Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.4% → 55.9%

Time: 10.5s

Alternatives: 19

Speedup: 5.2×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 24.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 55.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \left(\frac{1}{D} \cdot d\right)\\ t_1 := {\left(\left(-M\right) \cdot M\right)}^{4}\\ t_2 := \frac{c0}{2 \cdot w}\\ t_3 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_2 \cdot \left(t\_3 + \sqrt{t\_3 \cdot t\_3 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_2 \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{t\_1 \cdot t\_1}}}}}{w}\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ (* d c0) (* D (* h w))) (* (/ 1.0 D) d)))
        (t_1 (pow (* (- M) M) 4.0))
        (t_2 (/ c0 (* 2.0 w)))
        (t_3 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_2 (+ t_3 (sqrt (- (* t_3 t_3) (* M M))))) INFINITY)
     (* t_2 (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
     (* 0.5 (/ (* c0 (sqrt (sqrt (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 * c0) / (D * (h * w))) * ((1.0 / D) * d);
	double t_1 = pow((-M * M), 4.0);
	double t_2 = c0 / (2.0 * w);
	double t_3 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_2 * (t_3 + sqrt(((t_3 * t_3) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_2 * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	} else {
		tmp = 0.5 * ((c0 * sqrt(sqrt(sqrt(sqrt((t_1 * t_1)))))) / w);
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((d * c0) / (D * (h * w))) * ((1.0 / D) * d);
	double t_1 = Math.pow((-M * M), 4.0);
	double t_2 = c0 / (2.0 * w);
	double t_3 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_2 * (t_3 + Math.sqrt(((t_3 * t_3) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_2 * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
	} else {
		tmp = 0.5 * ((c0 * Math.sqrt(Math.sqrt(Math.sqrt(Math.sqrt((t_1 * t_1)))))) / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = ((d * c0) / (D * (h * w))) * ((1.0 / D) * d)
	t_1 = math.pow((-M * M), 4.0)
	t_2 = c0 / (2.0 * w)
	t_3 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if (t_2 * (t_3 + math.sqrt(((t_3 * t_3) - (M * M))))) <= math.inf:
		tmp = t_2 * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))
	else:
		tmp = 0.5 * ((c0 * math.sqrt(math.sqrt(math.sqrt(math.sqrt((t_1 * t_1)))))) / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(Float64(d * c0) / Float64(D * Float64(h * w))) * Float64(Float64(1.0 / D) * d))
	t_1 = Float64(Float64(-M) * M) ^ 4.0
	t_2 = Float64(c0 / Float64(2.0 * w))
	t_3 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_2 * Float64(t_3 + sqrt(Float64(Float64(t_3 * t_3) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_2 * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt(sqrt(sqrt(Float64(t_1 * t_1)))))) / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = ((d * c0) / (D * (h * w))) * ((1.0 / D) * d);
	t_1 = (-M * M) ^ 4.0;
	t_2 = c0 / (2.0 * w);
	t_3 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if ((t_2 * (t_3 + sqrt(((t_3 * t_3) - (M * M))))) <= Inf)
		tmp = t_2 * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	else
		tmp = 0.5 * ((c0 * sqrt(sqrt(sqrt(sqrt((t_1 * t_1)))))) / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(N[(d * c0), $MachinePrecision] / N[(D * N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / D), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[((-M) * M), $MachinePrecision], 4.0], $MachinePrecision]}, Block[{t$95$2 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$2 * N[(t$95$3 + N[Sqrt[N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$2 * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Sqrt[N[Sqrt[N[Sqrt[N[Sqrt[N[(t$95$1 * t$95$1), $MachinePrecision]], $MachinePrecision]], $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 24.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 23.8

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

   3. Applied rewrites 23.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 24.0

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

   5. Applied rewrites 24.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 34.6

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

   7. Applied rewrites 34.6%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 34.1

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

   9. Applied rewrites 34.1%

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

       1. lift-/.f64 N/A

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

       2. mult-flip N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 34.4

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

   11. Applied rewrites 34.4%

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

       1. lift-/.f64 N/A

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

       2. mult-flip N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 34.6

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

   13. Applied rewrites 34.6%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 24.4%

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

   2. Taylor expanded in c0 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-pow.f64 15.0

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

   4. Applied rewrites 15.0%

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

      1. lift-sqrt.f64 N/A

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

      2. 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 37.3%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 39.7

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. swap-sqr N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. sqr-neg-rev N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 39.7

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. swap-sqr N/A

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

   8. Applied rewrites 39.7%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 40.7

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

   10. Applied rewrites 40.7%

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

Alternative 2: 55.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\\ t_1 := {\left(\left(-M\right) \cdot M\right)}^{4}\\ t_2 := \frac{c0}{2 \cdot w}\\ t_3 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_2 \cdot \left(t\_3 + \sqrt{t\_3 \cdot t\_3 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_2 \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{t\_1 \cdot t\_1}}}}}{w}\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ (* d c0) (* D (* h w))) (/ d D)))
        (t_1 (pow (* (- M) M) 4.0))
        (t_2 (/ c0 (* 2.0 w)))
        (t_3 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_2 (+ t_3 (sqrt (- (* t_3 t_3) (* M M))))) INFINITY)
     (* t_2 (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
     (* 0.5 (/ (* c0 (sqrt (sqrt (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 * c0) / (D * (h * w))) * (d / D);
	double t_1 = pow((-M * M), 4.0);
	double t_2 = c0 / (2.0 * w);
	double t_3 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_2 * (t_3 + sqrt(((t_3 * t_3) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_2 * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	} else {
		tmp = 0.5 * ((c0 * sqrt(sqrt(sqrt(sqrt((t_1 * t_1)))))) / w);
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((d * c0) / (D * (h * w))) * (d / D);
	double t_1 = Math.pow((-M * M), 4.0);
	double t_2 = c0 / (2.0 * w);
	double t_3 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_2 * (t_3 + Math.sqrt(((t_3 * t_3) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_2 * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
	} else {
		tmp = 0.5 * ((c0 * Math.sqrt(Math.sqrt(Math.sqrt(Math.sqrt((t_1 * t_1)))))) / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = ((d * c0) / (D * (h * w))) * (d / D)
	t_1 = math.pow((-M * M), 4.0)
	t_2 = c0 / (2.0 * w)
	t_3 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if (t_2 * (t_3 + math.sqrt(((t_3 * t_3) - (M * M))))) <= math.inf:
		tmp = t_2 * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))
	else:
		tmp = 0.5 * ((c0 * math.sqrt(math.sqrt(math.sqrt(math.sqrt((t_1 * t_1)))))) / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(Float64(d * c0) / Float64(D * Float64(h * w))) * Float64(d / D))
	t_1 = Float64(Float64(-M) * M) ^ 4.0
	t_2 = Float64(c0 / Float64(2.0 * w))
	t_3 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_2 * Float64(t_3 + sqrt(Float64(Float64(t_3 * t_3) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_2 * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt(sqrt(sqrt(Float64(t_1 * t_1)))))) / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = ((d * c0) / (D * (h * w))) * (d / D);
	t_1 = (-M * M) ^ 4.0;
	t_2 = c0 / (2.0 * w);
	t_3 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if ((t_2 * (t_3 + sqrt(((t_3 * t_3) - (M * M))))) <= Inf)
		tmp = t_2 * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	else
		tmp = 0.5 * ((c0 * sqrt(sqrt(sqrt(sqrt((t_1 * t_1)))))) / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(N[(d * c0), $MachinePrecision] / N[(D * N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[((-M) * M), $MachinePrecision], 4.0], $MachinePrecision]}, Block[{t$95$2 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$2 * N[(t$95$3 + N[Sqrt[N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$2 * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Sqrt[N[Sqrt[N[Sqrt[N[Sqrt[N[(t$95$1 * t$95$1), $MachinePrecision]], $MachinePrecision]], $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 24.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 23.8

