Henrywood and Agarwal, Equation (13)

Percentage Accurate: 25.7% → 53.9%

Time: 10.2s

Alternatives: 11

Speedup: 3.2×

• could not determine a ground truth

  1. c0 = -1.1098105997106348e+220
  2. w = 1.3602507689287718e-237
  3. h = 4.238287495247169e-294
  4. D = -1.9280782713294762e-289
  5. d = 7.290685897496161e+302
  6. M = -7.18438959375844e-134

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: 25.7% 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: 53.9% accurate, 0.5× speedup

Math

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

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))))
        (t_2 (* (/ (* d c0) (* (* h D) w)) (/ d D)))
        (t_3 (* (- M) M)))
   (if (<= (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M))))) INFINITY)
     (* t_0 (+ t_2 (sqrt (- (* t_2 t_2) (* M M)))))
     (* 0.5 (/ (* c0 (sqrt (sqrt (* t_3 t_3)))) 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 t_2 = ((d * c0) / ((h * D) * w)) * (d / D);
	double t_3 = -M * M;
	double tmp;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_0 * (t_2 + sqrt(((t_2 * t_2) - (M * M))));
	} else {
		tmp = 0.5 * ((c0 * sqrt(sqrt((t_3 * t_3)))) / 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 t_2 = ((d * c0) / ((h * D) * w)) * (d / D);
	double t_3 = -M * M;
	double tmp;
	if ((t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * (t_2 + Math.sqrt(((t_2 * t_2) - (M * M))));
	} else {
		tmp = 0.5 * ((c0 * Math.sqrt(Math.sqrt((t_3 * t_3)))) / 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))
	t_2 = ((d * c0) / ((h * D) * w)) * (d / D)
	t_3 = -M * M
	tmp = 0
	if (t_0 * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))) <= math.inf:
		tmp = t_0 * (t_2 + math.sqrt(((t_2 * t_2) - (M * M))))
	else:
		tmp = 0.5 * ((c0 * math.sqrt(math.sqrt((t_3 * t_3)))) / 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)))
	t_2 = Float64(Float64(Float64(d * c0) / Float64(Float64(h * D) * w)) * Float64(d / D))
	t_3 = Float64(Float64(-M) * M)
	tmp = 0.0
	if (Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_0 * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M)))));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt(Float64(t_3 * t_3)))) / 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));
	t_2 = ((d * c0) / ((h * D) * w)) * (d / D);
	t_3 = -M * M;
	tmp = 0.0;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= Inf)
		tmp = t_0 * (t_2 + sqrt(((t_2 * t_2) - (M * M))));
	else
		tmp = 0.5 * ((c0 * sqrt(sqrt((t_3 * t_3)))) / 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]}, Block[{t$95$2 = N[(N[(N[(d * c0), $MachinePrecision] / N[(N[(h * D), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[((-M) * M), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 * N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Sqrt[N[Sqrt[N[(t$95$3 * t$95$3), $MachinePrecision]], $MachinePrecision]], $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 25.7%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 25.2%

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

   3. Applied rewrites 25.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 25.3%

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

   5. Applied rewrites 25.3%

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

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 33.6%

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

   9. Applied rewrites 33.6%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 33.6%

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

   11. Applied rewrites 33.6%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 35.6%

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

   13. Applied rewrites 35.6%

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

   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 25.7%

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

   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 14.3%

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

   4. Applied rewrites 14.3%

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

      1. lift-sqrt.f64 N/A

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

      2. sqrt-fabs-rev N/A

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

      3. lift-sqrt.f64 N/A

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

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

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

      5. lower-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. sqrt-unprod N/A

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

      17. lower-*.f32 N/A

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

      18. lower-unsound-*.f32 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lower-unsound-*.f64 N/A

