commutative N/A

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

       17. lower-*.f64 29.3%

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

   13. Applied rewrites 29.3%

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

       1. lift-*.f64 N/A

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

       2. count-2-rev N/A

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

       3. lower-+.f64 29.3%

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

   15. Applied rewrites 29.3%

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

   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 25.4%

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

   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 14.6%

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

   4. Applied rewrites 14.6%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   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. lower-pow.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. lift-*.f64 N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 22.2%

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

   6. Applied rewrites 22.2%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 44.4% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (c0 w h D d M)
  :precision binary64
  (let* ((t_0 (* (* (/ d (* (* (* D D) w) h)) c0) d))
       (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
  (if (<=
       (* (/ c0 (* 2.0 w)) (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))
       INFINITY)
    (* (/ c0 (+ w w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
    (* 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 = ((d / (((D * D) * w) * h)) * c0) * d;
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = (c0 / (w + w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	} 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 = ((d / (((D * D) * w) * h)) * c0) * d;
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = (c0 / (w + w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
	} 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 = ((d / (((D * D) * w) * h)) * c0) * d
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if ((c0 / (2.0 * w)) * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))) <= math.inf:
		tmp = (c0 / (w + w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))
	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(Float64(Float64(d / Float64(Float64(Float64(D * D) * w) * h)) * c0) * d)
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))))) <= Inf)
		tmp = Float64(Float64(c0 / Float64(w + w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))));
	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 = ((d / (((D * D) * w) * h)) * c0) * d;
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= Inf)
		tmp = (c0 / (w + w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	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[(N[(N[(d / N[(N[(N[(D * D), $MachinePrecision] * w), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision] * d), $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[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * 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[(N[(c0 / N[(w + 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], 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 25.4%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 24.7%

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

   3. Applied rewrites 24.7%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 25.0%

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

   5. Applied rewrites 25.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 35.5%

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

   7. Applied rewrites 35.5%

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

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lift-/.f64 N/A

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

      16. lower-*.f64 33.1%

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 33.1%

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

   9. Applied rewrites 33.1%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-/r* N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lift-*.f64 N/A

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

       9. associate-*r/ N/A

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

       10. lower-/.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. *-commutative N/A

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

       14. associate-/l* N/A

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

       15. lift-/.f64 N/A

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

       16. lower-*.f64 33.6%

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

       17. lift-*.f64 N/A

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

       18. *-commutative N/A

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

       19. lift-*.f64 33.6%

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

   11. Applied rewrites 33.6%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-/r* N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lift-*.f64 N/A

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

       9. associate-*r/ N/A

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

       10. lower-/.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. *-commutative N/A

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

       14. associate-/l* N/A

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

       15. lift-/.f64 N/A

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

       16. lower-*.f64 36.0%

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

       17. lift-*.f64 N/A

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

       18. *-commutative N/A

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

       19. lift-*.f64 36.0%

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

   13. Applied rewrites 36.0%

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

       1. lift-*.f64 N/A

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

       2. count-2-rev N/A

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

       3. lower-+.f64 36.0%

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

   15. Applied rewrites 36.0%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. associate-*r/ N/A

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

       7. *-commutative N/A

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

       8. lift-*.f64 N/A

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

       9. frac-times N/A

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

       10. lift-*.f64 N/A

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

       11. associate-*l* N/A

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

       12. lift-*.f64 N/A

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

       13. *-commutative N/A

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

       14. lift-*.f64 N/A

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

       15. lift-*.f64 N/A

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

       16. associate-/r* N/A

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

       17. lift-*.f64 N/A

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

       18. lift-*.f64 N/A

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

       19. *-commutative N/A

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

       20. lift-*.f64 N/A

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

       21. *-commutative N/A

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

       22. lift-*.f64 N/A

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

   17. Applied rewrites 30.3%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. associate-*r/ N/A

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

       7. *-commutative N/A

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

       8. lift-*.f64 N/A

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

       9. frac-times N/A

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

       10. lift-*.f64 N/A

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

       11. associate-*l* N/A

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

       12. lift-*.f64 N/A

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

       13. *-commutative N/A

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

       14. lift-*.f64 N/A

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

       15. lift-*.f64 N/A

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

       16. associate-/r* N/A

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

       17. lift-*.f64 N/A

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

       18. lift-*.f64 N/A

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

       19. *-commutative N/A

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

       20. lift-*.f64 N/A

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

       21. *-commutative N/A

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

       22. lift-*.f64 N/A

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

   19. Applied rewrites 28.2%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. associate-*r/ N/A

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

       7. *-commutative N/A

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

       8. lift-*.f64 N/A

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

       9. frac-times N/A

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

       10. lift-*.f64 N/A

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

       11. associate-*l* N/A

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

       12. lift-*.f64 N/A

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

       13. *-commutative N/A

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

       14. lift-*.f64 N/A

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

       15. lift-*.f64 N/A

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

       16. associate-/r* N/A

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

       17. lift-*.f64 N/A

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

       18. lift-*.f64 N/A

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

       19. *-commutative N/A

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

       20. lift-*.f64 N/A

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

       21. *-commutative N/A

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

       22. lift-*.f64 N/A

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

   21. Applied rewrites 31.3%

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

   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 25.4%

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

   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 14.6%

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

   4. Applied rewrites 14.6%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   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. lower-pow.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. lift-*.f64 N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 22.2%