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

   3. Applied rewrites 23.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 24.0

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

   5. Applied rewrites 24.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 34.6

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

   7. Applied rewrites 34.6%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 24.4%

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

   2. Taylor expanded in c0 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-pow.f64 15.0

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

   4. Applied rewrites 15.0%

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

      1. lift-sqrt.f64 N/A

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

      2. 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 37.3%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 39.7

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. swap-sqr N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. sqr-neg-rev N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 39.7

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. swap-sqr N/A

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

   8. Applied rewrites 39.7%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 40.7

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

   10. Applied rewrites 40.7%

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

Alternative 3: 55.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left(d \cdot c0\right) \cdot d}{\left(D \cdot \left(h \cdot w\right)\right) \cdot D}\\ t_1 := {\left(\left(-M\right) \cdot M\right)}^{4}\\ t_2 := \frac{c0}{2 \cdot w}\\ t_3 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_2 \cdot \left(t\_3 + \sqrt{t\_3 \cdot t\_3 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_2 \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{t\_1 \cdot t\_1}}}}}{w}\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* (* d c0) d) (* (* D (* h w)) D)))
        (t_1 (pow (* (- M) M) 4.0))
        (t_2 (/ c0 (* 2.0 w)))
        (t_3 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_2 (+ t_3 (sqrt (- (* t_3 t_3) (* M M))))) INFINITY)
     (* t_2 (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
     (* 0.5 (/ (* c0 (sqrt (sqrt (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 * c0) * d) / ((D * (h * w)) * D);
	double t_1 = pow((-M * M), 4.0);
	double t_2 = c0 / (2.0 * w);
	double t_3 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_2 * (t_3 + sqrt(((t_3 * t_3) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_2 * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	} else {
		tmp = 0.5 * ((c0 * sqrt(sqrt(sqrt(sqrt((t_1 * t_1)))))) / w);
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((d * c0) * d) / ((D * (h * w)) * D);
	double t_1 = Math.pow((-M * M), 4.0);
	double t_2 = c0 / (2.0 * w);
	double t_3 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_2 * (t_3 + Math.sqrt(((t_3 * t_3) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_2 * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
	} else {
		tmp = 0.5 * ((c0 * Math.sqrt(Math.sqrt(Math.sqrt(Math.sqrt((t_1 * t_1)))))) / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = ((d * c0) * d) / ((D * (h * w)) * D)
	t_1 = math.pow((-M * M), 4.0)
	t_2 = c0 / (2.0 * w)
	t_3 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if (t_2 * (t_3 + math.sqrt(((t_3 * t_3) - (M * M))))) <= math.inf:
		tmp = t_2 * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))
	else:
		tmp = 0.5 * ((c0 * math.sqrt(math.sqrt(math.sqrt(math.sqrt((t_1 * t_1)))))) / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(Float64(d * c0) * d) / Float64(Float64(D * Float64(h * w)) * D))
	t_1 = Float64(Float64(-M) * M) ^ 4.0
	t_2 = Float64(c0 / Float64(2.0 * w))
	t_3 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_2 * Float64(t_3 + sqrt(Float64(Float64(t_3 * t_3) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_2 * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt(sqrt(sqrt(Float64(t_1 * t_1)))))) / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = ((d * c0) * d) / ((D * (h * w)) * D);
	t_1 = (-M * M) ^ 4.0;
	t_2 = c0 / (2.0 * w);
	t_3 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if ((t_2 * (t_3 + sqrt(((t_3 * t_3) - (M * M))))) <= Inf)
		tmp = t_2 * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	else
		tmp = 0.5 * ((c0 * sqrt(sqrt(sqrt(sqrt((t_1 * t_1)))))) / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(N[(d * c0), $MachinePrecision] * d), $MachinePrecision] / N[(N[(D * N[(h * w), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[((-M) * M), $MachinePrecision], 4.0], $MachinePrecision]}, Block[{t$95$2 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$2 * N[(t$95$3 + N[Sqrt[N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$2 * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Sqrt[N[Sqrt[N[Sqrt[N[Sqrt[N[(t$95$1 * t$95$1), $MachinePrecision]], $MachinePrecision]], $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 24.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 24.1

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 24.1

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

   3. Applied rewrites 24.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 24.3

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 24.3

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

   5. Applied rewrites 24.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 27.5

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 27.5

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

   7. Applied rewrites 27.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 27.0

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

   9. Applied rewrites 27.0%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 N/A

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

       6. lower-*.f64 27.3

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

   11. Applied rewrites 27.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 N/A

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

       6. lower-*.f64 30.1

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

   13. Applied rewrites 30.1%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 24.4%

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

   2. Taylor expanded in c0 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-pow.f64 15.0

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

   4. Applied rewrites 15.0%

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

      1. lift-sqrt.f64 N/A

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

      2. 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 37.3%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 39.7

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. swap-sqr N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. sqr-neg-rev N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 39.7