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

   6. Applied rewrites 36.4%

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

Alternative 2: 53.3% accurate, 0.5× speedup

Math

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

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))))
        (t_2 (* (* (/ d (* (* (* D h) w) D)) d) c0))
        (t_3 (* (- M) M)))
   (if (<= (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M))))) INFINITY)
     (* t_0 (+ t_2 (sqrt (- (pow t_2 2.0) (* M M)))))
     (* 0.5 (/ (* c0 (sqrt (sqrt (* t_3 t_3)))) 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 t_2 = ((d / (((D * h) * w) * D)) * d) * c0;
	double t_3 = -M * M;
	double tmp;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_0 * (t_2 + sqrt((pow(t_2, 2.0) - (M * M))));
	} else {
		tmp = 0.5 * ((c0 * sqrt(sqrt((t_3 * t_3)))) / 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 t_2 = ((d / (((D * h) * w) * D)) * d) * c0;
	double t_3 = -M * M;
	double tmp;
	if ((t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * (t_2 + Math.sqrt((Math.pow(t_2, 2.0) - (M * M))));
	} else {
		tmp = 0.5 * ((c0 * Math.sqrt(Math.sqrt((t_3 * t_3)))) / 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))
	t_2 = ((d / (((D * h) * w) * D)) * d) * c0
	t_3 = -M * M
	tmp = 0
	if (t_0 * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))) <= math.inf:
		tmp = t_0 * (t_2 + math.sqrt((math.pow(t_2, 2.0) - (M * M))))
	else:
		tmp = 0.5 * ((c0 * math.sqrt(math.sqrt((t_3 * t_3)))) / 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)))
	t_2 = Float64(Float64(Float64(d / Float64(Float64(Float64(D * h) * w) * D)) * d) * c0)
	t_3 = Float64(Float64(-M) * M)
	tmp = 0.0
	if (Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_0 * Float64(t_2 + sqrt(Float64((t_2 ^ 2.0) - Float64(M * M)))));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt(Float64(t_3 * t_3)))) / 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));
	t_2 = ((d / (((D * h) * w) * D)) * d) * c0;
	t_3 = -M * M;
	tmp = 0.0;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= Inf)
		tmp = t_0 * (t_2 + sqrt(((t_2 ^ 2.0) - (M * M))));
	else
		tmp = 0.5 * ((c0 * sqrt(sqrt((t_3 * t_3)))) / 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]}, Block[{t$95$2 = N[(N[(N[(d / N[(N[(N[(D * h), $MachinePrecision] * w), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision] * c0), $MachinePrecision]}, Block[{t$95$3 = N[((-M) * M), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 * N[(t$95$2 + N[Sqrt[N[(N[Power[t$95$2, 2.0], $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Sqrt[N[Sqrt[N[(t$95$3 * t$95$3), $MachinePrecision]], $MachinePrecision]], $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 25.7%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 25.2%

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

   3. Applied rewrites 25.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 25.3%

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

   5. Applied rewrites 25.3%

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

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 33.6%

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

   9. Applied rewrites 33.6%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 33.6%

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

   11. Applied rewrites 33.6%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 35.6%

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

   13. Applied rewrites 35.6%

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

   15. Applied rewrites 34.0%

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

   17. Applied rewrites 33.6%

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

   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 25.7%

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

   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 14.3%

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

   4. Applied rewrites 14.3%

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

      1. lift-sqrt.f64 N/A

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

      2. sqrt-fabs-rev N/A

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

      3. lift-sqrt.f64 N/A

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

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

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

      5. lower-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. sqrt-unprod N/A

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

      17. lower-*.f32 N/A

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

      18. lower-unsound-*.f32 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lower-unsound-*.f64 N/A