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

   6. Applied rewrites 22.2%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 43.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left(d \cdot d\right) \cdot c0}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_1 \cdot \left(t\_2 + \sqrt{t\_2 \cdot t\_2 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_1 \cdot \left(t\_2 + \sqrt{\left(t\_0 - M\right) \cdot \left(M + t\_0\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w}\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
  :precision binary64
  (let* ((t_0 (/ (* (* d d) c0) (* (* (* D D) w) h)))
       (t_1 (/ c0 (* 2.0 w)))
       (t_2 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
  (if (<= (* t_1 (+ t_2 (sqrt (- (* t_2 t_2) (* M M))))) INFINITY)
    (* t_1 (+ t_2 (sqrt (* (- t_0 M) (+ M t_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 = ((d * d) * c0) / (((D * D) * w) * h);
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_1 * (t_2 + sqrt(((t_0 - M) * (M + t_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 = ((d * d) * c0) / (((D * D) * w) * h);
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_1 * (t_2 + Math.sqrt(((t_2 * t_2) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_1 * (t_2 + Math.sqrt(((t_0 - M) * (M + t_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 = ((d * d) * c0) / (((D * D) * w) * h)
	t_1 = c0 / (2.0 * w)
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if (t_1 * (t_2 + math.sqrt(((t_2 * t_2) - (M * M))))) <= math.inf:
		tmp = t_1 * (t_2 + math.sqrt(((t_0 - M) * (M + t_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(Float64(Float64(d * d) * c0) / Float64(Float64(Float64(D * D) * w) * h))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_1 * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_1 * Float64(t_2 + sqrt(Float64(Float64(t_0 - M) * Float64(M + t_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 = ((d * d) * c0) / (((D * D) * w) * h);
	t_1 = c0 / (2.0 * w);
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= Inf)
		tmp = t_1 * (t_2 + sqrt(((t_0 - M) * (M + t_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[(N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision] / N[(N[(N[(D * D), $MachinePrecision] * w), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 * N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$1 * N[(t$95$2 + N[Sqrt[N[(N[(t$95$0 - M), $MachinePrecision] * N[(M + t$95$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 25.4%

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

      1. lift--.f64 N/A

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

      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

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

      4. difference-of-squares N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   3. Applied rewrites 30.7%

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

   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 25.4%

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

   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 14.6%

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

   4. Applied rewrites 14.6%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   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. lower-pow.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. lift-*.f64 N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 22.2%