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. swap-sqr N/A

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

   8. Applied rewrites 39.7%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 40.7

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

   10. Applied rewrites 40.7%

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

Alternative 4: 55.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := c0 \cdot \left(d \cdot d\right)\\ t_2 := \frac{t\_1}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_3 := \frac{t\_1}{\left(D \cdot \left(h \cdot w\right)\right) \cdot D}\\ t_4 := {\left(\left(-M\right) \cdot M\right)}^{4}\\ \mathbf{if}\;t\_0 \cdot \left(t\_2 + \sqrt{t\_2 \cdot t\_2 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_0 \cdot \left(t\_3 + \sqrt{t\_3 \cdot t\_3 - M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{t\_4 \cdot t\_4}}}}}{w}\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w)))
        (t_1 (* c0 (* d d)))
        (t_2 (/ t_1 (* (* w h) (* D D))))
        (t_3 (/ t_1 (* (* D (* h w)) D)))
        (t_4 (pow (* (- M) M) 4.0)))
   (if (<= (* t_0 (+ t_2 (sqrt (- (* t_2 t_2) (* M M))))) INFINITY)
     (* t_0 (+ t_3 (sqrt (- (* t_3 t_3) (* M M)))))
     (* 0.5 (/ (* c0 (sqrt (sqrt (sqrt (sqrt (* t_4 t_4)))))) w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = c0 * (d * d);
	double t_2 = t_1 / ((w * h) * (D * D));
	double t_3 = t_1 / ((D * (h * w)) * D);
	double t_4 = pow((-M * M), 4.0);
	double tmp;
	if ((t_0 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_0 * (t_3 + sqrt(((t_3 * t_3) - (M * M))));
	} else {
		tmp = 0.5 * ((c0 * sqrt(sqrt(sqrt(sqrt((t_4 * t_4)))))) / w);
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = c0 * (d * d);
	double t_2 = t_1 / ((w * h) * (D * D));
	double t_3 = t_1 / ((D * (h * w)) * D);
	double t_4 = Math.pow((-M * M), 4.0);
	double tmp;
	if ((t_0 * (t_2 + Math.sqrt(((t_2 * t_2) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * (t_3 + Math.sqrt(((t_3 * t_3) - (M * M))));
	} else {
		tmp = 0.5 * ((c0 * Math.sqrt(Math.sqrt(Math.sqrt(Math.sqrt((t_4 * t_4)))))) / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	t_1 = c0 * (d * d)
	t_2 = t_1 / ((w * h) * (D * D))
	t_3 = t_1 / ((D * (h * w)) * D)
	t_4 = math.pow((-M * M), 4.0)
	tmp = 0
	if (t_0 * (t_2 + math.sqrt(((t_2 * t_2) - (M * M))))) <= math.inf:
		tmp = t_0 * (t_3 + math.sqrt(((t_3 * t_3) - (M * M))))
	else:
		tmp = 0.5 * ((c0 * math.sqrt(math.sqrt(math.sqrt(math.sqrt((t_4 * t_4)))))) / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(c0 * Float64(d * d))
	t_2 = Float64(t_1 / Float64(Float64(w * h) * Float64(D * D)))
	t_3 = Float64(t_1 / Float64(Float64(D * Float64(h * w)) * D))
	t_4 = Float64(Float64(-M) * M) ^ 4.0
	tmp = 0.0
	if (Float64(t_0 * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_0 * Float64(t_3 + sqrt(Float64(Float64(t_3 * t_3) - Float64(M * M)))));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt(sqrt(sqrt(Float64(t_4 * t_4)))))) / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	t_1 = c0 * (d * d);
	t_2 = t_1 / ((w * h) * (D * D));
	t_3 = t_1 / ((D * (h * w)) * D);
	t_4 = (-M * M) ^ 4.0;
	tmp = 0.0;
	if ((t_0 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= Inf)
		tmp = t_0 * (t_3 + sqrt(((t_3 * t_3) - (M * M))));
	else
		tmp = 0.5 * ((c0 * sqrt(sqrt(sqrt(sqrt((t_4 * t_4)))))) / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / N[(N[(D * N[(h * w), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[((-M) * M), $MachinePrecision], 4.0], $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 * N[(t$95$3 + N[Sqrt[N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Sqrt[N[Sqrt[N[Sqrt[N[Sqrt[N[(t$95$4 * t$95$4), $MachinePrecision]], $MachinePrecision]], $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 24.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 24.1

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 24.1

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

   3. Applied rewrites 24.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 24.3

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 24.3

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

   5. Applied rewrites 24.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 27.5

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 27.5

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

   7. Applied rewrites 27.5%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 24.4%

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

   2. Taylor expanded in c0 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-pow.f64 15.0

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

   4. Applied rewrites 15.0%

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

      1. lift-sqrt.f64 N/A

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

      2. 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 37.3%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 39.7

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. swap-sqr N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. sqr-neg-rev N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 39.7

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. swap-sqr N/A

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

   8. Applied rewrites 39.7%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 40.7

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

   10. Applied rewrites 40.7%

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

Alternative 5: 55.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\\ t_1 := {\left(\left(-M\right) \cdot M\right)}^{4}\\ t_2 := \frac{c0}{2 \cdot w}\\ t_3 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_2 \cdot \left(t\_3 + \sqrt{t\_3 \cdot t\_3 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_2 \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{t\_1 \cdot t\_1}}}}}{w}\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (* d c0) (/ d (* (* (* D D) w) h))))
        (t_1 (pow (* (- M) M) 4.0))
        (t_2 (/ c0 (* 2.0 w)))
        (t_3 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_2 (+ t_3 (sqrt (- (* t_3 t_3) (* M M))))) INFINITY)
     (* t_2 (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
     (* 0.5 (/ (* c0 (sqrt (sqrt (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 * c0) * (d / (((D * D) * w) * h));
	double t_1 = pow((-M * M), 4.0);
	double t_2 = c0 / (2.0 * w);
	double t_3 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_2 * (t_3 + sqrt(((t_3 * t_3) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_2 * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	} else {
		tmp = 0.5 * ((c0 * sqrt(sqrt(sqrt(sqrt((t_1 * t_1)))))) / w);
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (d * c0) * (d / (((D * D) * w) * h));
	double t_1 = Math.pow((-M * M), 4.0);
	double t_2 = c0 / (2.0 * w);
	double t_3 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_2 * (t_3 + Math.sqrt(((t_3 * t_3) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_2 * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
	} else {
		tmp = 0.5 * ((c0 * Math.sqrt(Math.sqrt(Math.sqrt(Math.sqrt((t_1 * t_1)))))) / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (d * c0) * (d / (((D * D) * w) * h))
	t_1 = math.pow((-M * M), 4.0)
	t_2 = c0 / (2.0 * w)
	t_3 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if (t_2 * (t_3 + math.sqrt(((t_3 * t_3) - (M * M))))) <= math.inf:
		tmp = t_2 * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))
	else:
		tmp = 0.5 * ((c0 * math.sqrt(math.sqrt(math.sqrt(math.sqrt((t_1 * t_1)))))) / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(d * c0) * Float64(d / Float64(Float64(Float64(D * D) * w) * h)))
	t_1 = Float64(Float64(-M) * M) ^ 4.0
	t_2 = Float64(c0 / Float64(2.0 * w))
	t_3 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_2 * Float64(t_3 + sqrt(Float64(Float64(t_3 * t_3) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_2 * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt(sqrt(sqrt(Float64(t_1 * t_1)))))) / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (d * c0) * (d / (((D * D) * w) * h));
	t_1 = (-M * M) ^ 4.0;
	t_2 = c0 / (2.0 * w);
	t_3 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if ((t_2 * (t_3 + sqrt(((t_3 * t_3) - (M * M))))) <= Inf)
		tmp = t_2 * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	else
		tmp = 0.5 * ((c0 * sqrt(sqrt(sqrt(sqrt((t_1 * t_1)))))) / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(d * c0), $MachinePrecision] * N[(d / N[(N[(N[(D * D), $MachinePrecision] * w), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[((-M) * M), $MachinePrecision], 4.0], $MachinePrecision]}, Block[{t$95$2 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$2 * N[(t$95$3 + N[Sqrt[N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$2 * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Sqrt[N[Sqrt[N[Sqrt[N[Sqrt[N[(t$95$1 * t$95$1), $MachinePrecision]], $MachinePrecision]], $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 24.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 23.9

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 23.3

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

   3. Applied rewrites 23.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 23.4

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 23.5

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

   5. Applied rewrites 23.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 27.7

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 29.5

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

   7. Applied rewrites 29.5%

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

   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 24.4%

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

   2. Taylor expanded in c0 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-pow.f64 15.0

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

   4. Applied rewrites 15.0%

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

      1. lift-sqrt.f64 N/A

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

      2. 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 37.3%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 39.7

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. swap-sqr N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. sqr-neg-rev N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 39.7