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

   6. Applied rewrites 36.4%

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

Alternative 3: 52.3% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (- M) M))
        (t_1 (/ d (* (* (* D h) w) D)))
        (t_2 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_2 (sqrt (- (* t_2 t_2) (* M M)))))
        INFINITY)
     (*
      c0
      (*
       (/ 0.5 w)
       (fma (* t_1 c0) d (sqrt (- (pow (* (* t_1 d) c0) 2.0) (* M M))))))
     (* 0.5 (/ (* c0 (sqrt (sqrt (* t_0 t_0)))) w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = -M * M;
	double t_1 = d / (((D * h) * w) * D);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= ((double) INFINITY)) {
		tmp = c0 * ((0.5 / w) * fma((t_1 * c0), d, sqrt((pow(((t_1 * d) * c0), 2.0) - (M * M)))));
	} else {
		tmp = 0.5 * ((c0 * sqrt(sqrt((t_0 * t_0)))) / w);
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(-M) * M)
	t_1 = Float64(d / Float64(Float64(Float64(D * h) * w) * D))
	t_2 = 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_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M))))) <= Inf)
		tmp = Float64(c0 * Float64(Float64(0.5 / w) * fma(Float64(t_1 * c0), d, sqrt(Float64((Float64(Float64(t_1 * d) * c0) ^ 2.0) - Float64(M * M))))));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt(Float64(t_0 * t_0)))) / w));
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[((-M) * M), $MachinePrecision]}, Block[{t$95$1 = N[(d / N[(N[(N[(D * h), $MachinePrecision] * w), $MachinePrecision] * D), $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[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * 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[(c0 * N[(N[(0.5 / w), $MachinePrecision] * N[(N[(t$95$1 * c0), $MachinePrecision] * d + N[Sqrt[N[(N[Power[N[(N[(t$95$1 * d), $MachinePrecision] * c0), $MachinePrecision], 2.0], $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Sqrt[N[Sqrt[N[(t$95$0 * t$95$0), $MachinePrecision]], $MachinePrecision]], $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 25.7%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 25.2%

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

   3. Applied rewrites 25.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 25.3%

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

   5. Applied rewrites 25.3%

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

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 33.6%

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

   9. Applied rewrites 33.6%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 33.6%

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

   11. Applied rewrites 33.6%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 35.6%

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

   13. Applied rewrites 35.6%

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

   15. Applied rewrites 34.0%

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

   17. Applied rewrites 32.4%

       \[\leadsto \color{blue}{c0 \cdot \left(\frac{0.5}{w} \cdot \mathsf{fma}\left(\frac{d}{\left(\left(D \cdot h\right) \cdot w\right) \cdot D} \cdot c0, d, \sqrt{{\left(\left(\frac{d}{\left(\left(D \cdot h\right) \cdot w\right) \cdot D} \cdot d\right) \cdot c0\right)}^{2} - M \cdot M}\right)\right)} \]

   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 25.7%

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

   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 14.3%

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

   4. Applied rewrites 14.3%

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

      1. lift-sqrt.f64 N/A

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

      2. sqrt-fabs-rev N/A

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

      3. lift-sqrt.f64 N/A

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

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

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

      5. lower-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. sqrt-unprod N/A

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

      17. lower-*.f32 N/A

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

      18. lower-unsound-*.f32 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lower-unsound-*.f64 N/A

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

   6. Applied rewrites 36.4%

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

Alternative 4: 52.3% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ d (* (* (* D h) w) D)))
        (t_1 (* (- M) M))
        (t_2 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_2 (sqrt (- (* t_2 t_2) (* M M)))))
        INFINITY)
     (*
      c0
      (/
       (fma (* t_0 c0) d (sqrt (- (pow (* (* t_0 d) c0) 2.0) (* M M))))
       (+ w w)))
     (* 0.5 (/ (* c0 (sqrt (sqrt (* t_1 t_1)))) w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = d / (((D * h) * w) * D);
	double t_1 = -M * M;
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= ((double) INFINITY)) {
		tmp = c0 * (fma((t_0 * c0), d, sqrt((pow(((t_0 * d) * c0), 2.0) - (M * M)))) / (w + w));
	} else {
		tmp = 0.5 * ((c0 * sqrt(sqrt((t_1 * t_1)))) / w);
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(d / Float64(Float64(Float64(D * h) * w) * D))
	t_1 = Float64(Float64(-M) * M)
	t_2 = 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_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M))))) <= Inf)
		tmp = Float64(c0 * Float64(fma(Float64(t_0 * c0), d, sqrt(Float64((Float64(Float64(t_0 * d) * c0) ^ 2.0) - Float64(M * M)))) / Float64(w + w)));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt(Float64(t_1 * t_1)))) / w));
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(d / N[(N[(N[(D * h), $MachinePrecision] * w), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-M) * M), $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[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * 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[(c0 * N[(N[(N[(t$95$0 * c0), $MachinePrecision] * d + N[Sqrt[N[(N[Power[N[(N[(t$95$0 * d), $MachinePrecision] * c0), $MachinePrecision], 2.0], $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(w + w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Sqrt[N[Sqrt[N[(t$95$1 * t$95$1), $MachinePrecision]], $MachinePrecision]], $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 25.7%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 25.2%