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

   6. Applied rewrites 22.2%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 41.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := c0 \cdot \left(d \cdot d\right)\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{t\_0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_1 \cdot \left(t\_2 + \sqrt{t\_2 \cdot t\_2 - \left|M\right| \cdot \left|M\right|}\right) \leq \infty:\\ \;\;\;\;t\_1 \cdot \left(\frac{t\_0}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} + \left(\left|M\right| \cdot \sqrt{\frac{1}{\left|M\right|}}\right) \cdot \sqrt{\frac{\left|d \cdot \left(\frac{d}{D} \cdot c0\right)\right|}{\left|\left(D \cdot w\right) \cdot h\right|} - \left|M\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot {\left(\left(-\left|M\right|\right) \cdot \left|M\right|\right)}^{0.5}}{w}\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
  :precision binary64
  (let* ((t_0 (* c0 (* d d)))
       (t_1 (/ c0 (* 2.0 w)))
       (t_2 (/ t_0 (* (* w h) (* D D)))))
  (if (<=
       (* t_1 (+ t_2 (sqrt (- (* t_2 t_2) (* (fabs M) (fabs M))))))
       INFINITY)
    (*
     t_1
     (+
      (/ t_0 (* (* h D) (* D w)))
      (*
       (* (fabs M) (sqrt (/ 1.0 (fabs M))))
       (sqrt
        (-
         (/ (fabs (* d (* (/ d D) c0))) (fabs (* (* D w) h)))
         (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 * (d * d);
	double t_1 = c0 / (2.0 * w);
	double t_2 = t_0 / ((w * h) * (D * D));
	double tmp;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (fabs(M) * fabs(M)))))) <= ((double) INFINITY)) {
		tmp = t_1 * ((t_0 / ((h * D) * (D * w))) + ((fabs(M) * sqrt((1.0 / fabs(M)))) * sqrt(((fabs((d * ((d / D) * c0))) / fabs(((D * w) * h))) - 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 * (d * d);
	double t_1 = c0 / (2.0 * w);
	double t_2 = t_0 / ((w * h) * (D * D));
	double tmp;
	if ((t_1 * (t_2 + Math.sqrt(((t_2 * t_2) - (Math.abs(M) * Math.abs(M)))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_1 * ((t_0 / ((h * D) * (D * w))) + ((Math.abs(M) * Math.sqrt((1.0 / Math.abs(M)))) * Math.sqrt(((Math.abs((d * ((d / D) * c0))) / Math.abs(((D * w) * h))) - 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 * (d * d)
	t_1 = c0 / (2.0 * w)
	t_2 = t_0 / ((w * h) * (D * D))
	tmp = 0
	if (t_1 * (t_2 + math.sqrt(((t_2 * t_2) - (math.fabs(M) * math.fabs(M)))))) <= math.inf:
		tmp = t_1 * ((t_0 / ((h * D) * (D * w))) + ((math.fabs(M) * math.sqrt((1.0 / math.fabs(M)))) * math.sqrt(((math.fabs((d * ((d / D) * c0))) / math.fabs(((D * w) * h))) - 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(d * d))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(t_0 / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_1 * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(abs(M) * abs(M)))))) <= Inf)
		tmp = Float64(t_1 * Float64(Float64(t_0 / Float64(Float64(h * D) * Float64(D * w))) + Float64(Float64(abs(M) * sqrt(Float64(1.0 / abs(M)))) * sqrt(Float64(Float64(abs(Float64(d * Float64(Float64(d / D) * c0))) / abs(Float64(Float64(D * w) * h))) - 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 * (d * d);
	t_1 = c0 / (2.0 * w);
	t_2 = t_0 / ((w * h) * (D * D));
	tmp = 0.0;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (abs(M) * abs(M)))))) <= Inf)
		tmp = t_1 * ((t_0 / ((h * D) * (D * w))) + ((abs(M) * sqrt((1.0 / abs(M)))) * sqrt(((abs((d * ((d / D) * c0))) / abs(((D * w) * h))) - 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[(d * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 * N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(N[Abs[M], $MachinePrecision] * N[Abs[M], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$1 * N[(N[(t$95$0 / N[(N[(h * D), $MachinePrecision] * N[(D * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Abs[M], $MachinePrecision] * N[Sqrt[N[(1.0 / N[Abs[M], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(N[Abs[N[(d * N[(N[(d / D), $MachinePrecision] * c0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[N[(N[(D * w), $MachinePrecision] * h), $MachinePrecision]], $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 25.4%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 31.3%

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

   4. Taylor expanded in M around inf

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \color{blue}{\left(M \cdot \sqrt{\frac{1}{M}}\right)} \cdot \sqrt{\left|\frac{\left(d \cdot d\right) \cdot c0}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right| - M}\right) \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 13.6%

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

   6. Applied rewrites 13.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \color{blue}{\left(M \cdot \sqrt{\frac{1}{M}}\right)} \cdot \sqrt{\left|\frac{\left(d \cdot d\right) \cdot c0}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right| - M}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 13.8%

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

   8. Applied rewrites 13.8%

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

   10. Applied rewrites 15.4%

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

   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 25.4%

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

   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 14.6%

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

   4. Applied rewrites 14.6%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   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. lower-pow.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. lift-*.f64 N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 22.2%