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. swap-sqr N/A

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

   8. Applied rewrites 39.7%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 40.7

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

   10. Applied rewrites 40.7%

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

Alternative 6: 55.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := {\left(\left(-M\right) \cdot M\right)}^{4}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_2 := t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot t\_2 \leq \infty:\\ \;\;\;\;\frac{c0}{w + w} \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{t\_0 \cdot t\_0}}}}}{w}\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (pow (* (- M) M) 4.0))
        (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_2 (+ t_1 (sqrt (- (* t_1 t_1) (* M M))))))
   (if (<= (* (/ c0 (* 2.0 w)) t_2) INFINITY)
     (* (/ c0 (+ w w)) t_2)
     (* 0.5 (/ (* c0 (sqrt (sqrt (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 = pow((-M * M), 4.0);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_2 = t_1 + sqrt(((t_1 * t_1) - (M * M)));
	double tmp;
	if (((c0 / (2.0 * w)) * t_2) <= ((double) INFINITY)) {
		tmp = (c0 / (w + w)) * t_2;
	} else {
		tmp = 0.5 * ((c0 * sqrt(sqrt(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.pow((-M * M), 4.0);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_2 = t_1 + Math.sqrt(((t_1 * t_1) - (M * M)));
	double tmp;
	if (((c0 / (2.0 * w)) * t_2) <= Double.POSITIVE_INFINITY) {
		tmp = (c0 / (w + w)) * t_2;
	} else {
		tmp = 0.5 * ((c0 * Math.sqrt(Math.sqrt(Math.sqrt(Math.sqrt((t_0 * t_0)))))) / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = math.pow((-M * M), 4.0)
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	t_2 = t_1 + math.sqrt(((t_1 * t_1) - (M * M)))
	tmp = 0
	if ((c0 / (2.0 * w)) * t_2) <= math.inf:
		tmp = (c0 / (w + w)) * t_2
	else:
		tmp = 0.5 * ((c0 * math.sqrt(math.sqrt(math.sqrt(math.sqrt((t_0 * t_0)))))) / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(-M) * M) ^ 4.0
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	t_2 = Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * t_2) <= Inf)
		tmp = Float64(Float64(c0 / Float64(w + w)) * t_2);
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt(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 = (-M * M) ^ 4.0;
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	t_2 = t_1 + sqrt(((t_1 * t_1) - (M * M)));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * t_2) <= Inf)
		tmp = (c0 / (w + w)) * t_2;
	else
		tmp = 0.5 * ((c0 * sqrt(sqrt(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[Power[N[((-M) * M), $MachinePrecision], 4.0], $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[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], Infinity], N[(N[(c0 / N[(w + w), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Sqrt[N[Sqrt[N[Sqrt[N[Sqrt[N[(t$95$0 * t$95$0), $MachinePrecision]], $MachinePrecision]], $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 24.4%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 24.4

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

   3. Applied rewrites 24.4%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 24.4%

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

   2. Taylor expanded in c0 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-pow.f64 15.0

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

   4. Applied rewrites 15.0%

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

      1. lift-sqrt.f64 N/A

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

      2. 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 37.3%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 39.7

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. swap-sqr N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. sqr-neg-rev N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 39.7

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. swap-sqr N/A

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

   8. Applied rewrites 39.7%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 40.7

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

   10. Applied rewrites 40.7%

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

Alternative 7: 54.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := {\left(\left(-M\right) \cdot M\right)}^{4}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_1 \cdot \left(t\_2 + \sqrt{t\_2 \cdot t\_2 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)}, \frac{d}{D}, \sqrt{{\left(\frac{\frac{d}{\left(D \cdot h\right) \cdot w} \cdot \left(d \cdot c0\right)}{D}\right)}^{2} - M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{t\_0 \cdot t\_0}}}}}{w}\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (pow (* (- M) M) 4.0))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_1 (+ t_2 (sqrt (- (* t_2 t_2) (* M M))))) INFINITY)
     (*
      t_1
      (fma
       (/ (* d c0) (* D (* h w)))
       (/ d D)
       (sqrt (- (pow (/ (* (/ d (* (* D h) w)) (* d c0)) D) 2.0) (* M M)))))
     (* 0.5 (/ (* c0 (sqrt (sqrt (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 = pow((-M * M), 4.0);
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_1 * fma(((d * c0) / (D * (h * w))), (d / D), sqrt((pow((((d / ((D * h) * w)) * (d * c0)) / D), 2.0) - (M * M))));
	} else {
		tmp = 0.5 * ((c0 * sqrt(sqrt(sqrt(sqrt((t_0 * t_0)))))) / w);
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(-M) * M) ^ 4.0
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_1 * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_1 * fma(Float64(Float64(d * c0) / Float64(D * Float64(h * w))), Float64(d / D), sqrt(Float64((Float64(Float64(Float64(d / Float64(Float64(D * h) * w)) * Float64(d * c0)) / D) ^ 2.0) - Float64(M * M)))));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt(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[Power[N[((-M) * M), $MachinePrecision], 4.0], $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 * N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$1 * N[(N[(N[(d * c0), $MachinePrecision] / N[(D * N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / D), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(N[(N[(d / N[(N[(D * h), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] * N[(d * c0), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision], 2.0], $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Sqrt[N[Sqrt[N[Sqrt[N[Sqrt[N[(t$95$0 * t$95$0), $MachinePrecision]], $MachinePrecision]], $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 24.4%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

   3. Applied rewrites 23.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 26.1

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 25.8

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

   5. Applied rewrites 25.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

   7. Applied rewrites 30.7%

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

   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 24.4%

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

   2. Taylor expanded in c0 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-pow.f64 15.0

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

   4. Applied rewrites 15.0%

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

      1. lift-sqrt.f64 N/A

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

      2. 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 37.3%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 39.7

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. swap-sqr N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. sqr-neg-rev N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 39.7

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. swap-sqr N/A

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

   8. Applied rewrites 39.7%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 40.7

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

   10. Applied rewrites 40.7%

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

Alternative 8: 53.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{d \cdot c0}{\left(D \cdot \left(D \cdot h\right)\right) \cdot w}\\ t_1 := {\left(\left(-M\right) \cdot M\right)}^{4}\\ t_2 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_2 + \sqrt{t\_2 \cdot t\_2 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(t\_0, d, \sqrt{{\left(t\_0 \cdot d\right)}^{2} - M \cdot M}\right)}{w + w}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{t\_1 \cdot t\_1}}}}}{w}\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* d c0) (* (* D (* D h)) w)))
        (t_1 (pow (* (- M) M) 4.0))
        (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 d (sqrt (- (pow (* t_0 d) 2.0) (* M M)))) (+ w w)))
     (* 0.5 (/ (* c0 (sqrt (sqrt (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 * c0) / ((D * (D * h)) * w);
	double t_1 = pow((-M * M), 4.0);
	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, d, sqrt((pow((t_0 * d), 2.0) - (M * M)))) / (w + w));
	} else {
		tmp = 0.5 * ((c0 * sqrt(sqrt(sqrt(sqrt((t_1 * t_1)))))) / w);
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(d * c0) / Float64(Float64(D * Float64(D * h)) * w))
	t_1 = Float64(Float64(-M) * M) ^ 4.0
	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(t_0, d, sqrt(Float64((Float64(t_0 * d) ^ 2.0) - Float64(M * M)))) / Float64(w + w)));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt(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[(N[(d * c0), $MachinePrecision] / N[(N[(D * N[(D * h), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[((-M) * M), $MachinePrecision], 4.0], $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[(t$95$0 * d + N[Sqrt[N[(N[Power[N[(t$95$0 * d), $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[Sqrt[N[Sqrt[N[(t$95$1 * t$95$1), $MachinePrecision]], $MachinePrecision]], $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 24.4%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

   3. Applied rewrites 23.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 26.1

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 25.8

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

   5. Applied rewrites 25.8%

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

   7. Applied rewrites 29.8%

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

   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 24.4%

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

   2. Taylor expanded in c0 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-pow.f64 15.0

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

   4. Applied rewrites 15.0%

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

      1. lift-sqrt.f64 N/A

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

      2. 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 37.3%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 39.7