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

   3. Applied rewrites 25.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 25.3%

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

   5. Applied rewrites 25.3%

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

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 33.6%

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

   9. Applied rewrites 33.6%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 33.6%

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

   11. Applied rewrites 33.6%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 35.6%

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

   13. Applied rewrites 35.6%

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

   15. Applied rewrites 34.0%

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

   17. Applied rewrites 32.4%

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

   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 25.7%

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

   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 14.3%

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

   4. Applied rewrites 14.3%

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

      1. lift-sqrt.f64 N/A

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

      2. sqrt-fabs-rev N/A

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

      3. lift-sqrt.f64 N/A

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

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

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

      5. lower-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. sqrt-unprod N/A

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

      17. lower-*.f32 N/A

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

      18. lower-unsound-*.f32 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lower-unsound-*.f64 N/A

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

   6. Applied rewrites 36.4%

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

Alternative 5: 52.0% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_1 (* (/ d (* (* (* D D) w) h)) d))
        (t_2 (* (- M) M)))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
        INFINITY)
     (* c0 (/ (fma t_1 c0 (sqrt (- (pow (* t_1 c0) 2.0) (* M M)))) (+ w w)))
     (* 0.5 (/ (* c0 (sqrt (sqrt (* t_2 t_2)))) w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_1 = (d / (((D * D) * w) * h)) * d;
	double t_2 = -M * M;
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= ((double) INFINITY)) {
		tmp = c0 * (fma(t_1, c0, sqrt((pow((t_1 * c0), 2.0) - (M * M)))) / (w + w));
	} else {
		tmp = 0.5 * ((c0 * sqrt(sqrt((t_2 * t_2)))) / w);
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	t_1 = Float64(Float64(d / Float64(Float64(Float64(D * D) * w) * h)) * d)
	t_2 = Float64(Float64(-M) * M)
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M))))) <= Inf)
		tmp = Float64(c0 * Float64(fma(t_1, c0, sqrt(Float64((Float64(t_1 * c0) ^ 2.0) - Float64(M * M)))) / Float64(w + w)));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt(Float64(t_2 * t_2)))) / w));
	end
	return tmp
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]}, Block[{t$95$1 = N[(N[(d / N[(N[(N[(D * D), $MachinePrecision] * w), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]}, Block[{t$95$2 = N[((-M) * M), $MachinePrecision]}, If[LessEqual[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], Infinity], N[(c0 * N[(N[(t$95$1 * c0 + N[Sqrt[N[(N[Power[N[(t$95$1 * c0), $MachinePrecision], 2.0], $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(w + w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Sqrt[N[Sqrt[N[(t$95$2 * t$95$2), $MachinePrecision]], $MachinePrecision]], $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 25.7%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 25.2%

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

   3. Applied rewrites 25.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 25.3%

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

   5. Applied rewrites 25.3%

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

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

   9. Applied rewrites 30.1%

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

   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 25.7%

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

   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 14.3%

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

   4. Applied rewrites 14.3%

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

      1. lift-sqrt.f64 N/A

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

      2. sqrt-fabs-rev N/A

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

      3. lift-sqrt.f64 N/A

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

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

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

      5. lower-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. sqrt-unprod N/A

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

      17. lower-*.f32 N/A

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

      18. lower-unsound-*.f32 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lower-unsound-*.f64 N/A