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

   6. Applied rewrites 22.2%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 41.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot \left(h \cdot D\right)\right) \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - \left|M\right| \cdot \left|M\right|}\right) \leq \infty:\\ \;\;\;\;c0 \cdot \frac{t\_0 + \sqrt{\left|t\_0\right| - \left|M\right|} \cdot \left(\sqrt{\frac{1}{\left|M\right|}} \cdot \left|M\right|\right)}{w + w}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot {\left(\left(-\left|M\right|\right) \cdot \left|M\right|\right)}^{0.5}}{w}\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
  :precision binary64
  (let* ((t_0 (/ (* (* d d) c0) (* (* D (* h D)) w)))
       (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
  (if (<=
       (*
        (/ c0 (* 2.0 w))
        (+ t_1 (sqrt (- (* t_1 t_1) (* (fabs M) (fabs M))))))
       INFINITY)
    (*
     c0
     (/
      (+
       t_0
       (*
        (sqrt (- (fabs t_0) (fabs M)))
        (* (sqrt (/ 1.0 (fabs M))) (fabs M))))
      (+ w w)))
    (* 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 = ((d * d) * c0) / ((D * (h * D)) * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (fabs(M) * fabs(M)))))) <= ((double) INFINITY)) {
		tmp = c0 * ((t_0 + (sqrt((fabs(t_0) - fabs(M))) * (sqrt((1.0 / fabs(M))) * fabs(M)))) / (w + w));
	} 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 = ((d * d) * c0) / ((D * (h * D)) * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + Math.sqrt(((t_1 * t_1) - (Math.abs(M) * Math.abs(M)))))) <= Double.POSITIVE_INFINITY) {
		tmp = c0 * ((t_0 + (Math.sqrt((Math.abs(t_0) - Math.abs(M))) * (Math.sqrt((1.0 / Math.abs(M))) * Math.abs(M)))) / (w + w));
	} 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 = ((d * d) * c0) / ((D * (h * D)) * w)
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if ((c0 / (2.0 * w)) * (t_1 + math.sqrt(((t_1 * t_1) - (math.fabs(M) * math.fabs(M)))))) <= math.inf:
		tmp = c0 * ((t_0 + (math.sqrt((math.fabs(t_0) - math.fabs(M))) * (math.sqrt((1.0 / math.fabs(M))) * math.fabs(M)))) / (w + w))
	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(Float64(Float64(d * d) * c0) / Float64(Float64(D * Float64(h * D)) * w))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(abs(M) * abs(M)))))) <= Inf)
		tmp = Float64(c0 * Float64(Float64(t_0 + Float64(sqrt(Float64(abs(t_0) - abs(M))) * Float64(sqrt(Float64(1.0 / abs(M))) * abs(M)))) / Float64(w + w)));
	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 = ((d * d) * c0) / ((D * (h * D)) * w);
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (abs(M) * abs(M)))))) <= Inf)
		tmp = c0 * ((t_0 + (sqrt((abs(t_0) - abs(M))) * (sqrt((1.0 / abs(M))) * abs(M)))) / (w + w));
	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[(N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision] / N[(N[(D * N[(h * D), $MachinePrecision]), $MachinePrecision] * 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[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * 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[(c0 * N[(N[(t$95$0 + N[(N[Sqrt[N[(N[Abs[t$95$0], $MachinePrecision] - N[Abs[M], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / N[Abs[M], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Abs[M], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(w + w), $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 25.4%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 31.3%

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

   4. Taylor expanded in M around inf

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \color{blue}{\left(M \cdot \sqrt{\frac{1}{M}}\right)} \cdot \sqrt{\left|\frac{\left(d \cdot d\right) \cdot c0}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right| - M}\right) \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 13.6%

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

   6. Applied rewrites 13.6%

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

   8. Applied rewrites 14.6%

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

   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 25.4%

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

   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 14.6%

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

   4. Applied rewrites 14.6%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   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. lower-pow.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. lift-*.f64 N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 22.2%

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

   6. Applied rewrites 22.2%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 40.9% accurate, 0.5× speedup

Math

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

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) c0) (* (* (* D D) w) h))) (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) * c0) / (((D * D) * w) * h))) - 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) * c0) / (((D * D) * w) * h))) - 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) * c0) / (((D * D) * w) * h))) - 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) * c0) / Float64(Float64(Float64(D * D) * w) * h))) - 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) * c0) / (((D * D) * w) * h))) - 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] * c0), $MachinePrecision] / N[(N[(N[(D * D), $MachinePrecision] * w), $MachinePrecision] * h), $MachinePrecision]), $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 25.4%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 31.3%

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

   4. Taylor expanded in M around inf

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \color{blue}{\left(M \cdot \sqrt{\frac{1}{M}}\right)} \cdot \sqrt{\left|\frac{\left(d \cdot d\right) \cdot c0}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right| - M}\right) \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 13.6%

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

   6. Applied rewrites 13.6%

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

   7. Taylor expanded in M around 0

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

      1. lower-sqrt.f64 13.6%

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

   9. Applied rewrites 13.6%

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

   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 25.4%

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

   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 14.6%

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

   4. Applied rewrites 14.6%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   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. lower-pow.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. lift-*.f64 N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 22.2%