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. swap-sqr N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. sqr-neg-rev N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 39.7

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. swap-sqr N/A

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

   8. Applied rewrites 39.7%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 40.7

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

   10. Applied rewrites 40.7%

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

Alternative 9: 43.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := {\left(\left(-M\right) \cdot M\right)}^{4}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_1 \cdot \left(t\_2 + \sqrt{t\_2 \cdot t\_2 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(d \cdot c0, \frac{d}{\left(\left(D \cdot h\right) \cdot D\right) \cdot w}, \left(d \cdot c0\right) \cdot \frac{d}{\left|\left(D \cdot \left(D \cdot h\right)\right) \cdot w\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{t\_0 \cdot t\_0}}}}}{w}\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (pow (* (- M) M) 4.0))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_1 (+ t_2 (sqrt (- (* t_2 t_2) (* M M))))) INFINITY)
     (*
      t_1
      (fma
       (* d c0)
       (/ d (* (* (* D h) D) w))
       (* (* d c0) (/ d (fabs (* (* D (* D h)) w))))))
     (* 0.5 (/ (* c0 (sqrt (sqrt (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 = pow((-M * M), 4.0);
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_1 * fma((d * c0), (d / (((D * h) * D) * w)), ((d * c0) * (d / fabs(((D * (D * h)) * w)))));
	} else {
		tmp = 0.5 * ((c0 * sqrt(sqrt(sqrt(sqrt((t_0 * t_0)))))) / w);
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(-M) * M) ^ 4.0
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_1 * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_1 * fma(Float64(d * c0), Float64(d / Float64(Float64(Float64(D * h) * D) * w)), Float64(Float64(d * c0) * Float64(d / abs(Float64(Float64(D * Float64(D * h)) * w))))));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt(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[Power[N[((-M) * M), $MachinePrecision], 4.0], $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 * N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$1 * N[(N[(d * c0), $MachinePrecision] * N[(d / N[(N[(N[(D * h), $MachinePrecision] * D), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] + N[(N[(d * c0), $MachinePrecision] * N[(d / N[Abs[N[(N[(D * N[(D * h), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Sqrt[N[Sqrt[N[Sqrt[N[Sqrt[N[(t$95$0 * t$95$0), $MachinePrecision]], $MachinePrecision]], $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 24.4%

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

   2. Taylor expanded in c0 around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \color{blue}{\sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

      9. lower-pow.f64 8.6

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

   4. Applied rewrites 8.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \color{blue}{c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}}\right) \]

   6. Applied rewrites 17.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 20.6

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

      11. lift-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

   8. Applied rewrites 22.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 22.7

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 22.7

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

   10. Applied rewrites 22.7%

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

   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 24.4%

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

   2. Taylor expanded in c0 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-pow.f64 15.0

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

   4. Applied rewrites 15.0%

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

      1. lift-sqrt.f64 N/A

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

      2. 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 37.3%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 39.7

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. swap-sqr N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. sqr-neg-rev N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 39.7

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. swap-sqr N/A

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

   8. Applied rewrites 39.7%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 40.7

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

   10. Applied rewrites 40.7%

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

Alternative 10: 42.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(d \cdot c0, \frac{d}{\left(\left(D \cdot h\right) \cdot D\right) \cdot w}, \left(d \cdot c0\right) \cdot \frac{d}{\left|\left(D \cdot \left(D \cdot h\right)\right) \cdot w\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{8}}}}}{w}\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w))) (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M))))) INFINITY)
     (*
      t_0
      (fma
       (* d c0)
       (/ d (* (* (* D h) D) w))
       (* (* d c0) (/ d (fabs (* (* D (* D h)) w))))))
     (* 0.5 (/ (* c0 (sqrt (sqrt (sqrt (pow M 8.0))))) w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_0 * fma((d * c0), (d / (((D * h) * D) * w)), ((d * c0) * (d / fabs(((D * (D * h)) * w)))));
	} else {
		tmp = 0.5 * ((c0 * sqrt(sqrt(sqrt(pow(M, 8.0))))) / w);
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_0 * fma(Float64(d * c0), Float64(d / Float64(Float64(Float64(D * h) * D) * w)), Float64(Float64(d * c0) * Float64(d / abs(Float64(Float64(D * Float64(D * h)) * w))))));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt(sqrt((M ^ 8.0))))) / 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]}, 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[(N[(d * c0), $MachinePrecision] * N[(d / N[(N[(N[(D * h), $MachinePrecision] * D), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] + N[(N[(d * c0), $MachinePrecision] * N[(d / N[Abs[N[(N[(D * N[(D * h), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Sqrt[N[Sqrt[N[Sqrt[N[Power[M, 8.0], $MachinePrecision]], $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 24.4%

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

   2. Taylor expanded in c0 around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \color{blue}{\sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

      9. lower-pow.f64 8.6

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

   4. Applied rewrites 8.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \color{blue}{c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}}\right) \]

   6. Applied rewrites 17.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 20.6

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

      11. lift-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

   8. Applied rewrites 22.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 22.7

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 22.7

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

   10. Applied rewrites 22.7%

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

   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 24.4%

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

   2. Taylor expanded in c0 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-pow.f64 15.0

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

   4. Applied rewrites 15.0%

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

      1. lift-sqrt.f64 N/A

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

      2. 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 37.3%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 39.7

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. swap-sqr N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. sqr-neg-rev N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 39.7

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. swap-sqr N/A

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

   8. Applied rewrites 39.7%

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

   9. Taylor expanded in M around 0

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{8}}}}}{w} \]
   10. Step-by-step derivation

       1. lower-pow.f64 39.7

          \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{8}}}}}{w} \]

   11. Applied rewrites 39.7%

       \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{8}}}}}{w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 42.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \left(D \cdot \left(D \cdot h\right)\right) \cdot w\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_1 \cdot \left(t\_2 + \sqrt{t\_2 \cdot t\_2 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(\frac{d}{t\_0}, d \cdot c0, \left(\frac{d}{\left|t\_0\right|} \cdot d\right) \cdot c0\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{8}}}}}{w}\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (* D (* D h)) w))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_1 (+ t_2 (sqrt (- (* t_2 t_2) (* M M))))) INFINITY)
     (* t_1 (fma (/ d t_0) (* d c0) (* (* (/ d (fabs t_0)) d) c0)))
     (* 0.5 (/ (* c0 (sqrt (sqrt (sqrt (pow M 8.0))))) w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (D * (D * h)) * w;
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_1 * fma((d / t_0), (d * c0), (((d / fabs(t_0)) * d) * c0));
	} else {
		tmp = 0.5 * ((c0 * sqrt(sqrt(sqrt(pow(M, 8.0))))) / w);
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(D * Float64(D * h)) * w)
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_1 * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_1 * fma(Float64(d / t_0), Float64(d * c0), Float64(Float64(Float64(d / abs(t_0)) * d) * c0)));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt(sqrt((M ^ 8.0))))) / w));
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(D * N[(D * h), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 * N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$1 * N[(N[(d / t$95$0), $MachinePrecision] * N[(d * c0), $MachinePrecision] + N[(N[(N[(d / N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision] * c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Sqrt[N[Sqrt[N[Sqrt[N[Power[M, 8.0], $MachinePrecision]], $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 24.4%

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

   2. Taylor expanded in c0 around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \color{blue}{\sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

      9. lower-pow.f64 8.6

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

   4. Applied rewrites 8.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \color{blue}{c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}}\right) \]

