Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.7% → 57.9%

Time: 35.1s

Alternatives: 9

Speedup: 21.6×

• could not determine a ground truth

  1. c0 = -3.1412019074855795e+283
  2. w = 1.4833311881827821e-260
  3. h = 3.503398654423981e-274
  4. D = -6.272981434886718e-221
  5. d = -2.9367041093663593e+249
  6. M = -3.848363928268092e-34

Specification

Math

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

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

real(8) function code(c0, w, h, d, d_1, m)
    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: 24.7% accurate, 1.0× speedup

Math

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

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

real(8) function code(c0, w, h, d, d_1, m)
    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: 57.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ \begin{array}{l} t_0 := {\left(\frac{d\_m}{D\_m}\right)}^{2}\\ t_1 := \frac{c0 \cdot t\_0}{w \cdot h}\\ t_2 := \frac{c0}{2 \cdot w}\\ t_3 := \frac{c0 \cdot \left(d\_m \cdot d\_m\right)}{\left(w \cdot h\right) \cdot \left(D\_m \cdot D\_m\right)}\\ t_4 := t\_2 \cdot \left(t\_3 + \sqrt{t\_3 \cdot t\_3 - M \cdot M}\right)\\ \mathbf{if}\;t\_4 \leq -2 \cdot 10^{-182}:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(c0, d\_m \cdot \frac{d\_m}{D\_m \cdot \left(w \cdot \left(h \cdot D\_m\right)\right)}, \left(\frac{c0}{h \cdot D\_m} \cdot \frac{d\_m}{w}\right) \cdot \frac{d\_m}{D\_m}\right)}{2 \cdot w}\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{-141}:\\ \;\;\;\;0.25 \cdot \frac{{D\_m}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d\_m}^{2}}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_2 \cdot \mathsf{fma}\left(\frac{d\_m}{D\_m} \cdot {\left(\sqrt{\sqrt{\frac{c0}{w \cdot h}}}\right)}^{2}, \sqrt{t\_1 - M}, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\frac{1}{c0 \cdot \left(2 \cdot \frac{t\_0}{h \cdot \left(w \cdot {M}^{2}\right)}\right)}}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
(FPCore (c0 w h D_m d_m M)
 :precision binary64
 (let* ((t_0 (pow (/ d_m D_m) 2.0))
        (t_1 (/ (* c0 t_0) (* w h)))
        (t_2 (/ c0 (* 2.0 w)))
        (t_3 (/ (* c0 (* d_m d_m)) (* (* w h) (* D_m D_m))))
        (t_4 (* t_2 (+ t_3 (sqrt (- (* t_3 t_3) (* M M)))))))
   (if (<= t_4 -2e-182)
     (*
      c0
      (/
       (fma
        c0
        (* d_m (/ d_m (* D_m (* w (* h D_m)))))
        (* (* (/ c0 (* h D_m)) (/ d_m w)) (/ d_m D_m)))
       (* 2.0 w)))
     (if (<= t_4 2e-141)
       (* 0.25 (/ (* (pow D_m 2.0) (* h (pow M 2.0))) (pow d_m 2.0)))
       (if (<= t_4 INFINITY)
         (*
          t_2
          (fma
           (* (/ d_m D_m) (pow (sqrt (sqrt (/ c0 (* w h)))) 2.0))
           (sqrt (- t_1 M))
           t_1))
         (*
          c0
          (/
           (/ 1.0 (* c0 (* 2.0 (/ t_0 (* h (* w (pow M 2.0)))))))
           (* 2.0 w))))))))

C

D_m = fabs(D);
d_m = fabs(d);
double code(double c0, double w, double h, double D_m, double d_m, double M) {
	double t_0 = pow((d_m / D_m), 2.0);
	double t_1 = (c0 * t_0) / (w * h);
	double t_2 = c0 / (2.0 * w);
	double t_3 = (c0 * (d_m * d_m)) / ((w * h) * (D_m * D_m));
	double t_4 = t_2 * (t_3 + sqrt(((t_3 * t_3) - (M * M))));
	double tmp;
	if (t_4 <= -2e-182) {
		tmp = c0 * (fma(c0, (d_m * (d_m / (D_m * (w * (h * D_m))))), (((c0 / (h * D_m)) * (d_m / w)) * (d_m / D_m))) / (2.0 * w));
	} else if (t_4 <= 2e-141) {
		tmp = 0.25 * ((pow(D_m, 2.0) * (h * pow(M, 2.0))) / pow(d_m, 2.0));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_2 * fma(((d_m / D_m) * pow(sqrt(sqrt((c0 / (w * h)))), 2.0)), sqrt((t_1 - M)), t_1);
	} else {
		tmp = c0 * ((1.0 / (c0 * (2.0 * (t_0 / (h * (w * pow(M, 2.0))))))) / (2.0 * w));
	}
	return tmp;
}

Julia

D_m = abs(D)
d_m = abs(d)
function code(c0, w, h, D_m, d_m, M)
	t_0 = Float64(d_m / D_m) ^ 2.0
	t_1 = Float64(Float64(c0 * t_0) / Float64(w * h))
	t_2 = Float64(c0 / Float64(2.0 * w))
	t_3 = Float64(Float64(c0 * Float64(d_m * d_m)) / Float64(Float64(w * h) * Float64(D_m * D_m)))
	t_4 = Float64(t_2 * Float64(t_3 + sqrt(Float64(Float64(t_3 * t_3) - Float64(M * M)))))
	tmp = 0.0
	if (t_4 <= -2e-182)
		tmp = Float64(c0 * Float64(fma(c0, Float64(d_m * Float64(d_m / Float64(D_m * Float64(w * Float64(h * D_m))))), Float64(Float64(Float64(c0 / Float64(h * D_m)) * Float64(d_m / w)) * Float64(d_m / D_m))) / Float64(2.0 * w)));
	elseif (t_4 <= 2e-141)
		tmp = Float64(0.25 * Float64(Float64((D_m ^ 2.0) * Float64(h * (M ^ 2.0))) / (d_m ^ 2.0)));
	elseif (t_4 <= Inf)
		tmp = Float64(t_2 * fma(Float64(Float64(d_m / D_m) * (sqrt(sqrt(Float64(c0 / Float64(w * h)))) ^ 2.0)), sqrt(Float64(t_1 - M)), t_1));
	else
		tmp = Float64(c0 * Float64(Float64(1.0 / Float64(c0 * Float64(2.0 * Float64(t_0 / Float64(h * Float64(w * (M ^ 2.0))))))) / Float64(2.0 * w)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
code[c0_, w_, h_, D$95$m_, d$95$m_, M_] := Block[{t$95$0 = N[Power[N[(d$95$m / D$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * t$95$0), $MachinePrecision] / N[(w * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(c0 * N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = 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]}, If[LessEqual[t$95$4, -2e-182], N[(c0 * N[(N[(c0 * N[(d$95$m * N[(d$95$m / N[(D$95$m * N[(w * N[(h * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(c0 / N[(h * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(d$95$m / w), $MachinePrecision]), $MachinePrecision] * N[(d$95$m / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2e-141], N[(0.25 * N[(N[(N[Power[D$95$m, 2.0], $MachinePrecision] * N[(h * N[Power[M, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[d$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(t$95$2 * N[(N[(N[(d$95$m / D$95$m), $MachinePrecision] * N[Power[N[Sqrt[N[Sqrt[N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$1 - M), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[(1.0 / N[(c0 * N[(2.0 * N[(t$95$0 / N[(h * N[(w * N[Power[M, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (/.f64 c0 (*.f64 2 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))))) < -2.0000000000000001e-182

   1. Initial program 81.5%

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

   3. Simplified 86.8%

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

   5. Taylor expanded in c0 around inf 81.6%

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

      1. *-commutative 81.6%

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

      2. *-commutative 81.6%

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

      3. associate-*r* 78.9%

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

      4. times-frac 79.0%

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

      5. *-commutative 79.0%

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

   7. Simplified 79.0%

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

      1. associate-*r/ 81.6%

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

      2. *-commutative 81.6%

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

   9. Applied egg-rr 81.6%

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

       1. times-frac 84.2%

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

       2. associate-/r* 84.2%

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

       3. pow2 84.2%

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

       4. pow2 84.2%

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

       5. frac-times 86.8%

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

       6. associate-*r* 89.4%

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

       7. *-commutative 89.4%

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

   11. Applied egg-rr 89.4%

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

   12. Taylor expanded in c0 around 0 89.4%

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

       1. associate-*r* 89.4%

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

       2. times-frac 89.5%

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

   14. Simplified 89.5%

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

   if -2.0000000000000001e-182 < (*.f64 (/.f64 c0 (*.f64 2 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))))) < 2.0000000000000001e-141

   1. Initial program 53.1%

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

   3. Simplified 16.6%

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

   6. Applied egg-rr 44.6%

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

      1. associate--r- 62.6%

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

      2. +-inverses 68.9%

         \[\leadsto c0 \cdot \frac{\frac{\color{blue}{0} + {M}^{2}}{\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h} - \sqrt{{\left(\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

      3. associate-/l* 62.8%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{\color{blue}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h}} - \sqrt{{\left(\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

      4. *-commutative 62.8%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{\color{blue}{h \cdot w}} - \sqrt{{\left(\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

      5. associate-/l* 62.7%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w} - \sqrt{{\color{blue}{\left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}}^{2} - {M}^{2}}}}{2 \cdot w} \]

      6. *-commutative 62.7%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w} - \sqrt{{\left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{\color{blue}{h \cdot w}}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

   8. Simplified 62.7%

      \[\leadsto c0 \cdot \frac{\color{blue}{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w} - \sqrt{{\left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w}\right)}^{2} - {M}^{2}}}}}{2 \cdot w} \]

   9. Taylor expanded in c0 around -inf 62.7%

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

       1. associate-*r/ 62.7%

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

       2. neg-mul-1 62.7%

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

       3. associate-*r/ 62.7%

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

       4. mul-1-neg 62.7%

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

   11. Simplified 62.7%

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

       1. clear-num 62.8%

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

       2. inv-pow 62.8%

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

       3. sub-div 62.8%

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

       4. *-commutative 62.8%

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

   13. Applied egg-rr 62.8%

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

       1. unpow-1 62.8%

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

       2. associate-/l* 62.8%

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

   15. Simplified 63.0%

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

   16. Taylor expanded in c0 around 0 74.9%

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

   if 2.0000000000000001e-141 < (*.f64 (/.f64 c0 (*.f64 2 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 85.8%

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

   3. Simplified 85.8%

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

   6. Applied egg-rr 94.9%

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

   7. Taylor expanded in c0 around inf 54.3%

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

      1. add-sqr-sqrt 54.1%

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

      2. pow2 54.1%

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

   9. Applied egg-rr 54.1%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 2 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 0.0%

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

   3. Simplified 20.7%

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

   6. Applied egg-rr 2.1%

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

      1. associate--r- 3.1%

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

      2. +-inverses 39.8%

         \[\leadsto c0 \cdot \frac{\frac{\color{blue}{0} + {M}^{2}}{\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h} - \sqrt{{\left(\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

      3. associate-/l* 40.4%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{\color{blue}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h}} - \sqrt{{\left(\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

      4. *-commutative 40.4%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{\color{blue}{h \cdot w}} - \sqrt{{\left(\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

      5. associate-/l* 39.7%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w} - \sqrt{{\color{blue}{\left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}}^{2} - {M}^{2}}}}{2 \cdot w} \]

      6. *-commutative 39.7%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w} - \sqrt{{\left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{\color{blue}{h \cdot w}}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

   8. Simplified 39.7%

      \[\leadsto c0 \cdot \frac{\color{blue}{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w} - \sqrt{{\left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w}\right)}^{2} - {M}^{2}}}}}{2 \cdot w} \]

   9. Taylor expanded in c0 around -inf 35.4%

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

       1. associate-*r/ 35.4%

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

       2. neg-mul-1 35.4%

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

       3. associate-*r/ 35.4%

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

       4. mul-1-neg 35.4%

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

   11. Simplified 35.4%

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

       1. clear-num 35.4%

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

       2. inv-pow 35.4%

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

       3. sub-div 35.4%

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

       4. *-commutative 35.4%

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

   13. Applied egg-rr 35.4%

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

       1. unpow-1 35.4%

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

       2. associate-/l* 36.0%

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

   15. Simplified 48.9%

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

   16. Taylor expanded in d around 0 36.6%

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

       1. associate-/r* 35.4%

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

       2. unpow2 35.4%

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

       3. unpow2 35.4%

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

       4. times-frac 44.9%

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

       5. unpow2 44.9%

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

       6. *-commutative 44.9%

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

       7. associate-*l* 50.5%

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

   18. Simplified 50.5%

       \[\leadsto c0 \cdot \frac{\frac{1}{c0 \cdot \color{blue}{\left(2 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot \left(w \cdot {M}^{2}\right)}\right)}}}{2 \cdot w} \]
3. Recombined 4 regimes into one program.