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

   6. Applied rewrites 36.4%

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

Alternative 6: 49.0% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (- (fabs M)) (fabs M)))
        (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) (* (fabs M) (fabs M))))))
        INFINITY)
     (*
      t_1
      (+
       (* (/ (* d c0) (* (* h D) w)) (/ d D))
       (*
        (sqrt (fabs M))
        (sqrt (- (* (/ (* d d) (* (* (* D h) w) D)) c0) (fabs M))))))
     (* 0.5 (/ (* c0 (sqrt (sqrt (* t_0 t_0)))) w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = -fabs(M) * fabs(M);
	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) - (fabs(M) * fabs(M)))))) <= ((double) INFINITY)) {
		tmp = t_1 * ((((d * c0) / ((h * D) * w)) * (d / D)) + (sqrt(fabs(M)) * sqrt(((((d * d) / (((D * h) * w) * D)) * c0) - fabs(M)))));
	} else {
		tmp = 0.5 * ((c0 * sqrt(sqrt((t_0 * t_0)))) / w);
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = -Math.abs(M) * Math.abs(M);
	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) - (Math.abs(M) * Math.abs(M)))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_1 * ((((d * c0) / ((h * D) * w)) * (d / D)) + (Math.sqrt(Math.abs(M)) * Math.sqrt(((((d * d) / (((D * h) * w) * D)) * c0) - Math.abs(M)))));
	} else {
		tmp = 0.5 * ((c0 * Math.sqrt(Math.sqrt((t_0 * t_0)))) / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = -math.fabs(M) * math.fabs(M)
	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) - (math.fabs(M) * math.fabs(M)))))) <= math.inf:
		tmp = t_1 * ((((d * c0) / ((h * D) * w)) * (d / D)) + (math.sqrt(math.fabs(M)) * math.sqrt(((((d * d) / (((D * h) * w) * D)) * c0) - math.fabs(M)))))
	else:
		tmp = 0.5 * ((c0 * math.sqrt(math.sqrt((t_0 * t_0)))) / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(-abs(M)) * abs(M))
	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(abs(M) * abs(M)))))) <= Inf)
		tmp = Float64(t_1 * Float64(Float64(Float64(Float64(d * c0) / Float64(Float64(h * D) * w)) * Float64(d / D)) + Float64(sqrt(abs(M)) * sqrt(Float64(Float64(Float64(Float64(d * d) / Float64(Float64(Float64(D * h) * w) * D)) * c0) - abs(M))))));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt(Float64(t_0 * t_0)))) / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = -abs(M) * abs(M);
	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) - (abs(M) * abs(M)))))) <= Inf)
		tmp = t_1 * ((((d * c0) / ((h * D) * w)) * (d / D)) + (sqrt(abs(M)) * sqrt(((((d * d) / (((D * h) * w) * D)) * c0) - abs(M)))));
	else
		tmp = 0.5 * ((c0 * sqrt(sqrt((t_0 * t_0)))) / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[((-N[Abs[M], $MachinePrecision]) * N[Abs[M], $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[(N[Abs[M], $MachinePrecision] * N[Abs[M], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$1 * N[(N[(N[(N[(d * c0), $MachinePrecision] / N[(N[(h * D), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[Abs[M], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(N[(N[(d * d), $MachinePrecision] / N[(N[(N[(D * h), $MachinePrecision] * w), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision] - N[Abs[M], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Sqrt[N[Sqrt[N[(t$95$0 * t$95$0), $MachinePrecision]], $MachinePrecision]], $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 25.7%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 25.2%

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

   3. Applied rewrites 25.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 25.3%

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

   5. Applied rewrites 25.3%

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

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 33.6%

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

   9. Applied rewrites 33.6%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 33.6%

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

   11. Applied rewrites 33.6%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 35.6%

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

   13. Applied rewrites 35.6%

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

   15. Applied rewrites 34.0%

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

   16. Taylor expanded in c0 around 0

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

   18. Applied rewrites 15.4%

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

   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 25.7%

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

   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 14.3%

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

   4. Applied rewrites 14.3%

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

      1. lift-sqrt.f64 N/A

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

      2. sqrt-fabs-rev N/A

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

      3. lift-sqrt.f64 N/A

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

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

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

      5. lower-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. sqrt-unprod N/A

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

      17. lower-*.f32 N/A

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

      18. lower-unsound-*.f32 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lower-unsound-*.f64 N/A

