Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.6% → 62.2%

Time: 38.9s

Alternatives: 14

Speedup: 151.0×

• could not determine a ground truth

  1. c0 = -5.4233147318356905e+234
  2. w = 2.5108267650882385e-294
  3. h = 3.580834937640173e+38
  4. D = 3.3489919665369163e-282
  5. d = -7.083329026995278e+288
  6. M = 3.1235850364264856e+205

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: 24.6% 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: 62.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{d}{D}\right)}^{2}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{\frac{c0}{w}}{h}\\ t_3 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\\ t_4 := t\_1 \cdot \left(t\_3 + \sqrt{t\_3 \cdot t\_3 - M \cdot M}\right)\\ t_5 := t\_0 \cdot t\_2\\ t_6 := \frac{t\_0 \cdot \frac{c0}{w}}{h}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(\frac{c0}{w \cdot h}, t\_0, M\right)\right)}^{0.25}\right)}^{2}, \sqrt{t\_5 - M}, t\_5\right)\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{d}{M}}{h}}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_1 \cdot \left(t\_6 + \sqrt{\mathsf{fma}\left(t\_2, t\_0, M\right)} \cdot \sqrt{t\_6 - M}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(h \cdot \frac{{D}^{2}}{{\left(\frac{d}{M}\right)}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (pow (/ d D) 2.0))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (/ (/ c0 w) h))
        (t_3 (/ (* c0 (* d d)) (* (* D D) (* w h))))
        (t_4 (* t_1 (+ t_3 (sqrt (- (* t_3 t_3) (* M M))))))
        (t_5 (* t_0 t_2))
        (t_6 (/ (* t_0 (/ c0 w)) h)))
   (if (<= t_4 (- INFINITY))
     (*
      t_1
      (fma
       (pow (pow (fma (/ c0 (* w h)) t_0 M) 0.25) 2.0)
       (sqrt (- t_5 M))
       t_5))
     (if (<= t_4 0.0)
       (* 0.25 (/ (pow D 2.0) (/ (* (/ d M) (/ d M)) h)))
       (if (<= t_4 INFINITY)
         (* t_1 (+ t_6 (* (sqrt (fma t_2 t_0 M)) (sqrt (- t_6 M)))))
         (* 0.25 (* h (/ (pow D 2.0) (pow (/ d M) 2.0)))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = pow((d / D), 2.0);
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 / w) / h;
	double t_3 = (c0 * (d * d)) / ((D * D) * (w * h));
	double t_4 = t_1 * (t_3 + sqrt(((t_3 * t_3) - (M * M))));
	double t_5 = t_0 * t_2;
	double t_6 = (t_0 * (c0 / w)) / h;
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = t_1 * fma(pow(pow(fma((c0 / (w * h)), t_0, M), 0.25), 2.0), sqrt((t_5 - M)), t_5);
	} else if (t_4 <= 0.0) {
		tmp = 0.25 * (pow(D, 2.0) / (((d / M) * (d / M)) / h));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_1 * (t_6 + (sqrt(fma(t_2, t_0, M)) * sqrt((t_6 - M))));
	} else {
		tmp = 0.25 * (h * (pow(D, 2.0) / pow((d / M), 2.0)));
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(d / D) ^ 2.0
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(Float64(c0 / w) / h)
	t_3 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(D * D) * Float64(w * h)))
	t_4 = Float64(t_1 * Float64(t_3 + sqrt(Float64(Float64(t_3 * t_3) - Float64(M * M)))))
	t_5 = Float64(t_0 * t_2)
	t_6 = Float64(Float64(t_0 * Float64(c0 / w)) / h)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(t_1 * fma(((fma(Float64(c0 / Float64(w * h)), t_0, M) ^ 0.25) ^ 2.0), sqrt(Float64(t_5 - M)), t_5));
	elseif (t_4 <= 0.0)
		tmp = Float64(0.25 * Float64((D ^ 2.0) / Float64(Float64(Float64(d / M) * Float64(d / M)) / h)));
	elseif (t_4 <= Inf)
		tmp = Float64(t_1 * Float64(t_6 + Float64(sqrt(fma(t_2, t_0, M)) * sqrt(Float64(t_6 - M)))));
	else
		tmp = Float64(0.25 * Float64(h * Float64((D ^ 2.0) / (Float64(d / M) ^ 2.0))));
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision]}, Block[{t$95$3 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 * N[(t$95$3 + N[Sqrt[N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$0 * t$95$2), $MachinePrecision]}, Block[{t$95$6 = N[(N[(t$95$0 * N[(c0 / w), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(t$95$1 * N[(N[Power[N[Power[N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] * t$95$0 + M), $MachinePrecision], 0.25], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[N[(t$95$5 - M), $MachinePrecision]], $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.0], N[(0.25 * N[(N[Power[D, 2.0], $MachinePrecision] / N[(N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(t$95$1 * N[(t$95$6 + N[(N[Sqrt[N[(t$95$2 * t$95$0 + M), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$6 - M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(h * N[(N[Power[D, 2.0], $MachinePrecision] / N[Power[N[(d / M), $MachinePrecision], 2.0], $MachinePrecision]), $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))))) < -inf.0

   1. Initial program 74.9%

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

      1. +-commutative 74.9%

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

      2. +-commutative 74.9%

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

      3. times-frac 74.9%

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

      4. fma-neg 74.9%

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

   3. Simplified 74.9%

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

   6. Applied egg-rr 87.0%

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

      1. associate-/r* 87.0%

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

      2. associate-/r* 87.1%

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

      3. associate-/r* 87.0%

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

   8. Simplified 87.0%

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

      1. fma-udef 87.0%

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

      2. associate-/r* 87.0%

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

      3. *-commutative 87.0%

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

   10. Applied egg-rr 87.0%

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

       1. add-sqr-sqrt 87.1%

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

       2. pow2 87.1%

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

       3. pow1/2 87.1%

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

       4. sqrt-pow1 87.2%

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

       5. *-commutative 87.2%

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

       6. fma-def 87.2%

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

       7. metadata-eval 87.2%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(\frac{c0}{w \cdot h}, {\left(\frac{d}{D}\right)}^{2}, M\right)\right)}^{\color{blue}{0.25}}\right)}^{2}, \sqrt{\frac{\frac{c0}{w}}{h} \cdot {\left(\frac{d}{D}\right)}^{2} - M}, \frac{\frac{c0}{w}}{h} \cdot {\left(\frac{d}{D}\right)}^{2}\right) \]