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq -2 \cdot 10^{-182}:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \left(\frac{c0}{h \cdot D} \cdot \frac{d}{w}\right) \cdot \frac{d}{D}\right)}{2 \cdot w}\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 2 \cdot 10^{-141}:\\ \;\;\;\;0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{d}{D} \cdot {\left(\sqrt{\sqrt{\frac{c0}{w \cdot h}}}\right)}^{2}, \sqrt{\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h} - M}, \frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\frac{1}{c0 \cdot \left(2 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot \left(w \cdot {M}^{2}\right)}\right)}}{2 \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 57.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ \begin{array}{l} t_0 := {\left(\frac{d\_m}{D\_m}\right)}^{2}\\ t_1 := \frac{c0 \cdot t\_0}{w \cdot h}\\ t_2 := \frac{c0}{2 \cdot w}\\ t_3 := \frac{c0 \cdot \left(d\_m \cdot d\_m\right)}{\left(w \cdot h\right) \cdot \left(D\_m \cdot D\_m\right)}\\ t_4 := t\_2 \cdot \left(t\_3 + \sqrt{t\_3 \cdot t\_3 - M \cdot M}\right)\\ \mathbf{if}\;t\_4 \leq -2 \cdot 10^{-182}:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(c0, d\_m \cdot \frac{d\_m}{D\_m \cdot \left(w \cdot \left(h \cdot D\_m\right)\right)}, \left(\frac{c0}{h \cdot D\_m} \cdot \frac{d\_m}{w}\right) \cdot \frac{d\_m}{D\_m}\right)}{2 \cdot w}\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{-141}:\\ \;\;\;\;0.25 \cdot \frac{{D\_m}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d\_m}^{2}}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_2 \cdot \mathsf{fma}\left(\frac{d\_m}{D\_m} \cdot \sqrt{\frac{c0}{w \cdot h}}, \sqrt{t\_1 - M}, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\frac{1}{c0 \cdot \left(2 \cdot \frac{t\_0}{h \cdot \left(w \cdot {M}^{2}\right)}\right)}}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
(FPCore (c0 w h D_m d_m M)
 :precision binary64
 (let* ((t_0 (pow (/ d_m D_m) 2.0))
        (t_1 (/ (* c0 t_0) (* w h)))
        (t_2 (/ c0 (* 2.0 w)))
        (t_3 (/ (* c0 (* d_m d_m)) (* (* w h) (* D_m D_m))))
        (t_4 (* t_2 (+ t_3 (sqrt (- (* t_3 t_3) (* M M)))))))
   (if (<= t_4 -2e-182)
     (*
      c0
      (/
       (fma
        c0
        (* d_m (/ d_m (* D_m (* w (* h D_m)))))
        (* (* (/ c0 (* h D_m)) (/ d_m w)) (/ d_m D_m)))
       (* 2.0 w)))
     (if (<= t_4 2e-141)
       (* 0.25 (/ (* (pow D_m 2.0) (* h (pow M 2.0))) (pow d_m 2.0)))
       (if (<= t_4 INFINITY)
         (*
          t_2
          (fma (* (/ d_m D_m) (sqrt (/ c0 (* w h)))) (sqrt (- t_1 M)) t_1))
         (*
          c0
          (/
           (/ 1.0 (* c0 (* 2.0 (/ t_0 (* h (* w (pow M 2.0)))))))
           (* 2.0 w))))))))

C

D_m = fabs(D);
d_m = fabs(d);
double code(double c0, double w, double h, double D_m, double d_m, double M) {
	double t_0 = pow((d_m / D_m), 2.0);
	double t_1 = (c0 * t_0) / (w * h);
	double t_2 = c0 / (2.0 * w);
	double t_3 = (c0 * (d_m * d_m)) / ((w * h) * (D_m * D_m));
	double t_4 = t_2 * (t_3 + sqrt(((t_3 * t_3) - (M * M))));
	double tmp;
	if (t_4 <= -2e-182) {
		tmp = c0 * (fma(c0, (d_m * (d_m / (D_m * (w * (h * D_m))))), (((c0 / (h * D_m)) * (d_m / w)) * (d_m / D_m))) / (2.0 * w));
	} else if (t_4 <= 2e-141) {
		tmp = 0.25 * ((pow(D_m, 2.0) * (h * pow(M, 2.0))) / pow(d_m, 2.0));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_2 * fma(((d_m / D_m) * sqrt((c0 / (w * h)))), sqrt((t_1 - M)), t_1);
	} else {
		tmp = c0 * ((1.0 / (c0 * (2.0 * (t_0 / (h * (w * pow(M, 2.0))))))) / (2.0 * w));
	}
	return tmp;
}

Julia

D_m = abs(D)
d_m = abs(d)
function code(c0, w, h, D_m, d_m, M)
	t_0 = Float64(d_m / D_m) ^ 2.0
	t_1 = Float64(Float64(c0 * t_0) / Float64(w * h))
	t_2 = Float64(c0 / Float64(2.0 * w))
	t_3 = Float64(Float64(c0 * Float64(d_m * d_m)) / Float64(Float64(w * h) * Float64(D_m * D_m)))
	t_4 = Float64(t_2 * Float64(t_3 + sqrt(Float64(Float64(t_3 * t_3) - Float64(M * M)))))
	tmp = 0.0
	if (t_4 <= -2e-182)
		tmp = Float64(c0 * Float64(fma(c0, Float64(d_m * Float64(d_m / Float64(D_m * Float64(w * Float64(h * D_m))))), Float64(Float64(Float64(c0 / Float64(h * D_m)) * Float64(d_m / w)) * Float64(d_m / D_m))) / Float64(2.0 * w)));
	elseif (t_4 <= 2e-141)
		tmp = Float64(0.25 * Float64(Float64((D_m ^ 2.0) * Float64(h * (M ^ 2.0))) / (d_m ^ 2.0)));
	elseif (t_4 <= Inf)
		tmp = Float64(t_2 * fma(Float64(Float64(d_m / D_m) * sqrt(Float64(c0 / Float64(w * h)))), sqrt(Float64(t_1 - M)), t_1));
	else
		tmp = Float64(c0 * Float64(Float64(1.0 / Float64(c0 * Float64(2.0 * Float64(t_0 / Float64(h * Float64(w * (M ^ 2.0))))))) / Float64(2.0 * w)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
code[c0_, w_, h_, D$95$m_, d$95$m_, M_] := Block[{t$95$0 = N[Power[N[(d$95$m / D$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * t$95$0), $MachinePrecision] / N[(w * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(c0 * N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = 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]}, If[LessEqual[t$95$4, -2e-182], N[(c0 * N[(N[(c0 * N[(d$95$m * N[(d$95$m / N[(D$95$m * N[(w * N[(h * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(c0 / N[(h * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(d$95$m / w), $MachinePrecision]), $MachinePrecision] * N[(d$95$m / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2e-141], N[(0.25 * N[(N[(N[Power[D$95$m, 2.0], $MachinePrecision] * N[(h * N[Power[M, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[d$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(t$95$2 * N[(N[(N[(d$95$m / D$95$m), $MachinePrecision] * N[Sqrt[N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$1 - M), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[(1.0 / N[(c0 * N[(2.0 * N[(t$95$0 / N[(h * N[(w * N[Power[M, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (/.f64 c0 (*.f64 2 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))))) < -2.0000000000000001e-182

   1. Initial program 81.5%

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

   3. Simplified 86.8%

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

   5. Taylor expanded in c0 around inf 81.6%

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

      1. *-commutative 81.6%

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

      2. *-commutative 81.6%

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

      3. associate-*r* 78.9%

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

      4. times-frac 79.0%

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

      5. *-commutative 79.0%

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

   7. Simplified 79.0%

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

      1. associate-*r/ 81.6%

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

      2. *-commutative 81.6%

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

   9. Applied egg-rr 81.6%

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

       1. times-frac 84.2%

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

       2. associate-/r* 84.2%

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

       3. pow2 84.2%

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

       4. pow2 84.2%

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

       5. frac-times 86.8%

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

       6. associate-*r* 89.4%

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

       7. *-commutative 89.4%

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

   11. Applied egg-rr 89.4%

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

   12. Taylor expanded in c0 around 0 89.4%

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

       1. associate-*r* 89.4%

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

       2. times-frac 89.5%

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

   14. Simplified 89.5%

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

   if -2.0000000000000001e-182 < (*.f64 (/.f64 c0 (*.f64 2 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))))) < 2.0000000000000001e-141

   1. Initial program 53.1%

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

   3. Simplified 16.6%

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

   6. Applied egg-rr 44.6%

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

      1. associate--r- 62.6%

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

      2. +-inverses 68.9%

         \[\leadsto c0 \cdot \frac{\frac{\color{blue}{0} + {M}^{2}}{\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h} - \sqrt{{\left(\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

      3. associate-/l* 62.8%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{\color{blue}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h}} - \sqrt{{\left(\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

      4. *-commutative 62.8%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{\color{blue}{h \cdot w}} - \sqrt{{\left(\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

      5. associate-/l* 62.7%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w} - \sqrt{{\color{blue}{\left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}}^{2} - {M}^{2}}}}{2 \cdot w} \]

      6. *-commutative 62.7%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w} - \sqrt{{\left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{\color{blue}{h \cdot w}}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

   8. Simplified 62.7%

      \[\leadsto c0 \cdot \frac{\color{blue}{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w} - \sqrt{{\left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w}\right)}^{2} - {M}^{2}}}}}{2 \cdot w} \]

   9. Taylor expanded in c0 around -inf 62.7%

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

       1. associate-*r/ 62.7%

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

       2. neg-mul-1 62.7%

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

       3. associate-*r/ 62.7%

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

       4. mul-1-neg 62.7%

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

   11. Simplified 62.7%

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

       1. clear-num 62.8%

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

       2. inv-pow 62.8%

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

       3. sub-div 62.8%

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

       4. *-commutative 62.8%

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

   13. Applied egg-rr 62.8%

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

       1. unpow-1 62.8%

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

       2. associate-/l* 62.8%

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

   15. Simplified 63.0%

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

   16. Taylor expanded in c0 around 0 74.9%

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

   if 2.0000000000000001e-141 < (*.f64 (/.f64 c0 (*.f64 2 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 85.8%

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

   3. Simplified 85.8%

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

   6. Applied egg-rr 94.9%

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

   7. Taylor expanded in c0 around inf 54.3%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 2 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 0.0%

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

   3. Simplified 20.7%

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

   6. Applied egg-rr 2.1%

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

      1. associate--r- 3.1%

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

      2. +-inverses 39.8%

         \[\leadsto c0 \cdot \frac{\frac{\color{blue}{0} + {M}^{2}}{\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h} - \sqrt{{\left(\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

      3. associate-/l* 40.4%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{\color{blue}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h}} - \sqrt{{\left(\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

      4. *-commutative 40.4%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{\color{blue}{h \cdot w}} - \sqrt{{\left(\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

      5. associate-/l* 39.7%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w} - \sqrt{{\color{blue}{\left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}}^{2} - {M}^{2}}}}{2 \cdot w} \]

      6. *-commutative 39.7%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w} - \sqrt{{\left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{\color{blue}{h \cdot w}}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

   8. Simplified 39.7%

      \[\leadsto c0 \cdot \frac{\color{blue}{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w} - \sqrt{{\left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w}\right)}^{2} - {M}^{2}}}}}{2 \cdot w} \]

   9. Taylor expanded in c0 around -inf 35.4%

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

       1. associate-*r/ 35.4%

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

       2. neg-mul-1 35.4%

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

       3. associate-*r/ 35.4%

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

       4. mul-1-neg 35.4%

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

   11. Simplified 35.4%

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

       1. clear-num 35.4%

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

       2. inv-pow 35.4%

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

       3. sub-div 35.4%

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

       4. *-commutative 35.4%

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

   13. Applied egg-rr 35.4%

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

       1. unpow-1 35.4%

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

       2. associate-/l* 36.0%

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

   15. Simplified 48.9%

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

   16. Taylor expanded in d around 0 36.6%

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

       1. associate-/r* 35.4%

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

       2. unpow2 35.4%

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

       3. unpow2 35.4%

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

       4. times-frac 44.9%

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

       5. unpow2 44.9%

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

       6. *-commutative 44.9%

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

       7. associate-*l* 50.5%

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

   18. Simplified 50.5%

       \[\leadsto c0 \cdot \frac{\frac{1}{c0 \cdot \color{blue}{\left(2 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot \left(w \cdot {M}^{2}\right)}\right)}}}{2 \cdot w} \]
3. Recombined 4 regimes into one program.