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

   6. Applied rewrites 22.2%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 39.5% accurate, 0.5× speedup

Math

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

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) c0) (* (* (* D D) w) h))) (fabs M))))))
    (* (pow (* (- (fabs M)) (fabs M)) 0.5) (/ c0 (+ w 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) * c0) / (((D * D) * w) * h))) - fabs(M)))));
	} else {
		tmp = pow((-fabs(M) * fabs(M)), 0.5) * (c0 / (w + 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) * c0) / (((D * D) * w) * h))) - Math.abs(M)))));
	} else {
		tmp = Math.pow((-Math.abs(M) * Math.abs(M)), 0.5) * (c0 / (w + 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) * c0) / (((D * D) * w) * h))) - math.fabs(M)))))
	else:
		tmp = math.pow((-math.fabs(M) * math.fabs(M)), 0.5) * (c0 / (w + 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) * c0) / Float64(Float64(Float64(D * D) * w) * h))) - abs(M))))));
	else
		tmp = Float64((Float64(Float64(-abs(M)) * abs(M)) ^ 0.5) * Float64(c0 / Float64(w + 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) * c0) / (((D * D) * w) * h))) - abs(M)))));
	else
		tmp = ((-abs(M) * abs(M)) ^ 0.5) * (c0 / (w + 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] * c0), $MachinePrecision] / N[(N[(N[(D * D), $MachinePrecision] * w), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Abs[M], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[((-N[Abs[M], $MachinePrecision]) * N[Abs[M], $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] * N[(c0 / N[(w + w), $MachinePrecision]), $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 25.4%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 31.3%

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

   4. Taylor expanded in M around inf

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \color{blue}{\left(M \cdot \sqrt{\frac{1}{M}}\right)} \cdot \sqrt{\left|\frac{\left(d \cdot d\right) \cdot c0}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right| - M}\right) \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 13.6%

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

   6. Applied rewrites 13.6%

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

   7. Taylor expanded in M around 0

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

      1. lower-sqrt.f64 13.6%

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

   9. Applied rewrites 13.6%

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

   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 25.4%

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

   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 14.6%

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

   4. Applied rewrites 14.6%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   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 14.6%

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

      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. distribute-lft-neg-out N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 14.6%

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

   6. Applied rewrites 14.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. mult-flip N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. associate-/r* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lower-*.f64 13.1%

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

      13. lift-*.f64 N/A

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

      14. count-2-rev N/A

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

      15. lower-+.f64 13.1%

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

   8. Applied rewrites 13.1%

      \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \color{blue}{\frac{c0}{w + w}} \]
   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 20.6%

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

   10. Applied rewrites 20.6%

       \[\leadsto {\left(\left(-M\right) \cdot M\right)}^{0.5} \cdot \frac{\color{blue}{c0}}{w + w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 36.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|M\right| \leq 2.75 \cdot 10^{-164}:\\ \;\;\;\;\frac{\sqrt{\left(-\left|M\right|\right) \cdot \left|M\right|} \cdot c0}{w + w}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} + \sqrt{\left|M\right|} \cdot \sqrt{\left|\frac{\left(d \cdot d\right) \cdot c0}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right| - \left|M\right|}\right)\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
  :precision binary64
  (if (<= (fabs M) 2.75e-164)
  (/ (* (sqrt (* (- (fabs M)) (fabs M))) c0) (+ w w))
  (*
   (/ c0 (* 2.0 w))
   (+
    (/ (* c0 (* d d)) (* (* h D) (* D w)))
    (*
     (sqrt (fabs M))
     (sqrt
      (- (fabs (/ (* (* d d) c0) (* (* (* D D) w) h))) (fabs M))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (fabs(M) <= 2.75e-164) {
		tmp = (sqrt((-fabs(M) * fabs(M))) * c0) / (w + w);
	} else {
		tmp = (c0 / (2.0 * w)) * (((c0 * (d * d)) / ((h * D) * (D * w))) + (sqrt(fabs(M)) * sqrt((fabs((((d * d) * c0) / (((D * D) * w) * h))) - fabs(M)))));
	}
	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) :: tmp
    if (abs(m) <= 2.75d-164) then
        tmp = (sqrt((-abs(m) * abs(m))) * c0) / (w + w)
    else
        tmp = (c0 / (2.0d0 * w)) * (((c0 * (d_1 * d_1)) / ((h * d) * (d * w))) + (sqrt(abs(m)) * sqrt((abs((((d_1 * d_1) * c0) / (((d * d) * w) * h))) - abs(m)))))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (Math.abs(M) <= 2.75e-164) {
		tmp = (Math.sqrt((-Math.abs(M) * Math.abs(M))) * c0) / (w + w);
	} else {
		tmp = (c0 / (2.0 * w)) * (((c0 * (d * d)) / ((h * D) * (D * w))) + (Math.sqrt(Math.abs(M)) * Math.sqrt((Math.abs((((d * d) * c0) / (((D * D) * w) * h))) - Math.abs(M)))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if math.fabs(M) <= 2.75e-164:
		tmp = (math.sqrt((-math.fabs(M) * math.fabs(M))) * c0) / (w + w)
	else:
		tmp = (c0 / (2.0 * w)) * (((c0 * (d * d)) / ((h * D) * (D * w))) + (math.sqrt(math.fabs(M)) * math.sqrt((math.fabs((((d * d) * c0) / (((D * D) * w) * h))) - math.fabs(M)))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (abs(M) <= 2.75e-164)
		tmp = Float64(Float64(sqrt(Float64(Float64(-abs(M)) * abs(M))) * c0) / Float64(w + w));
	else
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(h * D) * Float64(D * w))) + Float64(sqrt(abs(M)) * sqrt(Float64(abs(Float64(Float64(Float64(d * d) * c0) / Float64(Float64(Float64(D * D) * w) * h))) - abs(M))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (abs(M) <= 2.75e-164)
		tmp = (sqrt((-abs(M) * abs(M))) * c0) / (w + w);
	else
		tmp = (c0 / (2.0 * w)) * (((c0 * (d * d)) / ((h * D) * (D * w))) + (sqrt(abs(M)) * sqrt((abs((((d * d) * c0) / (((D * D) * w) * h))) - abs(M)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[N[Abs[M], $MachinePrecision], 2.75e-164], N[(N[(N[Sqrt[N[((-N[Abs[M], $MachinePrecision]) * N[Abs[M], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision] / N[(w + w), $MachinePrecision]), $MachinePrecision], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(h * D), $MachinePrecision] * N[(D * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[Abs[M], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[Abs[N[(N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision] / N[(N[(N[(D * D), $MachinePrecision] * w), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Abs[M], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.7500000000000001e-164