   12. Applied egg-rr 87.2%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\color{blue}{{\left({\left(\mathsf{fma}\left(\frac{c0}{w \cdot h}, {\left(\frac{d}{D}\right)}^{2}, M\right)\right)}^{0.25}\right)}^{2}}, \sqrt{\frac{\frac{c0}{w}}{h} \cdot {\left(\frac{d}{D}\right)}^{2} - M}, \frac{\frac{c0}{w}}{h} \cdot {\left(\frac{d}{D}\right)}^{2}\right) \]

   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))))) < -0.0

   1. Initial program 68.5%

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

      1. +-commutative 68.5%

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

      2. +-commutative 68.5%

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

      3. times-frac 48.6%

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

      4. fma-neg 48.6%

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

   3. Simplified 67.9%

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

   5. Taylor expanded in c0 around -inf 67.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) + 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right)} \]
   6. Step-by-step derivation

      1. fma-def 67.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(-1, c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right)} \]

      2. distribute-lft1-in 67.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}, 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      3. metadata-eval 67.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      4. associate-/r* 60.8%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \color{blue}{\frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      5. times-frac 60.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \color{blue}{\left(\frac{{D}^{2}}{c0} \cdot \frac{{M}^{2} \cdot \left(h \cdot w\right)}{{d}^{2}}\right)}\right) \]

      6. associate-*r* 60.6%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \left(\frac{{D}^{2}}{c0} \cdot \frac{\color{blue}{\left({M}^{2} \cdot h\right) \cdot w}}{{d}^{2}}\right)\right) \]

   7. Simplified 60.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \left(\frac{{D}^{2}}{c0} \cdot \frac{\left({M}^{2} \cdot h\right) \cdot w}{{d}^{2}}\right)\right)} \]

   8. Taylor expanded in c0 around 0 87.3%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. associate-/l* 87.2%

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

      2. associate-/r* 81.0%

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

   10. Simplified 81.0%

       \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2}}{\frac{\frac{{d}^{2}}{{M}^{2}}}{h}}} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\color{blue}{\sqrt{\frac{{d}^{2}}{{M}^{2}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}}{h}} \]

       2. sqrt-div 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{{M}^{2}}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       3. unpow2 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{{M}^{2}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       4. sqrt-prod 27.7%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{{M}^{2}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       5. add-sqr-sqrt 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{\color{blue}{d}}{\sqrt{{M}^{2}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       6. unpow2 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{\sqrt{\color{blue}{M \cdot M}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       7. sqrt-prod 34.4%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{\color{blue}{\sqrt{M} \cdot \sqrt{M}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       8. add-sqr-sqrt 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{\color{blue}{M}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       9. sqrt-div 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{{M}^{2}}}}}{h}} \]

       10. unpow2 80.9%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{{M}^{2}}}}{h}} \]

       11. sqrt-prod 27.7%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{{M}^{2}}}}{h}} \]

       12. add-sqr-sqrt 80.9%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{\color{blue}{d}}{\sqrt{{M}^{2}}}}{h}} \]

       13. unpow2 80.9%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{d}{\sqrt{\color{blue}{M \cdot M}}}}{h}} \]

       14. sqrt-prod 40.7%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{d}{\color{blue}{\sqrt{M} \cdot \sqrt{M}}}}{h}} \]

       15. add-sqr-sqrt 87.4%

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

   12. Applied egg-rr 87.4%

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

   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 73.8%

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

      1. +-commutative 73.8%

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

      2. +-commutative 73.8%

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

      3. times-frac 71.2%

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

      4. fma-neg 71.2%

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

   3. Simplified 71.2%

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

      1. associate-/r* 71.3%

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

      2. frac-times 71.2%

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

      3. *-commutative 71.2%

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

      4. associate-*r/ 71.2%

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

      5. pow2 71.2%

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

   6. Applied egg-rr 71.2%

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

   8. Applied egg-rr 79.9%

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

      1. *-commutative 79.9%

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

      2. associate-/r* 79.9%

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

      3. associate-/r* 82.0%

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

   10. Simplified 82.0%

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

       1. associate-*l/ 82.1%

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

   12. Applied egg-rr 82.1%

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

   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%

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

      1. +-commutative 0.0%

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

      2. +-commutative 0.0%

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

      3. times-frac 0.0%

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

      4. fma-neg 0.0%

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

   3. Simplified 3.9%

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

   5. Taylor expanded in c0 around -inf 1.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) + 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right)} \]
   6. Step-by-step derivation

      1. fma-def 1.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(-1, c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right)} \]

      2. distribute-lft1-in 1.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}, 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      3. metadata-eval 1.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      4. associate-/r* 1.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \color{blue}{\frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      5. times-frac 2.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \color{blue}{\left(\frac{{D}^{2}}{c0} \cdot \frac{{M}^{2} \cdot \left(h \cdot w\right)}{{d}^{2}}\right)}\right) \]

      6. associate-*r* 2.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \left(\frac{{D}^{2}}{c0} \cdot \frac{\color{blue}{\left({M}^{2} \cdot h\right) \cdot w}}{{d}^{2}}\right)\right) \]

   7. Simplified 2.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \left(\frac{{D}^{2}}{c0} \cdot \frac{\left({M}^{2} \cdot h\right) \cdot w}{{d}^{2}}\right)\right)} \]