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq -2 \cdot 10^{-182}:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \left(\frac{c0}{h \cdot D} \cdot \frac{d}{w}\right) \cdot \frac{d}{D}\right)}{2 \cdot w}\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 2 \cdot 10^{-141}:\\ \;\;\;\;0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{d}{D} \cdot \sqrt{\frac{c0}{w \cdot h}}, \sqrt{\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h} - M}, \frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\frac{1}{c0 \cdot \left(2 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot \left(w \cdot {M}^{2}\right)}\right)}}{2 \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 58.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ \begin{array}{l} t_0 := \frac{{\left(\frac{d\_m}{D\_m}\right)}^{2}}{h \cdot \left(w \cdot {M}^{2}\right)}\\ t_1 := \frac{c0 \cdot \left(d\_m \cdot d\_m\right)}{\left(w \cdot h\right) \cdot \left(D\_m \cdot D\_m\right)}\\ t_2 := \frac{c0}{2 \cdot w} \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right)\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-182}:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(c0, d\_m \cdot \frac{d\_m}{D\_m \cdot \left(w \cdot \left(h \cdot D\_m\right)\right)}, \left(\frac{c0}{h \cdot D\_m} \cdot \frac{d\_m}{w}\right) \cdot \frac{d\_m}{D\_m}\right)}{2 \cdot w}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;c0 \cdot \frac{\frac{0.5}{w}}{t\_0 \cdot \left(c0 \cdot 2\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;c0 \cdot \frac{2 \cdot \frac{d\_m \cdot \left(\frac{d\_m}{D\_m} \cdot \frac{c0}{w \cdot h}\right)}{D\_m}}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\frac{1}{c0 \cdot \left(2 \cdot t\_0\right)}}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
(FPCore (c0 w h D_m d_m M)
 :precision binary64
 (let* ((t_0 (/ (pow (/ d_m D_m) 2.0) (* h (* w (pow M 2.0)))))
        (t_1 (/ (* c0 (* d_m d_m)) (* (* w h) (* D_m D_m))))
        (t_2 (* (/ c0 (* 2.0 w)) (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))))
   (if (<= t_2 -2e-182)
     (*
      c0
      (/
       (fma
        c0
        (* d_m (/ d_m (* D_m (* w (* h D_m)))))
        (* (* (/ c0 (* h D_m)) (/ d_m w)) (/ d_m D_m)))
       (* 2.0 w)))
     (if (<= t_2 0.0)
       (* c0 (/ (/ 0.5 w) (* t_0 (* c0 2.0))))
       (if (<= t_2 INFINITY)
         (*
          c0
          (/ (* 2.0 (/ (* d_m (* (/ d_m D_m) (/ c0 (* w h)))) D_m)) (* 2.0 w)))
         (* c0 (/ (/ 1.0 (* c0 (* 2.0 t_0))) (* 2.0 w))))))))

C

D_m = fabs(D);
d_m = fabs(d);
double code(double c0, double w, double h, double D_m, double d_m, double M) {
	double t_0 = pow((d_m / D_m), 2.0) / (h * (w * pow(M, 2.0)));
	double t_1 = (c0 * (d_m * d_m)) / ((w * h) * (D_m * D_m));
	double t_2 = (c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= -2e-182) {
		tmp = c0 * (fma(c0, (d_m * (d_m / (D_m * (w * (h * D_m))))), (((c0 / (h * D_m)) * (d_m / w)) * (d_m / D_m))) / (2.0 * w));
	} else if (t_2 <= 0.0) {
		tmp = c0 * ((0.5 / w) / (t_0 * (c0 * 2.0)));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = c0 * ((2.0 * ((d_m * ((d_m / D_m) * (c0 / (w * h)))) / D_m)) / (2.0 * w));
	} else {
		tmp = c0 * ((1.0 / (c0 * (2.0 * t_0))) / (2.0 * w));
	}
	return tmp;
}

Julia

D_m = abs(D)
d_m = abs(d)
function code(c0, w, h, D_m, d_m, M)
	t_0 = Float64((Float64(d_m / D_m) ^ 2.0) / Float64(h * Float64(w * (M ^ 2.0))))
	t_1 = Float64(Float64(c0 * Float64(d_m * d_m)) / Float64(Float64(w * h) * Float64(D_m * D_m)))
	t_2 = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M)))))
	tmp = 0.0
	if (t_2 <= -2e-182)
		tmp = Float64(c0 * Float64(fma(c0, Float64(d_m * Float64(d_m / Float64(D_m * Float64(w * Float64(h * D_m))))), Float64(Float64(Float64(c0 / Float64(h * D_m)) * Float64(d_m / w)) * Float64(d_m / D_m))) / Float64(2.0 * w)));
	elseif (t_2 <= 0.0)
		tmp = Float64(c0 * Float64(Float64(0.5 / w) / Float64(t_0 * Float64(c0 * 2.0))));
	elseif (t_2 <= Inf)
		tmp = Float64(c0 * Float64(Float64(2.0 * Float64(Float64(d_m * Float64(Float64(d_m / D_m) * Float64(c0 / Float64(w * h)))) / D_m)) / Float64(2.0 * w)));
	else
		tmp = Float64(c0 * Float64(Float64(1.0 / Float64(c0 * Float64(2.0 * t_0))) / Float64(2.0 * w)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
code[c0_, w_, h_, D$95$m_, d$95$m_, M_] := Block[{t$95$0 = N[(N[Power[N[(d$95$m / D$95$m), $MachinePrecision], 2.0], $MachinePrecision] / N[(h * N[(w * N[Power[M, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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]}, If[LessEqual[t$95$2, -2e-182], N[(c0 * N[(N[(c0 * N[(d$95$m * N[(d$95$m / N[(D$95$m * N[(w * N[(h * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(c0 / N[(h * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(d$95$m / w), $MachinePrecision]), $MachinePrecision] * N[(d$95$m / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(c0 * N[(N[(0.5 / w), $MachinePrecision] / N[(t$95$0 * N[(c0 * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(c0 * N[(N[(2.0 * N[(N[(d$95$m * N[(N[(d$95$m / D$95$m), $MachinePrecision] * N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / D$95$m), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[(1.0 / N[(c0 * N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (/.f64 c0 (*.f64 2 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))))) < -2.0000000000000001e-182

   1. Initial program 81.5%

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

   3. Simplified 86.8%

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

   5. Taylor expanded in c0 around inf 81.6%

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

      1. *-commutative 81.6%

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

      2. *-commutative 81.6%

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

      3. associate-*r* 78.9%

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

      4. times-frac 79.0%

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

      5. *-commutative 79.0%

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

   7. Simplified 79.0%

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

      1. associate-*r/ 81.6%

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

      2. *-commutative 81.6%

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

   9. Applied egg-rr 81.6%

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

       1. times-frac 84.2%

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

       2. associate-/r* 84.2%

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

       3. pow2 84.2%

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

       4. pow2 84.2%

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

       5. frac-times 86.8%

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

       6. associate-*r* 89.4%

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

       7. *-commutative 89.4%

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

   11. Applied egg-rr 89.4%

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

   12. Taylor expanded in c0 around 0 89.4%

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

       1. associate-*r* 89.4%

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

       2. times-frac 89.5%

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

   14. Simplified 89.5%

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

   if -2.0000000000000001e-182 < (*.f64 (/.f64 c0 (*.f64 2 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))))) < -0.0

   1. Initial program 55.4%

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

   3. Simplified 17.4%

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

   6. Applied egg-rr 47.6%

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

      1. associate--r- 66.8%

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

      2. +-inverses 73.5%

         \[\leadsto c0 \cdot \frac{\frac{\color{blue}{0} + {M}^{2}}{\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h} - \sqrt{{\left(\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

      3. associate-/l* 67.0%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{\color{blue}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h}} - \sqrt{{\left(\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

      4. *-commutative 67.0%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{\color{blue}{h \cdot w}} - \sqrt{{\left(\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

      5. associate-/l* 66.9%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w} - \sqrt{{\color{blue}{\left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}}^{2} - {M}^{2}}}}{2 \cdot w} \]

      6. *-commutative 66.9%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w} - \sqrt{{\left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{\color{blue}{h \cdot w}}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

   8. Simplified 66.9%

      \[\leadsto c0 \cdot \frac{\color{blue}{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w} - \sqrt{{\left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w}\right)}^{2} - {M}^{2}}}}}{2 \cdot w} \]

   9. Taylor expanded in c0 around -inf 66.9%

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

       1. associate-*r/ 66.9%

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

       2. neg-mul-1 66.9%

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

       3. associate-*r/ 66.9%

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

       4. mul-1-neg 66.9%

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

   11. Simplified 66.9%

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

       1. clear-num 67.0%

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

       2. inv-pow 67.0%

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

       3. sub-div 67.0%

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

       4. *-commutative 67.0%

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

   13. Applied egg-rr 67.0%

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

       1. unpow-1 67.0%

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

       2. associate-/l* 67.0%

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

   15. Simplified 67.2%

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

       1. associate-*r/ 73.4%

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

       2. associate-/l* 73.3%

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

       3. associate-/l/ 73.3%

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

       4. *-commutative 73.3%

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

       5. *-commutative 73.3%

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

   17. Applied egg-rr 73.3%

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

   19. Simplified 73.6%

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

   if -0.0 < (*.f64 (/.f64 c0 (*.f64 2 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 84.2%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in c0 around inf 86.5%

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

      1. *-commutative 86.5%

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

      2. *-commutative 86.5%

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

      3. associate-*r* 79.4%

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

      4. times-frac 79.3%

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

      5. *-commutative 79.3%

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

   7. Simplified 79.3%

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

      1. associate-*r/ 79.3%

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

      2. *-commutative 79.3%

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

   9. Applied egg-rr 79.3%

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

   10. Taylor expanded in c0 around 0 89.7%

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

       1. *-commutative 89.7%

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

       2. times-frac 88.4%

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

       3. unpow2 88.4%

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

       4. unpow2 88.4%

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

       5. times-frac 90.7%

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

       6. unpow2 90.7%

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

       7. associate-/r* 83.6%

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

   12. Simplified 83.6%

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

       1. associate-/r* 90.7%

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

       2. unpow2 90.7%

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

       3. associate-*l* 91.9%

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

       4. associate-*r/ 92.0%

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

       5. *-commutative 92.0%

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

   14. Applied egg-rr 92.0%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 2 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 0.0%

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

   3. Simplified 20.7%

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

   6. Applied egg-rr 2.1%

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

      1. associate--r- 3.1%

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

      2. +-inverses 39.8%

         \[\leadsto c0 \cdot \frac{\frac{\color{blue}{0} + {M}^{2}}{\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h} - \sqrt{{\left(\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