   6. Applied rewrites 17.2%

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

   8. Applied rewrites 21.7%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 24.4%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. lower-pow.f64 15.0

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   4. Applied rewrites 15.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      2. 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 37.3%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w} \]
   7. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)} \cdot \sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}}{w} \]

      2. sqrt-unprod N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      4. lower-*.f64 39.7

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      8. swap-sqr N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      9. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      10. lift-neg.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(\mathsf{neg}\left(M\right)\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      11. sqr-neg-rev N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(M \cdot M\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(M \cdot M\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      13. associate-*r* N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(M \cdot M\right) \cdot M\right) \cdot M\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(M \cdot M\right) \cdot M\right) \cdot M\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      15. lower-*.f64 39.7

          \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(M \cdot M\right) \cdot M\right) \cdot M\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(M \cdot M\right) \cdot M\right) \cdot M\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(M \cdot M\right) \cdot M\right) \cdot M\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(M \cdot M\right) \cdot M\right) \cdot M\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      19. swap-sqr N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(M \cdot M\right) \cdot M\right) \cdot M\right) \cdot \left(\left(\left(-M\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

   8. Applied rewrites 39.7%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(M \cdot M\right) \cdot M\right) \cdot M\right) \cdot \left(\left(\left(M \cdot M\right) \cdot M\right) \cdot M\right)}}}}{w} \]

   9. Taylor expanded in M around 0

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{8}}}}}{w} \]
   10. Step-by-step derivation

       1. lower-pow.f64 39.7

          \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{8}}}}}{w} \]

   11. Applied rewrites 39.7%

       \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{8}}}}}{w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 42.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \left(D \cdot \left(D \cdot h\right)\right) \cdot w\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(\frac{d \cdot c0}{t\_0}, d, \left(\frac{d}{\left|t\_0\right|} \cdot d\right) \cdot c0\right)}{w + w}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{8}}}}}{w}\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (* D (* D h)) w)) (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))
        INFINITY)
     (* c0 (/ (fma (/ (* d c0) t_0) d (* (* (/ d (fabs t_0)) d) c0)) (+ w w)))
     (* 0.5 (/ (* c0 (sqrt (sqrt (sqrt (pow M 8.0))))) w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (D * (D * h)) * w;
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = c0 * (fma(((d * c0) / t_0), d, (((d / fabs(t_0)) * d) * c0)) / (w + w));
	} else {
		tmp = 0.5 * ((c0 * sqrt(sqrt(sqrt(pow(M, 8.0))))) / w);
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(D * Float64(D * h)) * w)
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))))) <= Inf)
		tmp = Float64(c0 * Float64(fma(Float64(Float64(d * c0) / t_0), d, Float64(Float64(Float64(d / abs(t_0)) * d) * c0)) / Float64(w + w)));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt(sqrt((M ^ 8.0))))) / w));
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(D * N[(D * h), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(c0 * N[(N[(N[(N[(d * c0), $MachinePrecision] / t$95$0), $MachinePrecision] * d + N[(N[(N[(d / N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision] * c0), $MachinePrecision]), $MachinePrecision] / N[(w + w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Sqrt[N[Sqrt[N[Sqrt[N[Power[M, 8.0], $MachinePrecision]], $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 24.4%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around inf

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \color{blue}{c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \color{blue}{\sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

      9. lower-pow.f64 8.6

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

   4. Applied rewrites 8.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \color{blue}{c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}}\right) \]

   6. Applied rewrites 17.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(d \cdot c0, \frac{1}{\left(h \cdot D\right) \cdot w} \cdot \frac{d}{D}, \frac{d \cdot d}{\sqrt{\left(\left(\left(\left(D \cdot D\right) \cdot w\right) \cdot h\right) \cdot \left(\left(D \cdot D\right) \cdot w\right)\right) \cdot h}} \cdot c0\right)} \]

   8. Applied rewrites 22.0%

      \[\leadsto \color{blue}{c0 \cdot \frac{\mathsf{fma}\left(\frac{d \cdot c0}{\left(D \cdot \left(D \cdot h\right)\right) \cdot w}, d, \left(\frac{d}{\left|\left(D \cdot \left(D \cdot h\right)\right) \cdot w\right|} \cdot d\right) \cdot c0\right)}{w + w}} \]

   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 24.4%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. lower-pow.f64 15.0

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   4. Applied rewrites 15.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      2. 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 37.3%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w} \]
   7. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)} \cdot \sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}}{w} \]

      2. sqrt-unprod N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      4. lower-*.f64 39.7

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      8. swap-sqr N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      9. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      10. lift-neg.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(\mathsf{neg}\left(M\right)\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      11. sqr-neg-rev N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(M \cdot M\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(M \cdot M\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      13. associate-*r* N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(M \cdot M\right) \cdot M\right) \cdot M\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(M \cdot M\right) \cdot M\right) \cdot M\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      15. lower-*.f64 39.7

          \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(M \cdot M\right) \cdot M\right) \cdot M\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(M \cdot M\right) \cdot M\right) \cdot M\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(M \cdot M\right) \cdot M\right) \cdot M\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(M \cdot M\right) \cdot M\right) \cdot M\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      19. swap-sqr N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(M \cdot M\right) \cdot M\right) \cdot M\right) \cdot \left(\left(\left(-M\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

   8. Applied rewrites 39.7%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(M \cdot M\right) \cdot M\right) \cdot M\right) \cdot \left(\left(\left(M \cdot M\right) \cdot M\right) \cdot M\right)}}}}{w} \]

   9. Taylor expanded in M around 0

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{8}}}}}{w} \]
   10. Step-by-step derivation

       1. lower-pow.f64 39.7

          \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{8}}}}}{w} \]

   11. Applied rewrites 39.7%

       \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{8}}}}}{w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 42.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \left(D \cdot \left(D \cdot h\right)\right) \cdot w\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{d \cdot c0}{t\_0}, d, \left(\frac{d}{\left|t\_0\right|} \cdot d\right) \cdot c0\right) \cdot \frac{c0}{w + w}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{8}}}}}{w}\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (* D (* D h)) w)) (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))
        INFINITY)
     (* (fma (/ (* d c0) t_0) d (* (* (/ d (fabs t_0)) d) c0)) (/ c0 (+ w w)))
     (* 0.5 (/ (* c0 (sqrt (sqrt (sqrt (pow M 8.0))))) w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (D * (D * h)) * w;
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = fma(((d * c0) / t_0), d, (((d / fabs(t_0)) * d) * c0)) * (c0 / (w + w));
	} else {
		tmp = 0.5 * ((c0 * sqrt(sqrt(sqrt(pow(M, 8.0))))) / w);
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(D * Float64(D * h)) * w)
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))))) <= Inf)
		tmp = Float64(fma(Float64(Float64(d * c0) / t_0), d, Float64(Float64(Float64(d / abs(t_0)) * d) * c0)) * Float64(c0 / Float64(w + w)));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt(sqrt((M ^ 8.0))))) / w));
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(D * N[(D * h), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(N[(d * c0), $MachinePrecision] / t$95$0), $MachinePrecision] * d + N[(N[(N[(d / N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision] * c0), $MachinePrecision]), $MachinePrecision] * N[(c0 / N[(w + w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Sqrt[N[Sqrt[N[Sqrt[N[Power[M, 8.0], $MachinePrecision]], $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 24.4%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around inf

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \color{blue}{c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \color{blue}{\sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

      9. lower-pow.f64 8.6

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}\right) \]