   8. Taylor expanded in c0 around 0 44.8%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. associate-/l* 44.2%

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

      2. associate-/r* 44.8%

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

   10. Simplified 44.8%

       \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2}}{\frac{\frac{{d}^{2}}{{M}^{2}}}{h}}} \]
   11. Step-by-step derivation

       1. associate-/r/ 45.5%

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

       2. add-sqr-sqrt 45.5%

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

       3. pow2 45.5%

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

       4. sqrt-div 45.5%

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

       5. unpow2 45.5%

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

       6. sqrt-prod 26.7%

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

       7. add-sqr-sqrt 53.7%

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

       8. unpow2 53.7%

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

       9. sqrt-prod 31.5%

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

       10. add-sqr-sqrt 59.7%

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

   12. Applied egg-rr 59.7%

       \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{{\left(\frac{d}{M}\right)}^{2}} \cdot h\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.2%

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

Alternative 2: 62.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{c0}{w}}{h}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\\ t_3 := t\_1 \cdot \left(t\_2 + \sqrt{t\_2 \cdot t\_2 - M \cdot M}\right)\\ t_4 := {\left(\frac{d}{D}\right)}^{2}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(t\_0, \frac{d}{D} \cdot \frac{d}{D}, \mathsf{fma}\left(\frac{c0}{w \cdot h}, t\_4, M\right)\right)\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{d}{M}}{h}}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_1 \cdot \left(\frac{t\_4 \cdot \frac{c0}{w}}{h} + \sqrt{t\_4 \cdot t\_0 - M} \cdot \sqrt{\mathsf{fma}\left(t\_0, t\_4, M\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(h \cdot \frac{{D}^{2}}{{\left(\frac{d}{M}\right)}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (/ c0 w) h))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (/ (* c0 (* d d)) (* (* D D) (* w h))))
        (t_3 (* t_1 (+ t_2 (sqrt (- (* t_2 t_2) (* M M))))))
        (t_4 (pow (/ d D) 2.0)))
   (if (<= t_3 (- INFINITY))
     (*
      (/ (/ c0 w) 2.0)
      (fma t_0 (* (/ d D) (/ d D)) (fma (/ c0 (* w h)) t_4 M)))
     (if (<= t_3 0.0)
       (* 0.25 (/ (pow D 2.0) (/ (* (/ d M) (/ d M)) h)))
       (if (<= t_3 INFINITY)
         (*
          t_1
          (+
           (/ (* t_4 (/ c0 w)) h)
           (* (sqrt (- (* t_4 t_0) M)) (sqrt (fma t_0 t_4 M)))))
         (* 0.25 (* h (/ (pow D 2.0) (pow (/ d M) 2.0)))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 / w) / h;
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((D * D) * (w * h));
	double t_3 = t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))));
	double t_4 = pow((d / D), 2.0);
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = ((c0 / w) / 2.0) * fma(t_0, ((d / D) * (d / D)), fma((c0 / (w * h)), t_4, M));
	} else if (t_3 <= 0.0) {
		tmp = 0.25 * (pow(D, 2.0) / (((d / M) * (d / M)) / h));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_1 * (((t_4 * (c0 / w)) / h) + (sqrt(((t_4 * t_0) - M)) * sqrt(fma(t_0, t_4, M))));
	} else {
		tmp = 0.25 * (h * (pow(D, 2.0) / pow((d / M), 2.0)));
	}
	return tmp;
}

Julia

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(N[(c0 / w), $MachinePrecision] / 2.0), $MachinePrecision] * N[(t$95$0 * N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] + N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] * t$95$4 + M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(0.25 * N[(N[Power[D, 2.0], $MachinePrecision] / N[(N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(t$95$1 * N[(N[(N[(t$95$4 * N[(c0 / w), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision] + N[(N[Sqrt[N[(N[(t$95$4 * t$95$0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$0 * t$95$4 + M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(h * N[(N[Power[D, 2.0], $MachinePrecision] / N[Power[N[(d / M), $MachinePrecision], 2.0], $MachinePrecision]), $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))))) < -inf.0

   1. Initial program 74.9%

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

   3. Simplified 74.5%

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

      1. times-frac 74.4%

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

   6. Applied egg-rr 74.4%

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

      1. sqrt-prod 79.2%

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

      2. pow2 79.2%

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

      3. associate-/l/ 79.2%

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

      4. frac-times 79.3%

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

      5. unpow2 79.3%

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

      6. add-sqr-sqrt 47.7%

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

      7. sqrt-unprod 79.3%

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

      8. sqr-neg 79.3%

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

      9. sqrt-prod 31.6%

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

      10. add-sqr-sqrt 79.3%

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

   8. Applied egg-rr 79.3%

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

   10. Simplified 87.2%

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

   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))))) < -0.0

   1. Initial program 68.5%

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

      1. +-commutative 68.5%

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

      2. +-commutative 68.5%

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

      3. times-frac 48.6%

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

      4. fma-neg 48.6%

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

   3. Simplified 67.9%

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

   5. Taylor expanded in c0 around -inf 67.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) + 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right)} \]
   6. Step-by-step derivation

      1. fma-def 67.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(-1, c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right)} \]

      2. distribute-lft1-in 67.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}, 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      3. metadata-eval 67.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      4. associate-/r* 60.8%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \color{blue}{\frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      5. times-frac 60.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \color{blue}{\left(\frac{{D}^{2}}{c0} \cdot \frac{{M}^{2} \cdot \left(h \cdot w\right)}{{d}^{2}}\right)}\right) \]

      6. associate-*r* 60.6%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \left(\frac{{D}^{2}}{c0} \cdot \frac{\color{blue}{\left({M}^{2} \cdot h\right) \cdot w}}{{d}^{2}}\right)\right) \]

   7. Simplified 60.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \left(\frac{{D}^{2}}{c0} \cdot \frac{\left({M}^{2} \cdot h\right) \cdot w}{{d}^{2}}\right)\right)} \]