      3. associate-/l* 40.4%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{\color{blue}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h}} - \sqrt{{\left(\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

      4. *-commutative 40.4%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{\color{blue}{h \cdot w}} - \sqrt{{\left(\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

      5. associate-/l* 39.7%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w} - \sqrt{{\color{blue}{\left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}}^{2} - {M}^{2}}}}{2 \cdot w} \]

      6. *-commutative 39.7%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w} - \sqrt{{\left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{\color{blue}{h \cdot w}}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

   8. Simplified 39.7%

      \[\leadsto c0 \cdot \frac{\color{blue}{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w} - \sqrt{{\left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w}\right)}^{2} - {M}^{2}}}}}{2 \cdot w} \]

   9. Taylor expanded in c0 around -inf 35.4%

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

       1. associate-*r/ 35.4%

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

       2. neg-mul-1 35.4%

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

       3. associate-*r/ 35.4%

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

       4. mul-1-neg 35.4%

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

   11. Simplified 35.4%

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

       1. clear-num 35.4%

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

       2. inv-pow 35.4%

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

       3. sub-div 35.4%

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

       4. *-commutative 35.4%

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

   13. Applied egg-rr 35.4%

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

       1. unpow-1 35.4%

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

       2. associate-/l* 36.0%

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

   15. Simplified 48.9%

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

   16. Taylor expanded in d around 0 36.6%

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

       1. associate-/r* 35.4%

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

       2. unpow2 35.4%

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

       3. unpow2 35.4%

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

       4. times-frac 44.9%

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

       5. unpow2 44.9%

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

       6. *-commutative 44.9%

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

       7. associate-*l* 50.5%

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

   18. Simplified 50.5%

       \[\leadsto c0 \cdot \frac{\frac{1}{c0 \cdot \color{blue}{\left(2 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot \left(w \cdot {M}^{2}\right)}\right)}}}{2 \cdot w} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq -2 \cdot 10^{-182}:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \left(\frac{c0}{h \cdot D} \cdot \frac{d}{w}\right) \cdot \frac{d}{D}\right)}{2 \cdot w}\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 0:\\ \;\;\;\;c0 \cdot \frac{\frac{0.5}{w}}{\frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot \left(w \cdot {M}^{2}\right)} \cdot \left(c0 \cdot 2\right)}\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;c0 \cdot \frac{2 \cdot \frac{d \cdot \left(\frac{d}{D} \cdot \frac{c0}{w \cdot h}\right)}{D}}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\frac{1}{c0 \cdot \left(2 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot \left(w \cdot {M}^{2}\right)}\right)}}{2 \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d\_m \cdot d\_m\right)}{\left(w \cdot h\right) \cdot \left(D\_m \cdot D\_m\right)}\\ t_1 := \frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-182}:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(c0, d\_m \cdot \frac{d\_m}{D\_m \cdot \left(w \cdot \left(h \cdot D\_m\right)\right)}, \left(\frac{c0}{h \cdot D\_m} \cdot \frac{d\_m}{w}\right) \cdot \frac{d\_m}{D\_m}\right)}{2 \cdot w}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;0.25 \cdot \frac{{D\_m}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d\_m}^{2}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;c0 \cdot \frac{2 \cdot \frac{d\_m \cdot \left(\frac{d\_m}{D\_m} \cdot \frac{c0}{w \cdot h}\right)}{D\_m}}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\frac{1}{c0 \cdot \left(2 \cdot \frac{{\left(\frac{d\_m}{D\_m}\right)}^{2}}{h \cdot \left(w \cdot {M}^{2}\right)}\right)}}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
(FPCore (c0 w h D_m d_m M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d_m d_m)) (* (* w h) (* D_m D_m))))
        (t_1 (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))))
   (if (<= t_1 -2e-182)
     (*
      c0
      (/
       (fma
        c0
        (* d_m (/ d_m (* D_m (* w (* h D_m)))))
        (* (* (/ c0 (* h D_m)) (/ d_m w)) (/ d_m D_m)))
       (* 2.0 w)))
     (if (<= t_1 0.0)
       (* 0.25 (/ (* (pow D_m 2.0) (* h (pow M 2.0))) (pow d_m 2.0)))
       (if (<= t_1 INFINITY)
         (*
          c0
          (/ (* 2.0 (/ (* d_m (* (/ d_m D_m) (/ c0 (* w h)))) D_m)) (* 2.0 w)))
         (*
          c0
          (/
           (/
            1.0
            (* c0 (* 2.0 (/ (pow (/ d_m D_m) 2.0) (* h (* w (pow M 2.0)))))))
           (* 2.0 w))))))))

C

D_m = fabs(D);
d_m = fabs(d);
double code(double c0, double w, double h, double D_m, double d_m, double M) {
	double t_0 = (c0 * (d_m * d_m)) / ((w * h) * (D_m * D_m));
	double t_1 = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	double tmp;
	if (t_1 <= -2e-182) {
		tmp = c0 * (fma(c0, (d_m * (d_m / (D_m * (w * (h * D_m))))), (((c0 / (h * D_m)) * (d_m / w)) * (d_m / D_m))) / (2.0 * w));
	} else if (t_1 <= 0.0) {
		tmp = 0.25 * ((pow(D_m, 2.0) * (h * pow(M, 2.0))) / pow(d_m, 2.0));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = c0 * ((2.0 * ((d_m * ((d_m / D_m) * (c0 / (w * h)))) / D_m)) / (2.0 * w));
	} else {
		tmp = c0 * ((1.0 / (c0 * (2.0 * (pow((d_m / D_m), 2.0) / (h * (w * pow(M, 2.0))))))) / (2.0 * w));
	}
	return tmp;
}

Julia

D_m = abs(D)
d_m = abs(d)
function code(c0, w, h, D_m, d_m, M)
	t_0 = Float64(Float64(c0 * Float64(d_m * d_m)) / Float64(Float64(w * h) * Float64(D_m * D_m)))
	t_1 = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))))
	tmp = 0.0
	if (t_1 <= -2e-182)
		tmp = Float64(c0 * Float64(fma(c0, Float64(d_m * Float64(d_m / Float64(D_m * Float64(w * Float64(h * D_m))))), Float64(Float64(Float64(c0 / Float64(h * D_m)) * Float64(d_m / w)) * Float64(d_m / D_m))) / Float64(2.0 * w)));
	elseif (t_1 <= 0.0)
		tmp = Float64(0.25 * Float64(Float64((D_m ^ 2.0) * Float64(h * (M ^ 2.0))) / (d_m ^ 2.0)));
	elseif (t_1 <= Inf)
		tmp = Float64(c0 * Float64(Float64(2.0 * Float64(Float64(d_m * Float64(Float64(d_m / D_m) * Float64(c0 / Float64(w * h)))) / D_m)) / Float64(2.0 * w)));
	else
		tmp = Float64(c0 * Float64(Float64(1.0 / Float64(c0 * Float64(2.0 * Float64((Float64(d_m / D_m) ^ 2.0) / Float64(h * Float64(w * (M ^ 2.0))))))) / Float64(2.0 * w)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
code[c0_, w_, h_, D$95$m_, d$95$m_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -2e-182], N[(c0 * N[(N[(c0 * N[(d$95$m * N[(d$95$m / N[(D$95$m * N[(w * N[(h * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(c0 / N[(h * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(d$95$m / w), $MachinePrecision]), $MachinePrecision] * N[(d$95$m / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(0.25 * N[(N[(N[Power[D$95$m, 2.0], $MachinePrecision] * N[(h * N[Power[M, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[d$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(c0 * N[(N[(2.0 * N[(N[(d$95$m * N[(N[(d$95$m / D$95$m), $MachinePrecision] * N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / D$95$m), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[(1.0 / N[(c0 * N[(2.0 * N[(N[Power[N[(d$95$m / D$95$m), $MachinePrecision], 2.0], $MachinePrecision] / N[(h * N[(w * N[Power[M, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (/.f64 c0 (*.f64 2 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))))) < -2.0000000000000001e-182

   1. Initial program 81.5%

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

   3. Simplified 86.8%

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

   5. Taylor expanded in c0 around inf 81.6%

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

      1. *-commutative 81.6%

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

      2. *-commutative 81.6%

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

      3. associate-*r* 78.9%

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

      4. times-frac 79.0%

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

      5. *-commutative 79.0%

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

   7. Simplified 79.0%

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

      1. associate-*r/ 81.6%

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

      2. *-commutative 81.6%

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

   9. Applied egg-rr 81.6%

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

       1. times-frac 84.2%

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

       2. associate-/r* 84.2%

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

       3. pow2 84.2%

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

       4. pow2 84.2%

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

       5. frac-times 86.8%

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

       6. associate-*r* 89.4%

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

       7. *-commutative 89.4%

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

   11. Applied egg-rr 89.4%

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

   12. Taylor expanded in c0 around 0 89.4%

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

       1. associate-*r* 89.4%

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

       2. times-frac 89.5%

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

   14. Simplified 89.5%

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

   if -2.0000000000000001e-182 < (*.f64 (/.f64 c0 (*.f64 2 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))))) < -0.0

   1. Initial program 55.4%

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

   3. Simplified 17.4%

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

   6. Applied egg-rr 47.6%

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

      1. associate--r- 66.8%

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

      2. +-inverses 73.5%

         \[\leadsto c0 \cdot \frac{\frac{\color{blue}{0} + {M}^{2}}{\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h} - \sqrt{{\left(\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

      3. associate-/l* 67.0%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{\color{blue}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h}} - \sqrt{{\left(\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

      4. *-commutative 67.0%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{\color{blue}{h \cdot w}} - \sqrt{{\left(\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

      5. associate-/l* 66.9%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w} - \sqrt{{\color{blue}{\left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}}^{2} - {M}^{2}}}}{2 \cdot w} \]

      6. *-commutative 66.9%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w} - \sqrt{{\left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{\color{blue}{h \cdot w}}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

   8. Simplified 66.9%

      \[\leadsto c0 \cdot \frac{\color{blue}{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w} - \sqrt{{\left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w}\right)}^{2} - {M}^{2}}}}}{2 \cdot w} \]

   9. Taylor expanded in c0 around -inf 66.9%

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

       1. associate-*r/ 66.9%

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

       2. neg-mul-1 66.9%

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

       3. associate-*r/ 66.9%

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

       4. mul-1-neg 66.9%

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

   11. Simplified 66.9%

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

       1. clear-num 67.0%

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

       2. inv-pow 67.0%

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

       3. sub-div 67.0%

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

       4. *-commutative 67.0%

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

   13. Applied egg-rr 67.0%

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

       1. unpow-1 67.0%

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

       2. associate-/l* 67.0%

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

   15. Simplified 67.2%

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

   16. Taylor expanded in c0 around 0 79.6%

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

   if -0.0 < (*.f64 (/.f64 c0 (*.f64 2 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 84.2%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in c0 around inf 86.5%

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

      1. *-commutative 86.5%

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

      2. *-commutative 86.5%

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

      3. associate-*r* 79.4%

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

      4. times-frac 79.3%

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

      5. *-commutative 79.3%

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

   7. Simplified 79.3%

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

      1. associate-*r/ 79.3%

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

      2. *-commutative 79.3%

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

   9. Applied egg-rr 79.3%

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

   10. Taylor expanded in c0 around 0 89.7%

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

       1. *-commutative 89.7%

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

       2. times-frac 88.4%

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

       3. unpow2 88.4%

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

       4. unpow2 88.4%

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

       5. times-frac 90.7%

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

       6. unpow2 90.7%

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

       7. associate-/r* 83.6%

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

   12. Simplified 83.6%

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

       1. associate-/r* 90.7%

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

       2. unpow2 90.7%

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

       3. associate-*l* 91.9%

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

       4. associate-*r/ 92.0%

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

       5. *-commutative 92.0%

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

   14. Applied egg-rr 92.0%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 2 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 0.0%

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

   3. Simplified 20.7%

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

   6. Applied egg-rr 2.1%

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

      1. associate--r- 3.1%

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

      2. +-inverses 39.8%

         \[\leadsto c0 \cdot \frac{\frac{\color{blue}{0} + {M}^{2}}{\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h} - \sqrt{{\left(\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