   1. Initial program 25.4%

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

   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 14.6%

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

   4. Applied rewrites 14.6%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   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 14.6%

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

      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. distribute-lft-neg-out N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 14.6%

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

   6. Applied rewrites 14.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. mult-flip N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. associate-/r* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lower-*.f64 13.1%

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

      13. lift-*.f64 N/A

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

      14. count-2-rev N/A

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

      15. lower-+.f64 13.1%

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

   8. Applied rewrites 13.1%

      \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \color{blue}{\frac{c0}{w + w}} \]
   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. lift-+.f64 N/A

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

      4. count-2-rev N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 14.6%

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lift-+.f64 14.6%

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

   10. Applied rewrites 14.6%

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

   if 2.7500000000000001e-164 < M

   1. Initial program 25.4%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 31.3%

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

   4. Taylor expanded in M around inf

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \color{blue}{\left(M \cdot \sqrt{\frac{1}{M}}\right)} \cdot \sqrt{\left|\frac{\left(d \cdot d\right) \cdot c0}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right| - M}\right) \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 13.6%

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

   6. Applied rewrites 13.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \color{blue}{\left(M \cdot \sqrt{\frac{1}{M}}\right)} \cdot \sqrt{\left|\frac{\left(d \cdot d\right) \cdot c0}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right| - M}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 13.8%