   8. Taylor expanded in c0 around 0 87.3%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. associate-/l* 87.2%

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

      2. associate-/r* 81.0%

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

   10. Simplified 81.0%

       \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2}}{\frac{\frac{{d}^{2}}{{M}^{2}}}{h}}} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\color{blue}{\sqrt{\frac{{d}^{2}}{{M}^{2}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}}{h}} \]

       2. sqrt-div 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{{M}^{2}}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       3. unpow2 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{{M}^{2}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       4. sqrt-prod 27.7%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{{M}^{2}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       5. add-sqr-sqrt 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{\color{blue}{d}}{\sqrt{{M}^{2}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       6. unpow2 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{\sqrt{\color{blue}{M \cdot M}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       7. sqrt-prod 34.4%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{\color{blue}{\sqrt{M} \cdot \sqrt{M}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       8. add-sqr-sqrt 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{\color{blue}{M}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       9. sqrt-div 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{{M}^{2}}}}}{h}} \]

       10. unpow2 80.9%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{{M}^{2}}}}{h}} \]

       11. sqrt-prod 27.7%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{{M}^{2}}}}{h}} \]

       12. add-sqr-sqrt 80.9%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{\color{blue}{d}}{\sqrt{{M}^{2}}}}{h}} \]

       13. unpow2 80.9%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{d}{\sqrt{\color{blue}{M \cdot M}}}}{h}} \]

       14. sqrt-prod 40.7%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{d}{\color{blue}{\sqrt{M} \cdot \sqrt{M}}}}{h}} \]

       15. add-sqr-sqrt 87.4%

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

   12. Applied egg-rr 87.4%

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

   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 73.8%

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

      1. +-commutative 73.8%

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

      2. +-commutative 73.8%

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

      3. times-frac 71.2%

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

      4. fma-neg 71.2%

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

   3. Simplified 71.2%

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

      1. associate-/r* 71.3%

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

      2. frac-times 71.2%

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

      3. *-commutative 71.2%

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

      4. associate-*r/ 71.2%

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

      5. pow2 71.2%

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

   6. Applied egg-rr 71.2%

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

   8. Applied egg-rr 79.9%

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

      1. *-commutative 79.9%

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

      2. associate-/r* 79.9%

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

      3. associate-/r* 82.0%

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

   10. Simplified 82.0%

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

   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%

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

      1. +-commutative 0.0%

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

      2. +-commutative 0.0%

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

      3. times-frac 0.0%

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

      4. fma-neg 0.0%

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

   3. Simplified 3.9%

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

   5. Taylor expanded in c0 around -inf 1.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) + 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right)} \]
   6. Step-by-step derivation

      1. fma-def 1.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(-1, c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right)} \]

      2. distribute-lft1-in 1.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}, 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      3. metadata-eval 1.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      4. associate-/r* 1.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \color{blue}{\frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      5. times-frac 2.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \color{blue}{\left(\frac{{D}^{2}}{c0} \cdot \frac{{M}^{2} \cdot \left(h \cdot w\right)}{{d}^{2}}\right)}\right) \]

      6. associate-*r* 2.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \left(\frac{{D}^{2}}{c0} \cdot \frac{\color{blue}{\left({M}^{2} \cdot h\right) \cdot w}}{{d}^{2}}\right)\right) \]

   7. Simplified 2.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \left(\frac{{D}^{2}}{c0} \cdot \frac{\left({M}^{2} \cdot h\right) \cdot w}{{d}^{2}}\right)\right)} \]

   8. Taylor expanded in c0 around 0 44.8%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. associate-/l* 44.2%

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

      2. associate-/r* 44.8%

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

   10. Simplified 44.8%

       \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2}}{\frac{\frac{{d}^{2}}{{M}^{2}}}{h}}} \]
   11. Step-by-step derivation

       1. associate-/r/ 45.5%

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

       2. add-sqr-sqrt 45.5%

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

       3. pow2 45.5%

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

       4. sqrt-div 45.5%

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

       5. unpow2 45.5%

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

       6. sqrt-prod 26.7%

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

       7. add-sqr-sqrt 53.7%

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

       8. unpow2 53.7%

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

       9. sqrt-prod 31.5%

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

       10. add-sqr-sqrt 59.7%

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

   12. Applied egg-rr 59.7%

       \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{{\left(\frac{d}{M}\right)}^{2}} \cdot h\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.2%