      3. associate-/l* 40.4%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{\color{blue}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h}} - \sqrt{{\left(\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

      4. *-commutative 40.4%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{\color{blue}{h \cdot w}} - \sqrt{{\left(\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

      5. associate-/l* 39.7%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w} - \sqrt{{\color{blue}{\left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}}^{2} - {M}^{2}}}}{2 \cdot w} \]

      6. *-commutative 39.7%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w} - \sqrt{{\left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{\color{blue}{h \cdot w}}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

   8. Simplified 39.7%

      \[\leadsto c0 \cdot \frac{\color{blue}{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w} - \sqrt{{\left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w}\right)}^{2} - {M}^{2}}}}}{2 \cdot w} \]

   9. Taylor expanded in c0 around -inf 35.4%

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

       1. associate-*r/ 35.4%

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

       2. neg-mul-1 35.4%

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

       3. associate-*r/ 35.4%

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

       4. mul-1-neg 35.4%

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

   11. Simplified 35.4%

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

       1. clear-num 35.4%

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

       2. inv-pow 35.4%

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

       3. sub-div 35.4%

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

       4. *-commutative 35.4%

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

   13. Applied egg-rr 35.4%

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

       1. unpow-1 35.4%

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

       2. associate-/l* 36.0%

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

   15. Simplified 48.9%

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

   16. Taylor expanded in d around 0 36.6%

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

       1. associate-/r* 35.4%

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

       2. unpow2 35.4%

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

       3. unpow2 35.4%

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

       4. times-frac 44.9%

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

       5. unpow2 44.9%

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

       6. *-commutative 44.9%

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

       7. associate-*l* 50.5%

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

   18. Simplified 50.5%

       \[\leadsto c0 \cdot \frac{\frac{1}{c0 \cdot \color{blue}{\left(2 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot \left(w \cdot {M}^{2}\right)}\right)}}}{2 \cdot w} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq -2 \cdot 10^{-182}:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \left(\frac{c0}{h \cdot D} \cdot \frac{d}{w}\right) \cdot \frac{d}{D}\right)}{2 \cdot w}\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 0:\\ \;\;\;\;0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;c0 \cdot \frac{2 \cdot \frac{d \cdot \left(\frac{d}{D} \cdot \frac{c0}{w \cdot h}\right)}{D}}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\frac{1}{c0 \cdot \left(2 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot \left(w \cdot {M}^{2}\right)}\right)}}{2 \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d\_m \cdot d\_m\right)}{\left(w \cdot h\right) \cdot \left(D\_m \cdot D\_m\right)}\\ t_1 := \frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-182}:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(c0, d\_m \cdot \frac{d\_m}{D\_m \cdot \left(w \cdot \left(h \cdot D\_m\right)\right)}, \left(\frac{c0}{h \cdot D\_m} \cdot \frac{d\_m}{w}\right) \cdot \frac{d\_m}{D\_m}\right)}{2 \cdot w}\\ \mathbf{elif}\;t\_1 \leq 0 \lor \neg \left(t\_1 \leq \infty\right):\\ \;\;\;\;c0 \cdot \frac{\frac{{M}^{2}}{c0} \cdot \frac{h}{2 \cdot \frac{{\left(\frac{d\_m}{D\_m}\right)}^{2}}{w}}}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{2 \cdot \frac{d\_m \cdot \left(\frac{d\_m}{D\_m} \cdot \frac{c0}{w \cdot h}\right)}{D\_m}}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
(FPCore (c0 w h D_m d_m M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d_m d_m)) (* (* w h) (* D_m D_m))))
        (t_1 (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))))
   (if (<= t_1 -2e-182)
     (*
      c0
      (/
       (fma
        c0
        (* d_m (/ d_m (* D_m (* w (* h D_m)))))
        (* (* (/ c0 (* h D_m)) (/ d_m w)) (/ d_m D_m)))
       (* 2.0 w)))
     (if (or (<= t_1 0.0) (not (<= t_1 INFINITY)))
       (*
        c0
        (/
         (* (/ (pow M 2.0) c0) (/ h (* 2.0 (/ (pow (/ d_m D_m) 2.0) w))))
         (* 2.0 w)))
       (*
        c0
        (/
         (* 2.0 (/ (* d_m (* (/ d_m D_m) (/ c0 (* w h)))) D_m))
         (* 2.0 w)))))))

C

D_m = fabs(D);
d_m = fabs(d);
double code(double c0, double w, double h, double D_m, double d_m, double M) {
	double t_0 = (c0 * (d_m * d_m)) / ((w * h) * (D_m * D_m));
	double t_1 = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	double tmp;
	if (t_1 <= -2e-182) {
		tmp = c0 * (fma(c0, (d_m * (d_m / (D_m * (w * (h * D_m))))), (((c0 / (h * D_m)) * (d_m / w)) * (d_m / D_m))) / (2.0 * w));
	} else if ((t_1 <= 0.0) || !(t_1 <= ((double) INFINITY))) {
		tmp = c0 * (((pow(M, 2.0) / c0) * (h / (2.0 * (pow((d_m / D_m), 2.0) / w)))) / (2.0 * w));
	} else {
		tmp = c0 * ((2.0 * ((d_m * ((d_m / D_m) * (c0 / (w * h)))) / D_m)) / (2.0 * w));
	}
	return tmp;
}

Julia

D_m = abs(D)
d_m = abs(d)
function code(c0, w, h, D_m, d_m, M)
	t_0 = Float64(Float64(c0 * Float64(d_m * d_m)) / Float64(Float64(w * h) * Float64(D_m * D_m)))
	t_1 = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))))
	tmp = 0.0
	if (t_1 <= -2e-182)
		tmp = Float64(c0 * Float64(fma(c0, Float64(d_m * Float64(d_m / Float64(D_m * Float64(w * Float64(h * D_m))))), Float64(Float64(Float64(c0 / Float64(h * D_m)) * Float64(d_m / w)) * Float64(d_m / D_m))) / Float64(2.0 * w)));
	elseif ((t_1 <= 0.0) || !(t_1 <= Inf))
		tmp = Float64(c0 * Float64(Float64(Float64((M ^ 2.0) / c0) * Float64(h / Float64(2.0 * Float64((Float64(d_m / D_m) ^ 2.0) / w)))) / Float64(2.0 * w)));
	else
		tmp = Float64(c0 * Float64(Float64(2.0 * Float64(Float64(d_m * Float64(Float64(d_m / D_m) * Float64(c0 / Float64(w * h)))) / D_m)) / Float64(2.0 * w)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
code[c0_, w_, h_, D$95$m_, d$95$m_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -2e-182], N[(c0 * N[(N[(c0 * N[(d$95$m * N[(d$95$m / N[(D$95$m * N[(w * N[(h * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(c0 / N[(h * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(d$95$m / w), $MachinePrecision]), $MachinePrecision] * N[(d$95$m / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$1, 0.0], N[Not[LessEqual[t$95$1, Infinity]], $MachinePrecision]], N[(c0 * N[(N[(N[(N[Power[M, 2.0], $MachinePrecision] / c0), $MachinePrecision] * N[(h / N[(2.0 * N[(N[Power[N[(d$95$m / D$95$m), $MachinePrecision], 2.0], $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[(2.0 * N[(N[(d$95$m * N[(N[(d$95$m / D$95$m), $MachinePrecision] * N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / D$95$m), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (/.f64 c0 (*.f64 2 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))))) < -2.0000000000000001e-182

   1. Initial program 81.5%

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

   3. Simplified 86.8%

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

   5. Taylor expanded in c0 around inf 81.6%

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

      1. *-commutative 81.6%

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

      2. *-commutative 81.6%

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

      3. associate-*r* 78.9%

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

      4. times-frac 79.0%

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

      5. *-commutative 79.0%

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

   7. Simplified 79.0%

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

      1. associate-*r/ 81.6%

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

      2. *-commutative 81.6%

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

   9. Applied egg-rr 81.6%

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

       1. times-frac 84.2%

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

       2. associate-/r* 84.2%

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

       3. pow2 84.2%

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

       4. pow2 84.2%

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

       5. frac-times 86.8%

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

       6. associate-*r* 89.4%

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

       7. *-commutative 89.4%

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

   11. Applied egg-rr 89.4%

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

   12. Taylor expanded in c0 around 0 89.4%

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

       1. associate-*r* 89.4%

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

       2. times-frac 89.5%

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

   14. Simplified 89.5%

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

   if -2.0000000000000001e-182 < (*.f64 (/.f64 c0 (*.f64 2 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))))) < -0.0 or +inf.0 < (*.f64 (/.f64 c0 (*.f64 2 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 4.7%

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

   3. Simplified 20.4%

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

   6. Applied egg-rr 5.9%

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

      1. associate--r- 8.5%

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

      2. +-inverses 42.7%

         \[\leadsto c0 \cdot \frac{\frac{\color{blue}{0} + {M}^{2}}{\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h} - \sqrt{{\left(\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

      3. associate-/l* 42.7%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{\color{blue}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h}} - \sqrt{{\left(\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

      4. *-commutative 42.7%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{\color{blue}{h \cdot w}} - \sqrt{{\left(\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

      5. associate-/l* 42.0%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w} - \sqrt{{\color{blue}{\left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}}^{2} - {M}^{2}}}}{2 \cdot w} \]

      6. *-commutative 42.0%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w} - \sqrt{{\left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{\color{blue}{h \cdot w}}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

   8. Simplified 42.0%

      \[\leadsto c0 \cdot \frac{\color{blue}{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w} - \sqrt{{\left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w}\right)}^{2} - {M}^{2}}}}}{2 \cdot w} \]

   9. Taylor expanded in c0 around -inf 38.0%

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

       1. associate-*r/ 38.0%

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

       2. neg-mul-1 38.0%

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

       3. associate-*r/ 38.0%

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

       4. mul-1-neg 38.0%

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

   11. Simplified 38.0%

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

   12. Taylor expanded in h around -inf 40.7%

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

       1. times-frac 40.4%

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

       2. cancel-sign-sub-inv 40.4%

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

       3. metadata-eval 40.4%

          \[\leadsto c0 \cdot \frac{\frac{{M}^{2}}{c0} \cdot \frac{h}{\frac{{d}^{2}}{{D}^{2} \cdot w} + \color{blue}{1} \cdot \frac{{d}^{2}}{{D}^{2} \cdot w}}}{2 \cdot w} \]

       4. distribute-rgt1-in 40.4%

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

       5. metadata-eval 40.4%

          \[\leadsto c0 \cdot \frac{\frac{{M}^{2}}{c0} \cdot \frac{h}{\color{blue}{2} \cdot \frac{{d}^{2}}{{D}^{2} \cdot w}}}{2 \cdot w} \]

       6. associate-/r* 41.5%

          \[\leadsto c0 \cdot \frac{\frac{{M}^{2}}{c0} \cdot \frac{h}{2 \cdot \color{blue}{\frac{\frac{{d}^{2}}{{D}^{2}}}{w}}}}{2 \cdot w} \]

       7. unpow2 41.5%

          \[\leadsto c0 \cdot \frac{\frac{{M}^{2}}{c0} \cdot \frac{h}{2 \cdot \frac{\frac{\color{blue}{d \cdot d}}{{D}^{2}}}{w}}}{2 \cdot w} \]

       8. unpow2 41.5%

          \[\leadsto c0 \cdot \frac{\frac{{M}^{2}}{c0} \cdot \frac{h}{2 \cdot \frac{\frac{d \cdot d}{\color{blue}{D \cdot D}}}{w}}}{2 \cdot w} \]

       9. times-frac 49.8%

          \[\leadsto c0 \cdot \frac{\frac{{M}^{2}}{c0} \cdot \frac{h}{2 \cdot \frac{\color{blue}{\frac{d}{D} \cdot \frac{d}{D}}}{w}}}{2 \cdot w} \]

       10. unpow2 49.8%

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

   14. Simplified 49.8%

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

   if -0.0 < (*.f64 (/.f64 c0 (*.f64 2 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 84.2%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in c0 around inf 86.5%