   4. Applied rewrites 8.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \color{blue}{c0 \cdot \sqrt{\frac{{d}^{4}}{{D}^{4} \cdot \left({h}^{2} \cdot {w}^{2}\right)}}}\right) \]

   6. Applied rewrites 17.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(d \cdot c0, \frac{1}{\left(h \cdot D\right) \cdot w} \cdot \frac{d}{D}, \frac{d \cdot d}{\sqrt{\left(\left(\left(\left(D \cdot D\right) \cdot w\right) \cdot h\right) \cdot \left(\left(D \cdot D\right) \cdot w\right)\right) \cdot h}} \cdot c0\right)} \]

   8. Applied rewrites 21.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{d \cdot c0}{\left(D \cdot \left(D \cdot h\right)\right) \cdot w}, d, \left(\frac{d}{\left|\left(D \cdot \left(D \cdot h\right)\right) \cdot w\right|} \cdot d\right) \cdot c0\right) \cdot \frac{c0}{w + w}} \]

   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 24.4%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. lower-pow.f64 15.0

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   4. Applied rewrites 15.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      2. 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 37.3%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w} \]
   7. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)} \cdot \sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}}{w} \]

      2. sqrt-unprod N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      4. lower-*.f64 39.7

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      8. swap-sqr N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      9. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      10. lift-neg.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(\mathsf{neg}\left(M\right)\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      11. sqr-neg-rev N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(M \cdot M\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(M \cdot M\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      13. associate-*r* N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(M \cdot M\right) \cdot M\right) \cdot M\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(M \cdot M\right) \cdot M\right) \cdot M\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      15. lower-*.f64 39.7

          \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(M \cdot M\right) \cdot M\right) \cdot M\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(M \cdot M\right) \cdot M\right) \cdot M\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(M \cdot M\right) \cdot M\right) \cdot M\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(M \cdot M\right) \cdot M\right) \cdot M\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      19. swap-sqr N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(M \cdot M\right) \cdot M\right) \cdot M\right) \cdot \left(\left(\left(-M\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

   8. Applied rewrites 39.7%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(M \cdot M\right) \cdot M\right) \cdot M\right) \cdot \left(\left(\left(M \cdot M\right) \cdot M\right) \cdot M\right)}}}}{w} \]

   9. Taylor expanded in M around 0

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{8}}}}}{w} \]
   10. Step-by-step derivation

       1. lower-pow.f64 39.7

          \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{8}}}}}{w} \]

   11. Applied rewrites 39.7%

       \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{8}}}}}{w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 39.7% accurate, 2.5× speedup

Math

\[0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{8}}}}}{w} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (* 0.5 (/ (* c0 (sqrt (sqrt (sqrt (pow M 8.0))))) w)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return 0.5 * ((c0 * sqrt(sqrt(sqrt(pow(M, 8.0))))) / 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
    code = 0.5d0 * ((c0 * sqrt(sqrt(sqrt((m ** 8.0d0))))) / w)
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return 0.5 * ((c0 * Math.sqrt(Math.sqrt(Math.sqrt(Math.pow(M, 8.0))))) / w);
}

Python

def code(c0, w, h, D, d, M):
	return 0.5 * ((c0 * math.sqrt(math.sqrt(math.sqrt(math.pow(M, 8.0))))) / w)

Julia

function code(c0, w, h, D, d, M)
	return Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt(sqrt((M ^ 8.0))))) / w))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = 0.5 * ((c0 * sqrt(sqrt(sqrt((M ^ 8.0))))) / w);
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(0.5 * N[(N[(c0 * N[Sqrt[N[Sqrt[N[Sqrt[N[Power[M, 8.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 24.4%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

2. Taylor expanded in c0 around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   5. lower-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   6. lower-pow.f64 15.0

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

4. Applied rewrites 15.0%

   \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
5. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   2. 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 37.3%

   \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w} \]
7. Step-by-step derivation

   1. rem-square-sqrt N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)} \cdot \sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}}{w} \]

   2. sqrt-unprod N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   4. lower-*.f64 39.7

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   8. swap-sqr N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   9. lift-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   10. lift-neg.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(\mathsf{neg}\left(M\right)\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   11. sqr-neg-rev N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(M \cdot M\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(M \cdot M\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   13. associate-*r* N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(M \cdot M\right) \cdot M\right) \cdot M\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(M \cdot M\right) \cdot M\right) \cdot M\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   15. lower-*.f64 39.7

       \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(M \cdot M\right) \cdot M\right) \cdot M\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   16. lift-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(M \cdot M\right) \cdot M\right) \cdot M\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   17. lift-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(M \cdot M\right) \cdot M\right) \cdot M\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   18. lift-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(M \cdot M\right) \cdot M\right) \cdot M\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   19. swap-sqr N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(M \cdot M\right) \cdot M\right) \cdot M\right) \cdot \left(\left(\left(-M\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

8. Applied rewrites 39.7%

   \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(M \cdot M\right) \cdot M\right) \cdot M\right) \cdot \left(\left(\left(M \cdot M\right) \cdot M\right) \cdot M\right)}}}}{w} \]

9. Taylor expanded in M around 0

   \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{8}}}}}{w} \]
10. Step-by-step derivation

    1. lower-pow.f64 39.7

       \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{8}}}}}{w} \]

11. Applied rewrites 39.7%

    \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{8}}}}}{w} \]
12. Add Preprocessing

Alternative 15: 37.3% accurate, 3.4× speedup

Math

\[0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(M \cdot M\right) \cdot M\right) \cdot M}}}{w} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (* 0.5 (/ (* c0 (sqrt (sqrt (* (* (* M M) M) M)))) w)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return 0.5 * ((c0 * sqrt(sqrt((((M * M) * M) * M)))) / w);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = 0.5d0 * ((c0 * sqrt(sqrt((((m * m) * m) * m)))) / w)
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return 0.5 * ((c0 * Math.sqrt(Math.sqrt((((M * M) * M) * M)))) / w);
}

Python

def code(c0, w, h, D, d, M):
	return 0.5 * ((c0 * math.sqrt(math.sqrt((((M * M) * M) * M)))) / w)

Julia

function code(c0, w, h, D, d, M)
	return Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt(Float64(Float64(Float64(M * M) * M) * M)))) / w))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = 0.5 * ((c0 * sqrt(sqrt((((M * M) * M) * M)))) / w);
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(0.5 * N[(N[(c0 * N[Sqrt[N[Sqrt[N[(N[(N[(M * M), $MachinePrecision] * M), $MachinePrecision] * M), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 24.4%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

2. Taylor expanded in c0 around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   5. lower-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   6. lower-pow.f64 15.0

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

4. Applied rewrites 15.0%

   \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
5. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   2. 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 37.3%

   \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w} \]

   4. swap-sqr N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)}}}{w} \]

   5. lift-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)}}}{w} \]

   6. lift-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(\mathsf{neg}\left(M\right)\right)\right) \cdot \left(M \cdot M\right)}}}{w} \]

   7. sqr-neg-rev N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(M \cdot M\right) \cdot \left(M \cdot M\right)}}}{w} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(M \cdot M\right) \cdot \left(M \cdot M\right)}}}{w} \]

   9. associate-*r* N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(M \cdot M\right) \cdot M\right) \cdot M}}}{w} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(M \cdot M\right) \cdot M\right) \cdot M}}}{w} \]

   11. lower-*.f64 37.3

       \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(M \cdot M\right) \cdot M\right) \cdot M}}}{w} \]