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

   8. Applied rewrites 13.8%

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

   9. Taylor expanded in M around 0

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

       1. lower-sqrt.f64 13.8%

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

   11. Applied rewrites 13.8%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} + \sqrt{M} \cdot \sqrt{\left|\frac{\left(d \cdot d\right) \cdot c0}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right| - M}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 25.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(D \cdot D\right) \cdot w\right) \cdot h\\ \mathbf{if}\;\left|M\right| \leq 3.9 \cdot 10^{-164}:\\ \;\;\;\;\frac{\sqrt{\left(-\left|M\right|\right) \cdot \left|M\right|} \cdot c0}{w + w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\frac{\sqrt{\left(d \cdot d\right) \cdot \left(d \cdot d\right)}}{\left|t\_0\right|} \cdot c0 + \frac{c0}{t\_0} \cdot \left(d \cdot d\right)}{w + w}\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
  :precision binary64
  (let* ((t_0 (* (* (* D D) w) h)))
  (if (<= (fabs M) 3.9e-164)
    (/ (* (sqrt (* (- (fabs M)) (fabs M))) c0) (+ w w))
    (*
     c0
     (/
      (+
       (* (/ (sqrt (* (* d d) (* d d))) (fabs t_0)) c0)
       (* (/ c0 t_0) (* d d)))
      (+ w w))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((D * D) * w) * h;
	double tmp;
	if (fabs(M) <= 3.9e-164) {
		tmp = (sqrt((-fabs(M) * fabs(M))) * c0) / (w + w);
	} else {
		tmp = c0 * ((((sqrt(((d * d) * (d * d))) / fabs(t_0)) * c0) + ((c0 / t_0) * (d * d))) / (w + 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) :: tmp
    t_0 = ((d * d) * w) * h
    if (abs(m) <= 3.9d-164) then
        tmp = (sqrt((-abs(m) * abs(m))) * c0) / (w + w)
    else
        tmp = c0 * ((((sqrt(((d_1 * d_1) * (d_1 * d_1))) / abs(t_0)) * c0) + ((c0 / t_0) * (d_1 * d_1))) / (w + 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 * D) * w) * h;
	double tmp;
	if (Math.abs(M) <= 3.9e-164) {
		tmp = (Math.sqrt((-Math.abs(M) * Math.abs(M))) * c0) / (w + w);
	} else {
		tmp = c0 * ((((Math.sqrt(((d * d) * (d * d))) / Math.abs(t_0)) * c0) + ((c0 / t_0) * (d * d))) / (w + w));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = ((D * D) * w) * h
	tmp = 0
	if math.fabs(M) <= 3.9e-164:
		tmp = (math.sqrt((-math.fabs(M) * math.fabs(M))) * c0) / (w + w)
	else:
		tmp = c0 * ((((math.sqrt(((d * d) * (d * d))) / math.fabs(t_0)) * c0) + ((c0 / t_0) * (d * d))) / (w + w))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(Float64(D * D) * w) * h)
	tmp = 0.0
	if (abs(M) <= 3.9e-164)
		tmp = Float64(Float64(sqrt(Float64(Float64(-abs(M)) * abs(M))) * c0) / Float64(w + w));
	else
		tmp = Float64(c0 * Float64(Float64(Float64(Float64(sqrt(Float64(Float64(d * d) * Float64(d * d))) / abs(t_0)) * c0) + Float64(Float64(c0 / t_0) * Float64(d * d))) / Float64(w + w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = ((D * D) * w) * h;
	tmp = 0.0;
	if (abs(M) <= 3.9e-164)
		tmp = (sqrt((-abs(M) * abs(M))) * c0) / (w + w);
	else
		tmp = c0 * ((((sqrt(((d * d) * (d * d))) / abs(t_0)) * c0) + ((c0 / t_0) * (d * d))) / (w + w));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(N[(D * D), $MachinePrecision] * w), $MachinePrecision] * h), $MachinePrecision]}, If[LessEqual[N[Abs[M], $MachinePrecision], 3.9e-164], N[(N[(N[Sqrt[N[((-N[Abs[M], $MachinePrecision]) * N[Abs[M], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision] / N[(w + w), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[(N[(N[(N[Sqrt[N[(N[(d * d), $MachinePrecision] * N[(d * d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision] + N[(N[(c0 / t$95$0), $MachinePrecision] * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(w + w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < 3.8999999999999999e-164

   1. Initial program 25.4%

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

   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 14.6%

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

   4. Applied rewrites 14.6%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   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 14.6%

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

      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. distribute-lft-neg-out N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 14.6%

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

   6. Applied rewrites 14.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. mult-flip N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. associate-/r* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lower-*.f64 13.1%

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

      13. lift-*.f64 N/A

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

      14. count-2-rev N/A

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

      15. lower-+.f64 13.1%

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

   8. Applied rewrites 13.1%

      \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \color{blue}{\frac{c0}{w + w}} \]
   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. lift-+.f64 N/A

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

      4. count-2-rev N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 14.6%

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lift-+.f64 14.6%

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

   10. Applied rewrites 14.6%

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

   if 3.8999999999999999e-164 < M

   1. Initial program 25.4%

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

   2. Taylor expanded in c0 around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \color{blue}{\sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

      9. lower-pow.f64 9.0%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

   4. Applied rewrites 9.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \color{blue}{c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}}\right) \]

   6. Applied rewrites 16.0%

      \[\leadsto \color{blue}{c0 \cdot \frac{\frac{\sqrt{\left(d \cdot d\right) \cdot \left(d \cdot d\right)}}{\left|\left(\left(D \cdot D\right) \cdot w\right) \cdot h\right|} \cdot c0 + \frac{c0}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} \cdot \left(d \cdot d\right)}{w + w}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 14.6% accurate, 4.2× 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 25.4%

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

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 14.6%

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

4. Applied rewrites 14.6%

   \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
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 14.6%

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

   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. distribute-lft-neg-out N/A

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

   13. lower-*.f64 N/A

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

   14. lower-neg.f64 14.6%

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

6. Applied rewrites 14.6%

   \[\leadsto \frac{\left(0.5 \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{\color{blue}{w}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{\color{blue}{w}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

   3. *-commutative N/A

      \[\leadsto \frac{\left(c0 \cdot \frac{1}{2}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

   4. metadata-eval N/A

      \[\leadsto \frac{\left(c0 \cdot \frac{1}{2}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

   5. mult-flip N/A

      \[\leadsto \frac{\frac{c0}{2} \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\frac{c0}{2} \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

   7. *-commutative N/A

      \[\leadsto \frac{\sqrt{\left(-M\right) \cdot M} \cdot \frac{c0}{2}}{w} \]

   8. associate-/l* N/A

      \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \color{blue}{\frac{\frac{c0}{2}}{w}} \]