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

      1. *-commutative 86.5%

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

      2. *-commutative 86.5%

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

      3. associate-*r* 79.4%

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

      4. times-frac 79.3%

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

      5. *-commutative 79.3%

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

   7. Simplified 79.3%

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

      1. associate-*r/ 79.3%

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

      2. *-commutative 79.3%

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

   9. Applied egg-rr 79.3%

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

   10. Taylor expanded in c0 around 0 89.7%

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

       1. *-commutative 89.7%

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

       2. times-frac 88.4%

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

       3. unpow2 88.4%

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

       4. unpow2 88.4%

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

       5. times-frac 90.7%

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

       6. unpow2 90.7%

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

       7. associate-/r* 83.6%

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

   12. Simplified 83.6%

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

       1. associate-/r* 90.7%

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

       2. unpow2 90.7%

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

       3. associate-*l* 91.9%

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

       4. associate-*r/ 92.0%

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

       5. *-commutative 92.0%

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

   14. Applied egg-rr 92.0%

       \[\leadsto c0 \cdot \frac{2 \cdot \color{blue}{\frac{\left(\frac{d}{D} \cdot \frac{c0}{h \cdot w}\right) \cdot d}{D}}}{2 \cdot w} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq -2 \cdot 10^{-182}:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \left(\frac{c0}{h \cdot D} \cdot \frac{d}{w}\right) \cdot \frac{d}{D}\right)}{2 \cdot w}\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 0 \lor \neg \left(\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty\right):\\ \;\;\;\;c0 \cdot \frac{\frac{{M}^{2}}{c0} \cdot \frac{h}{2 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w}}}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{2 \cdot \frac{d \cdot \left(\frac{d}{D} \cdot \frac{c0}{w \cdot h}\right)}{D}}{2 \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d\_m \cdot d\_m\right)}{\left(w \cdot h\right) \cdot \left(D\_m \cdot D\_m\right)}\\ t_1 := \frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-182}:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(c0, d\_m \cdot \frac{d\_m}{D\_m \cdot \left(w \cdot \left(h \cdot D\_m\right)\right)}, \left(\frac{c0}{h \cdot D\_m} \cdot \frac{d\_m}{w}\right) \cdot \frac{d\_m}{D\_m}\right)}{2 \cdot w}\\ \mathbf{elif}\;t\_1 \leq 0 \lor \neg \left(t\_1 \leq \infty\right):\\ \;\;\;\;c0 \cdot \frac{\frac{0.5}{w}}{\frac{{\left(\frac{d\_m}{D\_m}\right)}^{2}}{h \cdot \left(w \cdot {M}^{2}\right)} \cdot \left(c0 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{2 \cdot \frac{d\_m \cdot \left(\frac{d\_m}{D\_m} \cdot \frac{c0}{w \cdot h}\right)}{D\_m}}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
(FPCore (c0 w h D_m d_m M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d_m d_m)) (* (* w h) (* D_m D_m))))
        (t_1 (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))))
   (if (<= t_1 -2e-182)
     (*
      c0
      (/
       (fma
        c0
        (* d_m (/ d_m (* D_m (* w (* h D_m)))))
        (* (* (/ c0 (* h D_m)) (/ d_m w)) (/ d_m D_m)))
       (* 2.0 w)))
     (if (or (<= t_1 0.0) (not (<= t_1 INFINITY)))
       (*
        c0
        (/
         (/ 0.5 w)
         (* (/ (pow (/ d_m D_m) 2.0) (* h (* w (pow M 2.0)))) (* c0 2.0))))
       (*
        c0
        (/
         (* 2.0 (/ (* d_m (* (/ d_m D_m) (/ c0 (* w h)))) D_m))
         (* 2.0 w)))))))

C

D_m = fabs(D);
d_m = fabs(d);
double code(double c0, double w, double h, double D_m, double d_m, double M) {
	double t_0 = (c0 * (d_m * d_m)) / ((w * h) * (D_m * D_m));
	double t_1 = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	double tmp;
	if (t_1 <= -2e-182) {
		tmp = c0 * (fma(c0, (d_m * (d_m / (D_m * (w * (h * D_m))))), (((c0 / (h * D_m)) * (d_m / w)) * (d_m / D_m))) / (2.0 * w));
	} else if ((t_1 <= 0.0) || !(t_1 <= ((double) INFINITY))) {
		tmp = c0 * ((0.5 / w) / ((pow((d_m / D_m), 2.0) / (h * (w * pow(M, 2.0)))) * (c0 * 2.0)));
	} else {
		tmp = c0 * ((2.0 * ((d_m * ((d_m / D_m) * (c0 / (w * h)))) / D_m)) / (2.0 * w));
	}
	return tmp;
}

Julia

D_m = abs(D)
d_m = abs(d)
function code(c0, w, h, D_m, d_m, M)
	t_0 = Float64(Float64(c0 * Float64(d_m * d_m)) / Float64(Float64(w * h) * Float64(D_m * D_m)))
	t_1 = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))))
	tmp = 0.0
	if (t_1 <= -2e-182)
		tmp = Float64(c0 * Float64(fma(c0, Float64(d_m * Float64(d_m / Float64(D_m * Float64(w * Float64(h * D_m))))), Float64(Float64(Float64(c0 / Float64(h * D_m)) * Float64(d_m / w)) * Float64(d_m / D_m))) / Float64(2.0 * w)));
	elseif ((t_1 <= 0.0) || !(t_1 <= Inf))
		tmp = Float64(c0 * Float64(Float64(0.5 / w) / Float64(Float64((Float64(d_m / D_m) ^ 2.0) / Float64(h * Float64(w * (M ^ 2.0)))) * Float64(c0 * 2.0))));
	else
		tmp = Float64(c0 * Float64(Float64(2.0 * Float64(Float64(d_m * Float64(Float64(d_m / D_m) * Float64(c0 / Float64(w * h)))) / D_m)) / Float64(2.0 * w)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
code[c0_, w_, h_, D$95$m_, d$95$m_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -2e-182], N[(c0 * N[(N[(c0 * N[(d$95$m * N[(d$95$m / N[(D$95$m * N[(w * N[(h * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(c0 / N[(h * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(d$95$m / w), $MachinePrecision]), $MachinePrecision] * N[(d$95$m / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$1, 0.0], N[Not[LessEqual[t$95$1, Infinity]], $MachinePrecision]], N[(c0 * N[(N[(0.5 / w), $MachinePrecision] / N[(N[(N[Power[N[(d$95$m / D$95$m), $MachinePrecision], 2.0], $MachinePrecision] / N[(h * N[(w * N[Power[M, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c0 * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[(2.0 * N[(N[(d$95$m * N[(N[(d$95$m / D$95$m), $MachinePrecision] * N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / D$95$m), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (/.f64 c0 (*.f64 2 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))))) < -2.0000000000000001e-182

   1. Initial program 81.5%

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

   3. Simplified 86.8%

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

   5. Taylor expanded in c0 around inf 81.6%

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

      1. *-commutative 81.6%

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

      2. *-commutative 81.6%

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

      3. associate-*r* 78.9%

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

      4. times-frac 79.0%

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

      5. *-commutative 79.0%

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

   7. Simplified 79.0%

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

      1. associate-*r/ 81.6%

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

      2. *-commutative 81.6%

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

   9. Applied egg-rr 81.6%

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

       1. times-frac 84.2%

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

       2. associate-/r* 84.2%

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

       3. pow2 84.2%

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

       4. pow2 84.2%

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

       5. frac-times 86.8%

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

       6. associate-*r* 89.4%

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

       7. *-commutative 89.4%

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

   11. Applied egg-rr 89.4%

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

   12. Taylor expanded in c0 around 0 89.4%

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

       1. associate-*r* 89.4%

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

       2. times-frac 89.5%

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

   14. Simplified 89.5%

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

   if -2.0000000000000001e-182 < (*.f64 (/.f64 c0 (*.f64 2 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))))) < -0.0 or +inf.0 < (*.f64 (/.f64 c0 (*.f64 2 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 4.7%

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

   3. Simplified 20.4%

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

   6. Applied egg-rr 5.9%

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

      1. associate--r- 8.5%

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

      2. +-inverses 42.7%

         \[\leadsto c0 \cdot \frac{\frac{\color{blue}{0} + {M}^{2}}{\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h} - \sqrt{{\left(\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

      3. associate-/l* 42.7%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{\color{blue}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h}} - \sqrt{{\left(\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

      4. *-commutative 42.7%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{\color{blue}{h \cdot w}} - \sqrt{{\left(\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

      5. associate-/l* 42.0%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w} - \sqrt{{\color{blue}{\left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}}^{2} - {M}^{2}}}}{2 \cdot w} \]

      6. *-commutative 42.0%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w} - \sqrt{{\left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{\color{blue}{h \cdot w}}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

   8. Simplified 42.0%

      \[\leadsto c0 \cdot \frac{\color{blue}{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w} - \sqrt{{\left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w}\right)}^{2} - {M}^{2}}}}}{2 \cdot w} \]

   9. Taylor expanded in c0 around -inf 38.0%

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

       1. associate-*r/ 38.0%

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

       2. neg-mul-1 38.0%

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

       3. associate-*r/ 38.0%

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

       4. mul-1-neg 38.0%

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

   11. Simplified 38.0%

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

       1. clear-num 38.0%

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

       2. inv-pow 38.0%

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

       3. sub-div 38.0%

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

       4. *-commutative 38.0%

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

   13. Applied egg-rr 38.0%

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

       1. unpow-1 38.0%

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

       2. associate-/l* 38.6%

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

   15. Simplified 50.5%

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

       1. associate-*r/ 51.5%

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

       2. associate-/l* 51.5%

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

       3. associate-/l/ 47.3%

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

       4. *-commutative 47.3%

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

       5. *-commutative 47.3%

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

   17. Applied egg-rr 47.3%

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

   19. Simplified 52.5%

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

   if -0.0 < (*.f64 (/.f64 c0 (*.f64 2 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 84.2%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in c0 around inf 86.5%