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

Alternative 3: 62.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{c0}{w}}{h}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\\ t_3 := t\_1 \cdot \left(t\_2 + \sqrt{t\_2 \cdot t\_2 - M \cdot M}\right)\\ t_4 := {\left(\frac{d}{D}\right)}^{2}\\ t_5 := \frac{t\_4 \cdot \frac{c0}{w}}{h}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(t\_0, \frac{d}{D} \cdot \frac{d}{D}, \mathsf{fma}\left(\frac{c0}{w \cdot h}, t\_4, M\right)\right)\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{d}{M}}{h}}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_1 \cdot \left(t\_5 + \sqrt{\mathsf{fma}\left(t\_0, t\_4, M\right)} \cdot \sqrt{t\_5 - M}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(h \cdot \frac{{D}^{2}}{{\left(\frac{d}{M}\right)}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (/ c0 w) h))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (/ (* c0 (* d d)) (* (* D D) (* w h))))
        (t_3 (* t_1 (+ t_2 (sqrt (- (* t_2 t_2) (* M M))))))
        (t_4 (pow (/ d D) 2.0))
        (t_5 (/ (* t_4 (/ c0 w)) h)))
   (if (<= t_3 (- INFINITY))
     (*
      (/ (/ c0 w) 2.0)
      (fma t_0 (* (/ d D) (/ d D)) (fma (/ c0 (* w h)) t_4 M)))
     (if (<= t_3 0.0)
       (* 0.25 (/ (pow D 2.0) (/ (* (/ d M) (/ d M)) h)))
       (if (<= t_3 INFINITY)
         (* t_1 (+ t_5 (* (sqrt (fma t_0 t_4 M)) (sqrt (- t_5 M)))))
         (* 0.25 (* h (/ (pow D 2.0) (pow (/ d M) 2.0)))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 / w) / h;
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((D * D) * (w * h));
	double t_3 = t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))));
	double t_4 = pow((d / D), 2.0);
	double t_5 = (t_4 * (c0 / w)) / h;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = ((c0 / w) / 2.0) * fma(t_0, ((d / D) * (d / D)), fma((c0 / (w * h)), t_4, M));
	} else if (t_3 <= 0.0) {
		tmp = 0.25 * (pow(D, 2.0) / (((d / M) * (d / M)) / h));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_1 * (t_5 + (sqrt(fma(t_0, t_4, M)) * sqrt((t_5 - M))));
	} else {
		tmp = 0.25 * (h * (pow(D, 2.0) / pow((d / M), 2.0)));
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 / w) / h)
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(D * D) * Float64(w * h)))
	t_3 = Float64(t_1 * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M)))))
	t_4 = Float64(d / D) ^ 2.0
	t_5 = Float64(Float64(t_4 * Float64(c0 / w)) / h)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(c0 / w) / 2.0) * fma(t_0, Float64(Float64(d / D) * Float64(d / D)), fma(Float64(c0 / Float64(w * h)), t_4, M)));
	elseif (t_3 <= 0.0)
		tmp = Float64(0.25 * Float64((D ^ 2.0) / Float64(Float64(Float64(d / M) * Float64(d / M)) / h)));
	elseif (t_3 <= Inf)
		tmp = Float64(t_1 * Float64(t_5 + Float64(sqrt(fma(t_0, t_4, M)) * sqrt(Float64(t_5 - M)))));
	else
		tmp = Float64(0.25 * Float64(h * Float64((D ^ 2.0) / (Float64(d / M) ^ 2.0))));
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$4 * N[(c0 / w), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(N[(c0 / w), $MachinePrecision] / 2.0), $MachinePrecision] * N[(t$95$0 * N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] + N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] * t$95$4 + M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(0.25 * N[(N[Power[D, 2.0], $MachinePrecision] / N[(N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(t$95$1 * N[(t$95$5 + N[(N[Sqrt[N[(t$95$0 * t$95$4 + M), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$5 - M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(h * N[(N[Power[D, 2.0], $MachinePrecision] / N[Power[N[(d / M), $MachinePrecision], 2.0], $MachinePrecision]), $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))))) < -inf.0

   1. Initial program 74.9%

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

   3. Simplified 74.5%

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

      1. times-frac 74.4%

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

   6. Applied egg-rr 74.4%

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

      1. sqrt-prod 79.2%

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

      2. pow2 79.2%

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

      3. associate-/l/ 79.2%

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

      4. frac-times 79.3%

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

      5. unpow2 79.3%

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

      6. add-sqr-sqrt 47.7%

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

      7. sqrt-unprod 79.3%

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

      8. sqr-neg 79.3%

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

      9. sqrt-prod 31.6%

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

      10. add-sqr-sqrt 79.3%

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

   8. Applied egg-rr 79.3%

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

   10. Simplified 87.2%

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

   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))))) < -0.0

   1. Initial program 68.5%

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

      1. +-commutative 68.5%

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

      2. +-commutative 68.5%

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

      3. times-frac 48.6%

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

      4. fma-neg 48.6%

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

   3. Simplified 67.9%

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

   5. Taylor expanded in c0 around -inf 67.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) + 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right)} \]
   6. Step-by-step derivation

      1. fma-def 67.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(-1, c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right)} \]

      2. distribute-lft1-in 67.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}, 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      3. metadata-eval 67.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      4. associate-/r* 60.8%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \color{blue}{\frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      5. times-frac 60.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \color{blue}{\left(\frac{{D}^{2}}{c0} \cdot \frac{{M}^{2} \cdot \left(h \cdot w\right)}{{d}^{2}}\right)}\right) \]

      6. associate-*r* 60.6%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \left(\frac{{D}^{2}}{c0} \cdot \frac{\color{blue}{\left({M}^{2} \cdot h\right) \cdot w}}{{d}^{2}}\right)\right) \]

   7. Simplified 60.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \left(\frac{{D}^{2}}{c0} \cdot \frac{\left({M}^{2} \cdot h\right) \cdot w}{{d}^{2}}\right)\right)} \]

   8. Taylor expanded in c0 around 0 87.3%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. associate-/l* 87.2%

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

      2. associate-/r* 81.0%

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

   10. Simplified 81.0%

       \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2}}{\frac{\frac{{d}^{2}}{{M}^{2}}}{h}}} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\color{blue}{\sqrt{\frac{{d}^{2}}{{M}^{2}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}}{h}} \]

       2. sqrt-div 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{{M}^{2}}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       3. unpow2 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{{M}^{2}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       4. sqrt-prod 27.7%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{{M}^{2}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       5. add-sqr-sqrt 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{\color{blue}{d}}{\sqrt{{M}^{2}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       6. unpow2 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{\sqrt{\color{blue}{M \cdot M}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       7. sqrt-prod 34.4%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{\color{blue}{\sqrt{M} \cdot \sqrt{M}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       8. add-sqr-sqrt 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{\color{blue}{M}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       9. sqrt-div 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{{M}^{2}}}}}{h}} \]

       10. unpow2 80.9%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{{M}^{2}}}}{h}} \]

       11. sqrt-prod 27.7%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{{M}^{2}}}}{h}} \]

       12. add-sqr-sqrt 80.9%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{\color{blue}{d}}{\sqrt{{M}^{2}}}}{h}} \]

       13. unpow2 80.9%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{d}{\sqrt{\color{blue}{M \cdot M}}}}{h}} \]