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

   6. Applied rewrites 36.4%

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

Alternative 7: 36.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_2 := \left(-M\right) \cdot M\\ \mathbf{if}\;t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq -1 \cdot 10^{-289}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(d \cdot c0, \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}, \sqrt{-1 \cdot {M}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{t\_2 \cdot t\_2}}}{w}\\ \end{array} \]

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))))
        (t_2 (* (- M) M)))
   (if (<= (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M))))) -1e-289)
     (*
      t_0
      (fma (* d c0) (/ d (* (* (* D D) w) h)) (sqrt (* -1.0 (pow M 2.0)))))
     (* 0.5 (/ (* c0 (sqrt (sqrt (* t_2 t_2)))) 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 t_2 = -M * M;
	double tmp;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= -1e-289) {
		tmp = t_0 * fma((d * c0), (d / (((D * D) * w) * h)), sqrt((-1.0 * pow(M, 2.0))));
	} else {
		tmp = 0.5 * ((c0 * sqrt(sqrt((t_2 * t_2)))) / 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)))
	t_2 = Float64(Float64(-M) * M)
	tmp = 0.0
	if (Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))))) <= -1e-289)
		tmp = Float64(t_0 * fma(Float64(d * c0), Float64(d / Float64(Float64(Float64(D * D) * w) * h)), sqrt(Float64(-1.0 * (M ^ 2.0)))));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt(Float64(t_2 * t_2)))) / w));
	end
	return 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]}, Block[{t$95$2 = N[((-M) * M), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-289], N[(t$95$0 * N[(N[(d * c0), $MachinePrecision] * N[(d / N[(N[(N[(D * D), $MachinePrecision] * w), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(-1.0 * N[Power[M, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Sqrt[N[Sqrt[N[(t$95$2 * t$95$2), $MachinePrecision]], $MachinePrecision]], $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))))) < -1e-289

   1. Initial program 25.7%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 25.2%

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

   3. Applied rewrites 25.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 25.3%

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

   5. Applied rewrites 25.3%

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

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

   9. Applied rewrites 28.9%

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

   10. Taylor expanded in c0 around 0

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot c0, \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}, \sqrt{\color{blue}{-1 \cdot {M}^{2}}}\right) \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

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

       2. lower-pow.f64 10.8%

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

   12. Applied rewrites 10.8%

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

   if -1e-289 < (*.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 25.7%

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

   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 14.3%

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

   4. Applied rewrites 14.3%

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

      1. lift-sqrt.f64 N/A

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

      2. sqrt-fabs-rev N/A

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

      3. lift-sqrt.f64 N/A

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

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

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

      5. lower-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. sqrt-unprod N/A

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

      17. lower-*.f32 N/A

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

      18. lower-unsound-*.f32 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lower-unsound-*.f64 N/A