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

      1. *-commutative 86.5%

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

      2. *-commutative 86.5%

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

      3. associate-*r* 79.4%

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

      4. times-frac 79.3%

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

      5. *-commutative 79.3%

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

   7. Simplified 79.3%

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

      1. associate-*r/ 79.3%

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

      2. *-commutative 79.3%

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

   9. Applied egg-rr 79.3%

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

   10. Taylor expanded in c0 around 0 89.7%

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

       1. *-commutative 89.7%

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

       2. times-frac 88.4%

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

       3. unpow2 88.4%

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

       4. unpow2 88.4%

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

       5. times-frac 90.7%

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

       6. unpow2 90.7%

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

       7. associate-/r* 83.6%

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

   12. Simplified 83.6%

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

       1. associate-/r* 90.7%

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

       2. unpow2 90.7%

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

       3. associate-*l* 91.9%

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

       4. associate-*r/ 92.0%

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

       5. *-commutative 92.0%

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

   14. Applied egg-rr 92.0%

       \[\leadsto c0 \cdot \frac{2 \cdot \color{blue}{\frac{\left(\frac{d}{D} \cdot \frac{c0}{h \cdot w}\right) \cdot d}{D}}}{2 \cdot w} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq -2 \cdot 10^{-182}:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \left(\frac{c0}{h \cdot D} \cdot \frac{d}{w}\right) \cdot \frac{d}{D}\right)}{2 \cdot w}\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 0 \lor \neg \left(\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty\right):\\ \;\;\;\;c0 \cdot \frac{\frac{0.5}{w}}{\frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot \left(w \cdot {M}^{2}\right)} \cdot \left(c0 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{2 \cdot \frac{d \cdot \left(\frac{d}{D} \cdot \frac{c0}{w \cdot h}\right)}{D}}{2 \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ \begin{array}{l} t_0 := \frac{{M}^{2}}{c0}\\ t_1 := {\left(\frac{d\_m}{D\_m}\right)}^{2}\\ t_2 := \frac{c0 \cdot \left(d\_m \cdot d\_m\right)}{\left(w \cdot h\right) \cdot \left(D\_m \cdot D\_m\right)}\\ t_3 := \frac{c0}{2 \cdot w} \cdot \left(t\_2 + \sqrt{t\_2 \cdot t\_2 - M \cdot M}\right)\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{-182}:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(c0, d\_m \cdot \frac{d\_m}{D\_m \cdot \left(w \cdot \left(h \cdot D\_m\right)\right)}, \left(\frac{c0}{h \cdot D\_m} \cdot \frac{d\_m}{w}\right) \cdot \frac{d\_m}{D\_m}\right)}{2 \cdot w}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;c0 \cdot \frac{t\_0 \cdot \frac{w}{2 \cdot \frac{t\_1}{h}}}{2 \cdot w}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;c0 \cdot \frac{2 \cdot \frac{d\_m \cdot \left(\frac{d\_m}{D\_m} \cdot \frac{c0}{w \cdot h}\right)}{D\_m}}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{t\_0 \cdot \frac{h}{2 \cdot \frac{t\_1}{w}}}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
(FPCore (c0 w h D_m d_m M)
 :precision binary64
 (let* ((t_0 (/ (pow M 2.0) c0))
        (t_1 (pow (/ d_m D_m) 2.0))
        (t_2 (/ (* c0 (* d_m d_m)) (* (* w h) (* D_m D_m))))
        (t_3 (* (/ c0 (* 2.0 w)) (+ t_2 (sqrt (- (* t_2 t_2) (* M M)))))))
   (if (<= t_3 -2e-182)
     (*
      c0
      (/
       (fma
        c0
        (* d_m (/ d_m (* D_m (* w (* h D_m)))))
        (* (* (/ c0 (* h D_m)) (/ d_m w)) (/ d_m D_m)))
       (* 2.0 w)))
     (if (<= t_3 0.0)
       (* c0 (/ (* t_0 (/ w (* 2.0 (/ t_1 h)))) (* 2.0 w)))
       (if (<= t_3 INFINITY)
         (*
          c0
          (/ (* 2.0 (/ (* d_m (* (/ d_m D_m) (/ c0 (* w h)))) D_m)) (* 2.0 w)))
         (* c0 (/ (* t_0 (/ h (* 2.0 (/ t_1 w)))) (* 2.0 w))))))))

C

D_m = fabs(D);
d_m = fabs(d);
double code(double c0, double w, double h, double D_m, double d_m, double M) {
	double t_0 = pow(M, 2.0) / c0;
	double t_1 = pow((d_m / D_m), 2.0);
	double t_2 = (c0 * (d_m * d_m)) / ((w * h) * (D_m * D_m));
	double t_3 = (c0 / (2.0 * w)) * (t_2 + sqrt(((t_2 * t_2) - (M * M))));
	double tmp;
	if (t_3 <= -2e-182) {
		tmp = c0 * (fma(c0, (d_m * (d_m / (D_m * (w * (h * D_m))))), (((c0 / (h * D_m)) * (d_m / w)) * (d_m / D_m))) / (2.0 * w));
	} else if (t_3 <= 0.0) {
		tmp = c0 * ((t_0 * (w / (2.0 * (t_1 / h)))) / (2.0 * w));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = c0 * ((2.0 * ((d_m * ((d_m / D_m) * (c0 / (w * h)))) / D_m)) / (2.0 * w));
	} else {
		tmp = c0 * ((t_0 * (h / (2.0 * (t_1 / w)))) / (2.0 * w));
	}
	return tmp;
}

Julia

D_m = abs(D)
d_m = abs(d)
function code(c0, w, h, D_m, d_m, M)
	t_0 = Float64((M ^ 2.0) / c0)
	t_1 = Float64(d_m / D_m) ^ 2.0
	t_2 = Float64(Float64(c0 * Float64(d_m * d_m)) / Float64(Float64(w * h) * Float64(D_m * D_m)))
	t_3 = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M)))))
	tmp = 0.0
	if (t_3 <= -2e-182)
		tmp = Float64(c0 * Float64(fma(c0, Float64(d_m * Float64(d_m / Float64(D_m * Float64(w * Float64(h * D_m))))), Float64(Float64(Float64(c0 / Float64(h * D_m)) * Float64(d_m / w)) * Float64(d_m / D_m))) / Float64(2.0 * w)));
	elseif (t_3 <= 0.0)
		tmp = Float64(c0 * Float64(Float64(t_0 * Float64(w / Float64(2.0 * Float64(t_1 / h)))) / Float64(2.0 * w)));
	elseif (t_3 <= Inf)
		tmp = Float64(c0 * Float64(Float64(2.0 * Float64(Float64(d_m * Float64(Float64(d_m / D_m) * Float64(c0 / Float64(w * h)))) / D_m)) / Float64(2.0 * w)));
	else
		tmp = Float64(c0 * Float64(Float64(t_0 * Float64(h / Float64(2.0 * Float64(t_1 / w)))) / Float64(2.0 * w)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
code[c0_, w_, h_, D$95$m_, d$95$m_, M_] := Block[{t$95$0 = N[(N[Power[M, 2.0], $MachinePrecision] / c0), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(d$95$m / D$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(c0 * N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = 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]}, If[LessEqual[t$95$3, -2e-182], N[(c0 * N[(N[(c0 * N[(d$95$m * N[(d$95$m / N[(D$95$m * N[(w * N[(h * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(c0 / N[(h * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(d$95$m / w), $MachinePrecision]), $MachinePrecision] * N[(d$95$m / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(c0 * N[(N[(t$95$0 * N[(w / N[(2.0 * N[(t$95$1 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(c0 * N[(N[(2.0 * N[(N[(d$95$m * N[(N[(d$95$m / D$95$m), $MachinePrecision] * N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / D$95$m), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[(t$95$0 * N[(h / N[(2.0 * N[(t$95$1 / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (/.f64 c0 (*.f64 2 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))))) < -2.0000000000000001e-182

   1. Initial program 81.5%

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

   3. Simplified 86.8%

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

   5. Taylor expanded in c0 around inf 81.6%

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

      1. *-commutative 81.6%

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

      2. *-commutative 81.6%

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

      3. associate-*r* 78.9%

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

      4. times-frac 79.0%

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

      5. *-commutative 79.0%

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

   7. Simplified 79.0%

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

      1. associate-*r/ 81.6%

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

      2. *-commutative 81.6%

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

   9. Applied egg-rr 81.6%

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

       1. times-frac 84.2%

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

       2. associate-/r* 84.2%

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

       3. pow2 84.2%

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

       4. pow2 84.2%

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

       5. frac-times 86.8%

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

       6. associate-*r* 89.4%

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

       7. *-commutative 89.4%

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

   11. Applied egg-rr 89.4%

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

   12. Taylor expanded in c0 around 0 89.4%

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

       1. associate-*r* 89.4%

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

       2. times-frac 89.5%

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

   14. Simplified 89.5%

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

   if -2.0000000000000001e-182 < (*.f64 (/.f64 c0 (*.f64 2 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))))) < -0.0

   1. Initial program 55.4%

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

   3. Simplified 17.4%

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

   6. Applied egg-rr 47.6%

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

      1. associate--r- 66.8%

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

      2. +-inverses 73.5%

         \[\leadsto c0 \cdot \frac{\frac{\color{blue}{0} + {M}^{2}}{\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h} - \sqrt{{\left(\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

      3. associate-/l* 67.0%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{\color{blue}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h}} - \sqrt{{\left(\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

      4. *-commutative 67.0%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{\color{blue}{h \cdot w}} - \sqrt{{\left(\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

      5. associate-/l* 66.9%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w} - \sqrt{{\color{blue}{\left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}}^{2} - {M}^{2}}}}{2 \cdot w} \]

      6. *-commutative 66.9%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w} - \sqrt{{\left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{\color{blue}{h \cdot w}}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

   8. Simplified 66.9%

      \[\leadsto c0 \cdot \frac{\color{blue}{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w} - \sqrt{{\left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w}\right)}^{2} - {M}^{2}}}}}{2 \cdot w} \]

   9. Taylor expanded in c0 around -inf 66.9%

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

       1. associate-*r/ 66.9%

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

       2. neg-mul-1 66.9%

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

       3. associate-*r/ 66.9%

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

       4. mul-1-neg 66.9%

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

   11. Simplified 66.9%

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

   12. Taylor expanded in w around -inf 73.3%

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

       1. times-frac 66.9%

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

       2. cancel-sign-sub-inv 66.9%

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

       3. metadata-eval 66.9%

          \[\leadsto c0 \cdot \frac{\frac{{M}^{2}}{c0} \cdot \frac{w}{\frac{{d}^{2}}{{D}^{2} \cdot h} + \color{blue}{1} \cdot \frac{{d}^{2}}{{D}^{2} \cdot h}}}{2 \cdot w} \]

       4. distribute-rgt1-in 66.9%

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

       5. metadata-eval 66.9%

          \[\leadsto c0 \cdot \frac{\frac{{M}^{2}}{c0} \cdot \frac{w}{\color{blue}{2} \cdot \frac{{d}^{2}}{{D}^{2} \cdot h}}}{2 \cdot w} \]

       6. associate-/r* 66.8%

          \[\leadsto c0 \cdot \frac{\frac{{M}^{2}}{c0} \cdot \frac{w}{2 \cdot \color{blue}{\frac{\frac{{d}^{2}}{{D}^{2}}}{h}}}}{2 \cdot w} \]

       7. unpow2 66.8%

          \[\leadsto c0 \cdot \frac{\frac{{M}^{2}}{c0} \cdot \frac{w}{2 \cdot \frac{\frac{\color{blue}{d \cdot d}}{{D}^{2}}}{h}}}{2 \cdot w} \]

       8. unpow2 66.8%

          \[\leadsto c0 \cdot \frac{\frac{{M}^{2}}{c0} \cdot \frac{w}{2 \cdot \frac{\frac{d \cdot d}{\color{blue}{D \cdot D}}}{h}}}{2 \cdot w} \]

       9. times-frac 67.0%

          \[\leadsto c0 \cdot \frac{\frac{{M}^{2}}{c0} \cdot \frac{w}{2 \cdot \frac{\color{blue}{\frac{d}{D} \cdot \frac{d}{D}}}{h}}}{2 \cdot w} \]

       10. unpow2 67.0%

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

   14. Simplified 67.0%

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

   if -0.0 < (*.f64 (/.f64 c0 (*.f64 2 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 84.2%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in c0 around inf 86.5%

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

      1. *-commutative 86.5%

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

      2. *-commutative 86.5%

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

      3. associate-*r* 79.4%

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

      4. times-frac 79.3%

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

      5. *-commutative 79.3%

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

   7. Simplified 79.3%

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

      1. associate-*r/ 79.3%

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

      2. *-commutative 79.3%

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

   9. Applied egg-rr 79.3%

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

   10. Taylor expanded in c0 around 0 89.7%

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

       1. *-commutative 89.7%

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

       2. times-frac 88.4%

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

       3. unpow2 88.4%

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

       4. unpow2 88.4%

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

       5. times-frac 90.7%

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

       6. unpow2 90.7%

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

       7. associate-/r* 83.6%

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

   12. Simplified 83.6%

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

       1. associate-/r* 90.7%

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

       2. unpow2 90.7%

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

       3. associate-*l* 91.9%

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

       4. associate-*r/ 92.0%

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

       5. *-commutative 92.0%

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

   14. Applied egg-rr 92.0%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 2 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 0.0%

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

   3. Simplified 20.7%

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

   6. Applied egg-rr 2.1%

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

      1. associate--r- 3.1%

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

      2. +-inverses 39.8%

         \[\leadsto c0 \cdot \frac{\frac{\color{blue}{0} + {M}^{2}}{\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h} - \sqrt{{\left(\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

      3. associate-/l* 40.4%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{\color{blue}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h}} - \sqrt{{\left(\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

      4. *-commutative 40.4%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{\color{blue}{h \cdot w}} - \sqrt{{\left(\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