       14. sqrt-prod 40.7%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{d}{\color{blue}{\sqrt{M} \cdot \sqrt{M}}}}{h}} \]

       15. add-sqr-sqrt 87.4%

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

   12. Applied egg-rr 87.4%

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

   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 73.8%

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

      1. +-commutative 73.8%

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

      2. +-commutative 73.8%

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

      3. times-frac 71.2%

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

      4. fma-neg 71.2%

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

   3. Simplified 71.2%

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

      1. associate-/r* 71.3%

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

      2. frac-times 71.2%

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

      3. *-commutative 71.2%

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

      4. associate-*r/ 71.2%

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

      5. pow2 71.2%

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

   6. Applied egg-rr 71.2%

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

   8. Applied egg-rr 79.9%

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

      1. *-commutative 79.9%

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

      2. associate-/r* 79.9%

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

      3. associate-/r* 82.0%

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

   10. Simplified 82.0%

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

       1. associate-*l/ 82.1%

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

   12. Applied egg-rr 82.1%

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

   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%

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

      1. +-commutative 0.0%

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

      2. +-commutative 0.0%

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

      3. times-frac 0.0%

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

      4. fma-neg 0.0%

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

   3. Simplified 3.9%

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

   5. Taylor expanded in c0 around -inf 1.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) + 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right)} \]
   6. Step-by-step derivation

      1. fma-def 1.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(-1, c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right)} \]

      2. distribute-lft1-in 1.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}, 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      3. metadata-eval 1.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      4. associate-/r* 1.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \color{blue}{\frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      5. times-frac 2.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \color{blue}{\left(\frac{{D}^{2}}{c0} \cdot \frac{{M}^{2} \cdot \left(h \cdot w\right)}{{d}^{2}}\right)}\right) \]

      6. associate-*r* 2.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \left(\frac{{D}^{2}}{c0} \cdot \frac{\color{blue}{\left({M}^{2} \cdot h\right) \cdot w}}{{d}^{2}}\right)\right) \]

   7. Simplified 2.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \left(\frac{{D}^{2}}{c0} \cdot \frac{\left({M}^{2} \cdot h\right) \cdot w}{{d}^{2}}\right)\right)} \]

   8. Taylor expanded in c0 around 0 44.8%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. associate-/l* 44.2%

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

      2. associate-/r* 44.8%

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

   10. Simplified 44.8%

       \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2}}{\frac{\frac{{d}^{2}}{{M}^{2}}}{h}}} \]
   11. Step-by-step derivation

       1. associate-/r/ 45.5%

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

       2. add-sqr-sqrt 45.5%

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

       3. pow2 45.5%

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

       4. sqrt-div 45.5%

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

       5. unpow2 45.5%

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

       6. sqrt-prod 26.7%

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

       7. add-sqr-sqrt 53.7%

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

       8. unpow2 53.7%

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

       9. sqrt-prod 31.5%

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

       10. add-sqr-sqrt 59.7%

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

   12. Applied egg-rr 59.7%

       \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{{\left(\frac{d}{M}\right)}^{2}} \cdot h\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.2%

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

Alternative 4: 56.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{c0}{w}}{h}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\\ t_3 := t\_1 \cdot \left(t\_2 + \sqrt{t\_2 \cdot t\_2 - M \cdot M}\right)\\ t_4 := {\left(\frac{d}{D}\right)}^{2}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(t\_0, \frac{d}{D} \cdot \frac{d}{D}, \mathsf{fma}\left(\frac{c0}{w \cdot h}, t\_4, M\right)\right)\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{d}{M}}{h}}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(t\_0, t\_4, M\right)}, \frac{d}{D} \cdot \frac{1}{\sqrt{\frac{h}{\frac{c0}{w}}}}, \frac{t\_4 \cdot \frac{c0}{w}}{h}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(h \cdot \frac{{D}^{2}}{{\left(\frac{d}{M}\right)}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (/ c0 w) h))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (/ (* c0 (* d d)) (* (* D D) (* w h))))
        (t_3 (* t_1 (+ t_2 (sqrt (- (* t_2 t_2) (* M M))))))
        (t_4 (pow (/ d D) 2.0)))
   (if (<= t_3 (- INFINITY))
     (*
      (/ (/ c0 w) 2.0)
      (fma t_0 (* (/ d D) (/ d D)) (fma (/ c0 (* w h)) t_4 M)))
     (if (<= t_3 0.0)
       (* 0.25 (/ (pow D 2.0) (/ (* (/ d M) (/ d M)) h)))
       (if (<= t_3 INFINITY)
         (*
          t_1
          (fma
           (sqrt (fma t_0 t_4 M))
           (* (/ d D) (/ 1.0 (sqrt (/ h (/ c0 w)))))
           (/ (* t_4 (/ c0 w)) h)))
         (* 0.25 (* h (/ (pow D 2.0) (pow (/ d M) 2.0)))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 / w) / h;
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((D * D) * (w * h));
	double t_3 = t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))));
	double t_4 = pow((d / D), 2.0);
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = ((c0 / w) / 2.0) * fma(t_0, ((d / D) * (d / D)), fma((c0 / (w * h)), t_4, M));
	} else if (t_3 <= 0.0) {
		tmp = 0.25 * (pow(D, 2.0) / (((d / M) * (d / M)) / h));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_1 * fma(sqrt(fma(t_0, t_4, M)), ((d / D) * (1.0 / sqrt((h / (c0 / w))))), ((t_4 * (c0 / w)) / h));
	} else {
		tmp = 0.25 * (h * (pow(D, 2.0) / pow((d / M), 2.0)));
	}
	return tmp;
}