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

   6. Applied rewrites 36.4%

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

Alternative 8: 36.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} t_0 := \left(-\left|M\right|\right) \cdot \left|M\right|\\ \mathbf{if}\;\left|M\right| \leq 4.5 \cdot 10^{-178}:\\ \;\;\;\;\frac{\sqrt{t\_0} \cdot c0}{w + w}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{t\_0 \cdot t\_0}} \cdot \frac{c0}{w + w}\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (- (fabs M)) (fabs M))))
   (if (<= (fabs M) 4.5e-178)
     (/ (* (sqrt t_0) c0) (+ w w))
     (* (sqrt (sqrt (* t_0 t_0))) (/ c0 (+ w w))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = -fabs(M) * fabs(M);
	double tmp;
	if (fabs(M) <= 4.5e-178) {
		tmp = (sqrt(t_0) * c0) / (w + w);
	} else {
		tmp = sqrt(sqrt((t_0 * t_0))) * (c0 / (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 = -abs(m) * abs(m)
    if (abs(m) <= 4.5d-178) then
        tmp = (sqrt(t_0) * c0) / (w + w)
    else
        tmp = sqrt(sqrt((t_0 * t_0))) * (c0 / (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 = -Math.abs(M) * Math.abs(M);
	double tmp;
	if (Math.abs(M) <= 4.5e-178) {
		tmp = (Math.sqrt(t_0) * c0) / (w + w);
	} else {
		tmp = Math.sqrt(Math.sqrt((t_0 * t_0))) * (c0 / (w + w));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = -math.fabs(M) * math.fabs(M)
	tmp = 0
	if math.fabs(M) <= 4.5e-178:
		tmp = (math.sqrt(t_0) * c0) / (w + w)
	else:
		tmp = math.sqrt(math.sqrt((t_0 * t_0))) * (c0 / (w + w))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(-abs(M)) * abs(M))
	tmp = 0.0
	if (abs(M) <= 4.5e-178)
		tmp = Float64(Float64(sqrt(t_0) * c0) / Float64(w + w));
	else
		tmp = Float64(sqrt(sqrt(Float64(t_0 * t_0))) * Float64(c0 / Float64(w + w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = -abs(M) * abs(M);
	tmp = 0.0;
	if (abs(M) <= 4.5e-178)
		tmp = (sqrt(t_0) * c0) / (w + w);
	else
		tmp = sqrt(sqrt((t_0 * t_0))) * (c0 / (w + w));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[((-N[Abs[M], $MachinePrecision]) * N[Abs[M], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[M], $MachinePrecision], 4.5e-178], N[(N[(N[Sqrt[t$95$0], $MachinePrecision] * c0), $MachinePrecision] / N[(w + w), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[Sqrt[N[(t$95$0 * t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(c0 / N[(w + w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < 4.4999999999999998e-178

   1. Initial program 25.7%

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

   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 14.3%

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

   4. Applied rewrites 14.3%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   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 14.3%

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

      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 14.3%

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

   6. Applied rewrites 14.3%

      \[\leadsto \frac{\left(0.5 \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{\color{blue}{w}} \]
   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. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. mult-flip N/A

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

      10. associate-/l* N/A

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

      11. associate-/r* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lower-*.f64 12.4%

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

      15. lift-*.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 12.4%

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

   8. Applied rewrites 12.4%

      \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \color{blue}{\frac{c0}{w + w}} \]
   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. mult-flip N/A

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

      4. lift-+.f64 N/A

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

      5. count-2-rev N/A

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

      6. lift-*.f64 N/A

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

      7. mult-flip N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 14.3%

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

      11. lift-*.f64 N/A

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

      12. count-2-rev N/A

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

      13. lift-+.f64 14.3%

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

   10. Applied rewrites 14.3%

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

   if 4.4999999999999998e-178 < M

   1. Initial program 25.7%

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

   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 14.3%

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

   4. Applied rewrites 14.3%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   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 14.3%

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

      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 14.3%

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

   6. Applied rewrites 14.3%

      \[\leadsto \frac{\left(0.5 \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{\color{blue}{w}} \]
   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. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. mult-flip N/A

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

      10. associate-/l* N/A

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

      11. associate-/r* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lower-*.f64 12.4%

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

      15. lift-*.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 12.4%

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

   8. Applied rewrites 12.4%

      \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \color{blue}{\frac{c0}{w + w}} \]
   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. sqrt-fabs-rev N/A

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

      3. lift-sqrt.f64 N/A

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

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

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

      5. lower-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. sqrt-unprod N/A

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

      9. lower-*.f32 N/A

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

      10. lower-unsound-*.f32 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-unsound-*.f64 33.8%