      5. associate-/l* 39.7%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w} - \sqrt{{\color{blue}{\left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}}^{2} - {M}^{2}}}}{2 \cdot w} \]

      6. *-commutative 39.7%

         \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w} - \sqrt{{\left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{\color{blue}{h \cdot w}}\right)}^{2} - {M}^{2}}}}{2 \cdot w} \]

   8. Simplified 39.7%

      \[\leadsto c0 \cdot \frac{\color{blue}{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w} - \sqrt{{\left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w}\right)}^{2} - {M}^{2}}}}}{2 \cdot w} \]

   9. Taylor expanded in c0 around -inf 35.4%

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

       1. associate-*r/ 35.4%

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

       2. neg-mul-1 35.4%

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

       3. associate-*r/ 35.4%

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

       4. mul-1-neg 35.4%

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

   11. Simplified 35.4%

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

   12. Taylor expanded in h around -inf 38.3%

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

       1. times-frac 38.6%

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

       2. cancel-sign-sub-inv 38.6%

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

       3. metadata-eval 38.6%

          \[\leadsto c0 \cdot \frac{\frac{{M}^{2}}{c0} \cdot \frac{h}{\frac{{d}^{2}}{{D}^{2} \cdot w} + \color{blue}{1} \cdot \frac{{d}^{2}}{{D}^{2} \cdot w}}}{2 \cdot w} \]

       4. distribute-rgt1-in 38.6%

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

       5. metadata-eval 38.6%

          \[\leadsto c0 \cdot \frac{\frac{{M}^{2}}{c0} \cdot \frac{h}{\color{blue}{2} \cdot \frac{{d}^{2}}{{D}^{2} \cdot w}}}{2 \cdot w} \]

       6. associate-/r* 39.2%

          \[\leadsto c0 \cdot \frac{\frac{{M}^{2}}{c0} \cdot \frac{h}{2 \cdot \color{blue}{\frac{\frac{{d}^{2}}{{D}^{2}}}{w}}}}{2 \cdot w} \]

       7. unpow2 39.2%

          \[\leadsto c0 \cdot \frac{\frac{{M}^{2}}{c0} \cdot \frac{h}{2 \cdot \frac{\frac{\color{blue}{d \cdot d}}{{D}^{2}}}{w}}}{2 \cdot w} \]

       8. unpow2 39.2%

          \[\leadsto c0 \cdot \frac{\frac{{M}^{2}}{c0} \cdot \frac{h}{2 \cdot \frac{\frac{d \cdot d}{\color{blue}{D \cdot D}}}{w}}}{2 \cdot w} \]

       9. times-frac 48.3%

          \[\leadsto c0 \cdot \frac{\frac{{M}^{2}}{c0} \cdot \frac{h}{2 \cdot \frac{\color{blue}{\frac{d}{D} \cdot \frac{d}{D}}}{w}}}{2 \cdot w} \]

       10. unpow2 48.3%

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

   14. Simplified 48.3%

       \[\leadsto c0 \cdot \frac{\color{blue}{\frac{{M}^{2}}{c0} \cdot \frac{h}{2 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w}}}}{2 \cdot w} \]
3. Recombined 4 regimes into one program.

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq -2 \cdot 10^{-182}:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \left(\frac{c0}{h \cdot D} \cdot \frac{d}{w}\right) \cdot \frac{d}{D}\right)}{2 \cdot w}\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 0:\\ \;\;\;\;c0 \cdot \frac{\frac{{M}^{2}}{c0} \cdot \frac{w}{2 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h}}}{2 \cdot w}\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;c0 \cdot \frac{2 \cdot \frac{d \cdot \left(\frac{d}{D} \cdot \frac{c0}{w \cdot h}\right)}{D}}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\frac{{M}^{2}}{c0} \cdot \frac{h}{2 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w}}}{2 \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d\_m \cdot d\_m\right)}{\left(w \cdot h\right) \cdot \left(D\_m \cdot D\_m\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;c0 \cdot \frac{2 \cdot \frac{d\_m \cdot \left(\frac{d\_m}{D\_m} \cdot \frac{c0}{w \cdot h}\right)}{D\_m}}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
(FPCore (c0 w h D_m d_m M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d_m d_m)) (* (* w h) (* D_m D_m)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
        INFINITY)
     (*
      c0
      (/ (* 2.0 (/ (* d_m (* (/ d_m D_m) (/ c0 (* w h)))) D_m)) (* 2.0 w)))
     (* c0 (/ 0.0 (* 2.0 w))))))

C

D_m = fabs(D);
d_m = fabs(d);
double code(double c0, double w, double h, double D_m, double d_m, double M) {
	double t_0 = (c0 * (d_m * d_m)) / ((w * h) * (D_m * D_m));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= ((double) INFINITY)) {
		tmp = c0 * ((2.0 * ((d_m * ((d_m / D_m) * (c0 / (w * h)))) / D_m)) / (2.0 * w));
	} else {
		tmp = c0 * (0.0 / (2.0 * w));
	}
	return tmp;
}

Java

D_m = Math.abs(D);
d_m = Math.abs(d);
public static double code(double c0, double w, double h, double D_m, double d_m, double M) {
	double t_0 = (c0 * (d_m * d_m)) / ((w * h) * (D_m * D_m));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = c0 * ((2.0 * ((d_m * ((d_m / D_m) * (c0 / (w * h)))) / D_m)) / (2.0 * w));
	} else {
		tmp = c0 * (0.0 / (2.0 * w));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
d_m = math.fabs(d)
def code(c0, w, h, D_m, d_m, M):
	t_0 = (c0 * (d_m * d_m)) / ((w * h) * (D_m * D_m))
	tmp = 0
	if ((c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))) <= math.inf:
		tmp = c0 * ((2.0 * ((d_m * ((d_m / D_m) * (c0 / (w * h)))) / D_m)) / (2.0 * w))
	else:
		tmp = c0 * (0.0 / (2.0 * w))
	return tmp

Julia

D_m = abs(D)
d_m = abs(d)
function code(c0, w, h, D_m, d_m, M)
	t_0 = Float64(Float64(c0 * Float64(d_m * d_m)) / Float64(Float64(w * h) * Float64(D_m * D_m)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M))))) <= Inf)
		tmp = Float64(c0 * Float64(Float64(2.0 * Float64(Float64(d_m * Float64(Float64(d_m / D_m) * Float64(c0 / Float64(w * h)))) / D_m)) / Float64(2.0 * w)));
	else
		tmp = Float64(c0 * Float64(0.0 / Float64(2.0 * w)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d_m = abs(d);
function tmp_2 = code(c0, w, h, D_m, d_m, M)
	t_0 = (c0 * (d_m * d_m)) / ((w * h) * (D_m * D_m));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= Inf)
		tmp = c0 * ((2.0 * ((d_m * ((d_m / D_m) * (c0 / (w * h)))) / D_m)) / (2.0 * w));
	else
		tmp = c0 * (0.0 / (2.0 * w));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
code[c0_, w_, h_, D$95$m_, d$95$m_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(c0 * N[(N[(2.0 * N[(N[(d$95$m * N[(N[(d$95$m / D$95$m), $MachinePrecision] * N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / D$95$m), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(0.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 2 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 78.6%

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

   3. Simplified 72.3%

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

   5. Taylor expanded in c0 around inf 72.7%

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

      1. *-commutative 72.7%

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

      2. *-commutative 72.7%

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

      3. associate-*r* 68.4%

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

      4. times-frac 68.4%

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

      5. *-commutative 68.4%

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

   7. Simplified 68.4%

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

      1. associate-*r/ 69.2%

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

      2. *-commutative 69.2%

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

   9. Applied egg-rr 69.2%

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

   10. Taylor expanded in c0 around 0 74.0%

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

       1. *-commutative 74.0%

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

       2. times-frac 74.4%

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

       3. unpow2 74.4%

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

       4. unpow2 74.4%

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

       5. times-frac 78.4%

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

       6. unpow2 78.4%

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

       7. associate-/r* 75.0%

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

   12. Simplified 75.0%

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

       1. associate-/r* 78.4%

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

       2. unpow2 78.4%

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

       3. associate-*l* 80.0%

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

       4. associate-*r/ 79.1%

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

       5. *-commutative 79.1%

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

   14. Applied egg-rr 79.1%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 2 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 0.0%

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

   3. Simplified 20.7%

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

   5. Taylor expanded in c0 around -inf 1.4%

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

      1. mul-1-neg 1.4%

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

      2. distribute-lft-in 0.1%

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

      3. mul-1-neg 0.1%

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

      4. distribute-rgt-neg-in 0.1%

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

      5. associate-/l* 0.1%

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

      6. mul-1-neg 0.1%

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

      7. associate-/l* 0.1%

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

      8. distribute-lft1-in 0.1%

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

      9. metadata-eval 0.1%

         \[\leadsto c0 \cdot \frac{-\color{blue}{0} \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{2 \cdot w} \]

      10. mul0-lft 46.6%

          \[\leadsto c0 \cdot \frac{-\color{blue}{0}}{2 \cdot w} \]

      11. metadata-eval 46.6%

          \[\leadsto c0 \cdot \frac{\color{blue}{0}}{2 \cdot w} \]

   7. Simplified 46.6%

      \[\leadsto c0 \cdot \frac{\color{blue}{0}}{2 \cdot w} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;c0 \cdot \frac{2 \cdot \frac{d \cdot \left(\frac{d}{D} \cdot \frac{c0}{w \cdot h}\right)}{D}}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 33.7% accurate, 21.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ c0 \cdot \frac{0}{2 \cdot w} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
(FPCore (c0 w h D_m d_m M) :precision binary64 (* c0 (/ 0.0 (* 2.0 w))))

C

D_m = fabs(D);
d_m = fabs(d);
double code(double c0, double w, double h, double D_m, double d_m, double M) {
	return c0 * (0.0 / (2.0 * w));
}

Fortran

D_m = abs(d)
d_m = abs(d)
real(8) function code(c0, w, h, d_m, d_m_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d_m
    real(8), intent (in) :: d_m_1
    real(8), intent (in) :: m
    code = c0 * (0.0d0 / (2.0d0 * w))
end function

Java

D_m = Math.abs(D);
d_m = Math.abs(d);
public static double code(double c0, double w, double h, double D_m, double d_m, double M) {
	return c0 * (0.0 / (2.0 * w));
}

Python

D_m = math.fabs(D)
d_m = math.fabs(d)
def code(c0, w, h, D_m, d_m, M):
	return c0 * (0.0 / (2.0 * w))

Julia

D_m = abs(D)
d_m = abs(d)
function code(c0, w, h, D_m, d_m, M)
	return Float64(c0 * Float64(0.0 / Float64(2.0 * w)))
end

MATLAB

D_m = abs(D);
d_m = abs(d);
function tmp = code(c0, w, h, D_m, d_m, M)
	tmp = c0 * (0.0 / (2.0 * w));
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
code[c0_, w_, h_, D$95$m_, d$95$m_, M_] := N[(c0 * N[(0.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.8%

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

3. Simplified 39.6%

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

5. Taylor expanded in c0 around -inf 4.0%

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

   1. mul-1-neg 4.0%

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

   2. distribute-lft-in 3.1%

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

   3. mul-1-neg 3.1%

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

   4. distribute-rgt-neg-in 3.1%

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

   5. associate-/l* 2.4%

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

   6. mul-1-neg 2.4%

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

   7. associate-/l* 3.5%

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

   8. distribute-lft1-in 3.5%

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

   9. metadata-eval 3.5%

      \[\leadsto c0 \cdot \frac{-\color{blue}{0} \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{2 \cdot w} \]

   10. mul0-lft 33.4%

       \[\leadsto c0 \cdot \frac{-\color{blue}{0}}{2 \cdot w} \]

   11. metadata-eval 33.4%

       \[\leadsto c0 \cdot \frac{\color{blue}{0}}{2 \cdot w} \]

7. Simplified 33.4%

   \[\leadsto c0 \cdot \frac{\color{blue}{0}}{2 \cdot w} \]

8. Final simplification 33.4%

   \[\leadsto c0 \cdot \frac{0}{2 \cdot w} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024051 
(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))))))