Julia

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(N[(c0 / w), $MachinePrecision] / 2.0), $MachinePrecision] * N[(t$95$0 * N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] + N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] * t$95$4 + M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(0.25 * N[(N[Power[D, 2.0], $MachinePrecision] / N[(N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(t$95$1 * N[(N[Sqrt[N[(t$95$0 * t$95$4 + M), $MachinePrecision]], $MachinePrecision] * N[(N[(d / D), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(h / N[(c0 / w), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$4 * N[(c0 / w), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(h * N[(N[Power[D, 2.0], $MachinePrecision] / N[Power[N[(d / M), $MachinePrecision], 2.0], $MachinePrecision]), $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))))) < -inf.0

   1. Initial program 74.9%

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

   3. Simplified 74.5%

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

      1. times-frac 74.4%

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

   6. Applied egg-rr 74.4%

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

      1. sqrt-prod 79.2%

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

      2. pow2 79.2%

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

      3. associate-/l/ 79.2%

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

      4. frac-times 79.3%

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

      5. unpow2 79.3%

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

      6. add-sqr-sqrt 47.7%

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

      7. sqrt-unprod 79.3%

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

      8. sqr-neg 79.3%

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

      9. sqrt-prod 31.6%

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

      10. add-sqr-sqrt 79.3%

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

   8. Applied egg-rr 79.3%

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

   10. Simplified 87.2%

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

   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))))) < -0.0

   1. Initial program 68.5%

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

      1. +-commutative 68.5%

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

      2. +-commutative 68.5%

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

      3. times-frac 48.6%

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

      4. fma-neg 48.6%

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

   3. Simplified 67.9%

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

   5. Taylor expanded in c0 around -inf 67.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) + 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right)} \]
   6. Step-by-step derivation

      1. fma-def 67.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(-1, c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right)} \]

      2. distribute-lft1-in 67.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}, 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      3. metadata-eval 67.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      4. associate-/r* 60.8%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \color{blue}{\frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      5. times-frac 60.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \color{blue}{\left(\frac{{D}^{2}}{c0} \cdot \frac{{M}^{2} \cdot \left(h \cdot w\right)}{{d}^{2}}\right)}\right) \]

      6. associate-*r* 60.6%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \left(\frac{{D}^{2}}{c0} \cdot \frac{\color{blue}{\left({M}^{2} \cdot h\right) \cdot w}}{{d}^{2}}\right)\right) \]

   7. Simplified 60.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \left(\frac{{D}^{2}}{c0} \cdot \frac{\left({M}^{2} \cdot h\right) \cdot w}{{d}^{2}}\right)\right)} \]

   8. Taylor expanded in c0 around 0 87.3%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. associate-/l* 87.2%

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

      2. associate-/r* 81.0%

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

   10. Simplified 81.0%

       \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2}}{\frac{\frac{{d}^{2}}{{M}^{2}}}{h}}} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\color{blue}{\sqrt{\frac{{d}^{2}}{{M}^{2}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}}{h}} \]

       2. sqrt-div 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{{M}^{2}}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       3. unpow2 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{{M}^{2}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       4. sqrt-prod 27.7%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{{M}^{2}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       5. add-sqr-sqrt 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{\color{blue}{d}}{\sqrt{{M}^{2}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       6. unpow2 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{\sqrt{\color{blue}{M \cdot M}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       7. sqrt-prod 34.4%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{\color{blue}{\sqrt{M} \cdot \sqrt{M}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       8. add-sqr-sqrt 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{\color{blue}{M}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       9. sqrt-div 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{{M}^{2}}}}}{h}} \]

       10. unpow2 80.9%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{{M}^{2}}}}{h}} \]

       11. sqrt-prod 27.7%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{{M}^{2}}}}{h}} \]

       12. add-sqr-sqrt 80.9%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{\color{blue}{d}}{\sqrt{{M}^{2}}}}{h}} \]

       13. unpow2 80.9%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{d}{\sqrt{\color{blue}{M \cdot M}}}}{h}} \]

       14. sqrt-prod 40.7%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{d}{\color{blue}{\sqrt{M} \cdot \sqrt{M}}}}{h}} \]

       15. add-sqr-sqrt 87.4%

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

   12. Applied egg-rr 87.4%

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

   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 73.8%

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

      1. +-commutative 73.8%

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

      2. +-commutative 73.8%

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

      3. times-frac 71.2%

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

      4. fma-neg 71.2%

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

   3. Simplified 71.2%

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

   6. Applied egg-rr 79.8%

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

      1. associate-/r* 79.9%

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

      2. associate-/r* 79.9%

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

      3. associate-/r* 82.0%

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

   8. Simplified 82.0%

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

      1. associate-*l/ 82.1%

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

   10. Applied egg-rr 82.0%

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

   11. Taylor expanded in d around inf 40.0%

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

       1. clear-num 40.0%

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

       2. sqrt-div 39.8%

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

       3. metadata-eval 39.8%

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

   13. Applied egg-rr 39.8%

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

       1. associate-/l* 40.0%

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

   15. Simplified 40.0%

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

   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%

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

      1. +-commutative 0.0%

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

      2. +-commutative 0.0%

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

      3. times-frac 0.0%

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

      4. fma-neg 0.0%

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

   3. Simplified 3.9%

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

   5. Taylor expanded in c0 around -inf 1.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) + 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right)} \]
   6. Step-by-step derivation

      1. fma-def 1.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(-1, c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right)} \]

      2. distribute-lft1-in 1.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}, 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      3. metadata-eval 1.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      4. associate-/r* 1.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \color{blue}{\frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      5. times-frac 2.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \color{blue}{\left(\frac{{D}^{2}}{c0} \cdot \frac{{M}^{2} \cdot \left(h \cdot w\right)}{{d}^{2}}\right)}\right) \]

      6. associate-*r* 2.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \left(\frac{{D}^{2}}{c0} \cdot \frac{\color{blue}{\left({M}^{2} \cdot h\right) \cdot w}}{{d}^{2}}\right)\right) \]

   7. Simplified 2.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \left(\frac{{D}^{2}}{c0} \cdot \frac{\left({M}^{2} \cdot h\right) \cdot w}{{d}^{2}}\right)\right)} \]