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

   10. Applied rewrites 33.8%

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

Alternative 9: 35.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} t_0 := \left(-M\right) \cdot M\\ 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{t\_0 \cdot t\_0}}}{w} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (- M) M))) (* 0.5 (/ (* c0 (sqrt (sqrt (* t_0 t_0)))) w))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = -M * M;
	return 0.5 * ((c0 * sqrt(sqrt((t_0 * t_0)))) / 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
    real(8) :: t_0
    t_0 = -m * m
    code = 0.5d0 * ((c0 * sqrt(sqrt((t_0 * t_0)))) / w)
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = -M * M;
	return 0.5 * ((c0 * Math.sqrt(Math.sqrt((t_0 * t_0)))) / w);
}

Python

def code(c0, w, h, D, d, M):
	t_0 = -M * M
	return 0.5 * ((c0 * math.sqrt(math.sqrt((t_0 * t_0)))) / w)

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(-M) * M)
	return Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt(Float64(t_0 * t_0)))) / w))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	t_0 = -M * M;
	tmp = 0.5 * ((c0 * sqrt(sqrt((t_0 * t_0)))) / w);
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[((-M) * M), $MachinePrecision]}, N[(0.5 * N[(N[(c0 * N[Sqrt[N[Sqrt[N[(t$95$0 * t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 25.7%

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

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 14.3%

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

4. Applied rewrites 14.3%

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

   1. lift-sqrt.f64 N/A

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

   2. sqrt-fabs-rev N/A

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

   3. lift-sqrt.f64 N/A

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

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

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

   5. lower-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-neg.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. pow2 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. lift-neg.f64 N/A

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

   13. lift-pow.f64 N/A

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

   14. pow2 N/A

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

   15. lift-*.f64 N/A

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

   16. sqrt-unprod N/A

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

   17. lower-*.f32 N/A

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

   18. lower-unsound-*.f32 N/A

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

   19. lower-sqrt.f64 N/A

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

   20. lower-unsound-*.f64 N/A

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

6. Applied rewrites 36.4%

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

Alternative 10: 14.3% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(N[(N[Sqrt[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision] / N[(w + w), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 25.7%

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

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 14.3%

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

4. Applied rewrites 14.3%

   \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
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 14.3%

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

   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 14.3%

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

6. Applied rewrites 14.3%

   \[\leadsto \frac{\left(0.5 \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{\color{blue}{w}} \]
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. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. metadata-eval N/A

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

   9. mult-flip N/A

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

   10. associate-/l* N/A

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

   11. associate-/r* N/A

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

   12. lift-*.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. lower-*.f64 12.4%

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

   15. lift-*.f64 N/A

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

   16. count-2-rev N/A

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

   17. lower-+.f64 12.4%

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

8. Applied rewrites 12.4%

   \[\leadsto \sqrt{\left(-M\right) \cdot M} \cdot \color{blue}{\frac{c0}{w + w}} \]
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. mult-flip N/A

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

   4. lift-+.f64 N/A

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

   5. count-2-rev N/A

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

   6. lift-*.f64 N/A

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

   7. mult-flip N/A

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

   8. associate-*r/ N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 14.3%

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

   11. lift-*.f64 N/A

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

   12. count-2-rev N/A

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

   13. lift-+.f64 14.3%

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

10. Applied rewrites 14.3%

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

Alternative 11: 12.4% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(N[Sqrt[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * N[(c0 / N[(w + w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 25.7%

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

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 14.3%

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

4. Applied rewrites 14.3%

   \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
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 14.3%

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

   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 14.3%

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

6. Applied rewrites 14.3%

   \[\leadsto \frac{\left(0.5 \cdot c0\right) \cdot \sqrt{\left(-M\right) \cdot M}}{\color{blue}{w}} \]
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. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. metadata-eval N/A

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

   9. mult-flip N/A

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

   10. associate-/l* N/A

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

   11. associate-/r* N/A

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

   12. lift-*.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. lower-*.f64 12.4%

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

   15. lift-*.f64 N/A

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

   16. count-2-rev N/A

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

   17. lower-+.f64 12.4%

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

8. Applied rewrites 12.4%

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

Reproduce

herbie shell --seed 2025189 
(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))))))