8. Applied rewrites 37.3%

   \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(M \cdot M\right) \cdot M\right) \cdot M}}}{w} \]
9. Add Preprocessing

Alternative 16: 32.4% accurate, 5.2× speedup

Math

\[0.5 \cdot \frac{c0 \cdot \sqrt{M \cdot M}}{w} \]

FPCore

(FPCore (c0 w h D d M) :precision binary64 (* 0.5 (/ (* c0 (sqrt (* M M))) w)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return 0.5 * ((c0 * sqrt((M * M))) / w);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = 0.5d0 * ((c0 * sqrt((m * m))) / w)
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return 0.5 * ((c0 * Math.sqrt((M * M))) / w);
}

Python

def code(c0, w, h, D, d, M):
	return 0.5 * ((c0 * math.sqrt((M * M))) / w)

Julia

function code(c0, w, h, D, d, M)
	return Float64(0.5 * Float64(Float64(c0 * sqrt(Float64(M * M))) / w))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = 0.5 * ((c0 * sqrt((M * M))) / w);
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(0.5 * N[(N[(c0 * N[Sqrt[N[(M * M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 24.4%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

2. Taylor expanded in c0 around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   5. lower-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   6. lower-pow.f64 15.0

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

4. Applied rewrites 15.0%

   \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
5. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   2. 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 37.3%

   \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w} \]
7. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w} \]

   3. rem-sqrt-square N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\left|\left(-M\right) \cdot M\right|}}{w} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\left|\left(-M\right) \cdot M\right|}}{w} \]

   5. fabs-mul N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\left|-M\right| \cdot \left|M\right|}}{w} \]

   6. lift-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\left|\mathsf{neg}\left(M\right)\right| \cdot \left|M\right|}}{w} \]

   7. neg-fabs N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\left|M\right| \cdot \left|M\right|}}{w} \]

   8. sqr-abs-rev N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{M \cdot M}}{w} \]

   9. lift-*.f64 32.4

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{M \cdot M}}{w} \]

8. Applied rewrites 32.4%

   \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{M \cdot M}}{w} \]
9. Add Preprocessing

Alternative 17: 24.3% accurate, 6.8× speedup

Math

\[0.5 \cdot \frac{c0 \cdot \left|M\right|}{w} \]

FPCore

(FPCore (c0 w h D d M) :precision binary64 (* 0.5 (/ (* c0 (fabs M)) w)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return 0.5 * ((c0 * fabs(M)) / w);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = 0.5d0 * ((c0 * abs(m)) / w)
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return 0.5 * ((c0 * Math.abs(M)) / w);
}

Python

def code(c0, w, h, D, d, M):
	return 0.5 * ((c0 * math.fabs(M)) / w)

Julia

function code(c0, w, h, D, d, M)
	return Float64(0.5 * Float64(Float64(c0 * abs(M)) / w))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = 0.5 * ((c0 * abs(M)) / w);
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(0.5 * N[(N[(c0 * N[Abs[M], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 24.4%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

2. Taylor expanded in c0 around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   5. lower-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   6. lower-pow.f64 15.0

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

4. Applied rewrites 15.0%

   \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
5. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   2. 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 37.3%

   \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w} \]

7. Taylor expanded in M around 0

   \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot M}{w} \]
8. Step-by-step derivation

9. Applied rewrites 18.9%

   \[\leadsto 0.5 \cdot \frac{c0 \cdot M}{w} \]
10. Add Preprocessing

Alternative 18: 19.5% accurate, 7.6× speedup

Math

\[-0.5 \cdot \frac{M \cdot c0}{w} \]

FPCore

(FPCore (c0 w h D d M) :precision binary64 (* -0.5 (/ (* M c0) w)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return -0.5 * ((M * c0) / w);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = (-0.5d0) * ((m * c0) / w)
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return -0.5 * ((M * c0) / w);
}

Python

def code(c0, w, h, D, d, M):
	return -0.5 * ((M * c0) / w)

Julia

function code(c0, w, h, D, d, M)
	return Float64(-0.5 * Float64(Float64(M * c0) / w))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = -0.5 * ((M * c0) / w);
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(-0.5 * N[(N[(M * c0), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 24.4%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

2. Taylor expanded in c0 around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   5. lower-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   6. lower-pow.f64 15.0

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

4. Applied rewrites 15.0%

   \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
5. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   2. 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 37.3%

   \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w} \]

7. Taylor expanded in M around -inf

   \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{M \cdot c0}{w}} \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{-1}{2} \cdot \frac{M \cdot c0}{\color{blue}{w}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{-1}{2} \cdot \frac{M \cdot c0}{w} \]

   3. lower-*.f64 19.5

      \[\leadsto -0.5 \cdot \frac{M \cdot c0}{w} \]

9. Applied rewrites 19.5%

   \[\leadsto -0.5 \cdot \color{blue}{\frac{M \cdot c0}{w}} \]
10. Add Preprocessing

Alternative 19: 19.3% accurate, 7.6× speedup

Math

\[-0.5 \cdot \left(c0 \cdot \frac{M}{w}\right) \]

FPCore

(FPCore (c0 w h D d M) :precision binary64 (* -0.5 (* c0 (/ M w))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return -0.5 * (c0 * (M / w));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = (-0.5d0) * (c0 * (m / w))
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return -0.5 * (c0 * (M / w));
}

Python

def code(c0, w, h, D, d, M):
	return -0.5 * (c0 * (M / w))

Julia

function code(c0, w, h, D, d, M)
	return Float64(-0.5 * Float64(c0 * Float64(M / w)))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = -0.5 * (c0 * (M / w));
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(-0.5 * N[(c0 * N[(M / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 24.4%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

2. Taylor expanded in c0 around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   5. lower-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   6. lower-pow.f64 15.0

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

4. Applied rewrites 15.0%

   \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
5. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   2. 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 37.3%

   \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w} \]

7. Taylor expanded in M around -inf

   \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{M \cdot c0}{w}} \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{-1}{2} \cdot \frac{M \cdot c0}{\color{blue}{w}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{-1}{2} \cdot \frac{M \cdot c0}{w} \]

   3. lower-*.f64 19.5

      \[\leadsto -0.5 \cdot \frac{M \cdot c0}{w} \]

9. Applied rewrites 19.5%

   \[\leadsto -0.5 \cdot \color{blue}{\frac{M \cdot c0}{w}} \]
10. Step-by-step derivation

    1. lift-/.f64 N/A

       \[\leadsto \frac{-1}{2} \cdot \frac{M \cdot c0}{w} \]

    2. lift-*.f64 N/A

       \[\leadsto \frac{-1}{2} \cdot \frac{M \cdot c0}{w} \]

    3. *-commutative N/A

       \[\leadsto \frac{-1}{2} \cdot \frac{c0 \cdot M}{w} \]

    4. associate-/l* N/A

       \[\leadsto \frac{-1}{2} \cdot \left(c0 \cdot \frac{M}{\color{blue}{w}}\right) \]

    5. lower-*.f64 N/A

       \[\leadsto \frac{-1}{2} \cdot \left(c0 \cdot \frac{M}{\color{blue}{w}}\right) \]

    6. lower-/.f64 19.3

       \[\leadsto -0.5 \cdot \left(c0 \cdot \frac{M}{w}\right) \]

11. Applied rewrites 19.3%

    \[\leadsto -0.5 \cdot \left(c0 \cdot \frac{M}{\color{blue}{w}}\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025178 
(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))))))