   9. associate-/r* N/A

      \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \frac{c0}{\color{blue}{2 \cdot w}} \]

   10. lift-*.f64 N/A

       \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \frac{c0}{2 \cdot \color{blue}{w}} \]

   11. lift-/.f64 N/A

       \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \frac{c0}{\color{blue}{2 \cdot w}} \]

   12. lower-*.f64 13.1%

       \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \color{blue}{\frac{c0}{2 \cdot w}} \]

   13. lift-*.f64 N/A

       \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \frac{c0}{2 \cdot \color{blue}{w}} \]

   14. count-2-rev N/A

       \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \frac{c0}{w + \color{blue}{w}} \]

   15. lower-+.f64 13.1%

       \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \frac{c0}{w + \color{blue}{w}} \]

8. Applied rewrites 13.1%

   \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \color{blue}{\frac{c0}{w + w}} \]
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. lift-+.f64 N/A

      \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \frac{c0}{w + \color{blue}{w}} \]

   4. count-2-rev N/A

      \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \frac{c0}{2 \cdot \color{blue}{w}} \]

   5. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \frac{c0}{2 \cdot \color{blue}{w}} \]

   6. associate-*r/ N/A

      \[\leadsto \frac{\sqrt{\left(-M\right) \cdot M} \cdot c0}{\color{blue}{2 \cdot w}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\sqrt{\left(-M\right) \cdot M} \cdot c0}{\color{blue}{2 \cdot w}} \]

   8. lower-*.f64 14.6%

      \[\leadsto \frac{\sqrt{\left(-M\right) \cdot M} \cdot c0}{\color{blue}{2} \cdot w} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\sqrt{\left(-M\right) \cdot M} \cdot c0}{2 \cdot \color{blue}{w}} \]

   10. count-2-rev N/A

       \[\leadsto \frac{\sqrt{\left(-M\right) \cdot M} \cdot c0}{w + \color{blue}{w}} \]

   11. lift-+.f64 14.6%

       \[\leadsto \frac{\sqrt{\left(-M\right) \cdot M} \cdot c0}{w + \color{blue}{w}} \]

10. Applied rewrites 14.6%

    \[\leadsto \frac{\sqrt{\left(-M\right) \cdot M} \cdot c0}{\color{blue}{w + w}} \]
11. Add Preprocessing

Alternative 16: 13.1% accurate, 4.2× 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 25.4%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

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 14.6%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

4. Applied rewrites 14.6%

   \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
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 14.6%

      \[\leadsto \frac{\left(0.5 \cdot c0\right) \cdot \sqrt{-{M}^{2}}}{w} \]

   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. distribute-lft-neg-out N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\left(\mathsf{neg}\left(M\right)\right) \cdot M}}{w} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\left(\mathsf{neg}\left(M\right)\right) \cdot M}}{w} \]

   14. lower-neg.f64 14.6%

       \[\leadsto \frac{\left(0.5 \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

6. Applied rewrites 14.6%

   \[\leadsto \frac{\left(0.5 \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{\color{blue}{w}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{\color{blue}{w}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\left(\frac{1}{2} \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

   3. *-commutative N/A

      \[\leadsto \frac{\left(c0 \cdot \frac{1}{2}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

   4. metadata-eval N/A

      \[\leadsto \frac{\left(c0 \cdot \frac{1}{2}\right) \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

   5. mult-flip N/A

      \[\leadsto \frac{\frac{c0}{2} \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\frac{c0}{2} \cdot \sqrt{\left(-M\right) \cdot M}}{w} \]

   7. *-commutative N/A

      \[\leadsto \frac{\sqrt{\left(-M\right) \cdot M} \cdot \frac{c0}{2}}{w} \]

   8. associate-/l* N/A

      \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \color{blue}{\frac{\frac{c0}{2}}{w}} \]

   9. associate-/r* N/A

      \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \frac{c0}{\color{blue}{2 \cdot w}} \]

   10. lift-*.f64 N/A

       \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \frac{c0}{2 \cdot \color{blue}{w}} \]

   11. lift-/.f64 N/A

       \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \frac{c0}{\color{blue}{2 \cdot w}} \]

   12. lower-*.f64 13.1%

       \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \color{blue}{\frac{c0}{2 \cdot w}} \]

   13. lift-*.f64 N/A

       \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \frac{c0}{2 \cdot \color{blue}{w}} \]

   14. count-2-rev N/A

       \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \frac{c0}{w + \color{blue}{w}} \]

   15. lower-+.f64 13.1%

       \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \frac{c0}{w + \color{blue}{w}} \]

8. Applied rewrites 13.1%

   \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \color{blue}{\frac{c0}{w + w}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025258 
(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))))))