   8. Taylor expanded in c0 around 0 44.8%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. associate-/l* 44.2%

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

      2. associate-/r* 44.8%

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

   10. Simplified 44.8%

       \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2}}{\frac{\frac{{d}^{2}}{{M}^{2}}}{h}}} \]
   11. Step-by-step derivation

       1. associate-/r/ 45.5%

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

       2. add-sqr-sqrt 45.5%

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

       3. pow2 45.5%

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

       4. sqrt-div 45.5%

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

       5. unpow2 45.5%

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

       6. sqrt-prod 26.7%

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

       7. add-sqr-sqrt 53.7%

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

       8. unpow2 53.7%

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

       9. sqrt-prod 31.5%

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

       10. add-sqr-sqrt 59.7%

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

   12. Applied egg-rr 59.7%

       \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{{\left(\frac{d}{M}\right)}^{2}} \cdot h\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 62.0%

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

Alternative 5: 62.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{w \cdot h}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := {\left(\frac{d}{D}\right)}^{2}\\ t_3 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\\ t_4 := t\_1 \cdot \left(t\_3 + \sqrt{t\_3 \cdot t\_3 - M \cdot M}\right)\\ t_5 := \frac{d}{D} \cdot \sqrt{t\_0}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \mathsf{fma}\left(t\_0, t\_2, M\right)\right)\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{d}{M}}{h}}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(0.5 \cdot \left(\frac{D \cdot M}{d} \cdot \sqrt{\frac{w \cdot h}{c0}}\right) + t\_5, t\_5, \frac{t\_2 \cdot \frac{c0}{w}}{h}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(h \cdot \frac{{D}^{2}}{{\left(\frac{d}{M}\right)}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* w h)))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (pow (/ d D) 2.0))
        (t_3 (/ (* c0 (* d d)) (* (* D D) (* w h))))
        (t_4 (* t_1 (+ t_3 (sqrt (- (* t_3 t_3) (* M M))))))
        (t_5 (* (/ d D) (sqrt t_0))))
   (if (<= t_4 (- INFINITY))
     (*
      (/ (/ c0 w) 2.0)
      (fma (/ (/ c0 w) h) (* (/ d D) (/ d D)) (fma t_0 t_2 M)))
     (if (<= t_4 0.0)
       (* 0.25 (/ (pow D 2.0) (/ (* (/ d M) (/ d M)) h)))
       (if (<= t_4 INFINITY)
         (*
          t_1
          (fma
           (+ (* 0.5 (* (/ (* D M) d) (sqrt (/ (* w h) c0)))) t_5)
           t_5
           (/ (* t_2 (/ c0 w)) h)))
         (* 0.25 (* h (/ (pow D 2.0) (pow (/ d M) 2.0)))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (w * h);
	double t_1 = c0 / (2.0 * w);
	double t_2 = pow((d / D), 2.0);
	double t_3 = (c0 * (d * d)) / ((D * D) * (w * h));
	double t_4 = t_1 * (t_3 + sqrt(((t_3 * t_3) - (M * M))));
	double t_5 = (d / D) * sqrt(t_0);
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = ((c0 / w) / 2.0) * fma(((c0 / w) / h), ((d / D) * (d / D)), fma(t_0, t_2, M));
	} else if (t_4 <= 0.0) {
		tmp = 0.25 * (pow(D, 2.0) / (((d / M) * (d / M)) / h));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_1 * fma(((0.5 * (((D * M) / d) * sqrt(((w * h) / c0)))) + t_5), t_5, ((t_2 * (c0 / w)) / h));
	} else {
		tmp = 0.25 * (h * (pow(D, 2.0) / pow((d / M), 2.0)));
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(w * h))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(d / D) ^ 2.0
	t_3 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(D * D) * Float64(w * h)))
	t_4 = Float64(t_1 * Float64(t_3 + sqrt(Float64(Float64(t_3 * t_3) - Float64(M * M)))))
	t_5 = Float64(Float64(d / D) * sqrt(t_0))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(c0 / w) / 2.0) * fma(Float64(Float64(c0 / w) / h), Float64(Float64(d / D) * Float64(d / D)), fma(t_0, t_2, M)));
	elseif (t_4 <= 0.0)
		tmp = Float64(0.25 * Float64((D ^ 2.0) / Float64(Float64(Float64(d / M) * Float64(d / M)) / h)));
	elseif (t_4 <= Inf)
		tmp = Float64(t_1 * fma(Float64(Float64(0.5 * Float64(Float64(Float64(D * M) / d) * sqrt(Float64(Float64(w * h) / c0)))) + t_5), t_5, Float64(Float64(t_2 * Float64(c0 / w)) / h)));
	else
		tmp = Float64(0.25 * Float64(h * Float64((D ^ 2.0) / (Float64(d / M) ^ 2.0))));
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 * N[(t$95$3 + N[Sqrt[N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(d / D), $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(N[(N[(c0 / w), $MachinePrecision] / 2.0), $MachinePrecision] * N[(N[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision] * N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * t$95$2 + M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.0], N[(0.25 * N[(N[Power[D, 2.0], $MachinePrecision] / N[(N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(t$95$1 * N[(N[(N[(0.5 * N[(N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision] * N[Sqrt[N[(N[(w * h), $MachinePrecision] / c0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision] * t$95$5 + N[(N[(t$95$2 * N[(c0 / w), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(h * N[(N[Power[D, 2.0], $MachinePrecision] / N[Power[N[(d / M), $MachinePrecision], 2.0], $MachinePrecision]), $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))))) < -inf.0

   1. Initial program 74.9%

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

   3. Simplified 74.5%

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

      1. times-frac 74.4%

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

   6. Applied egg-rr 74.4%

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

      1. sqrt-prod 79.2%

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

      2. pow2 79.2%

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

      3. associate-/l/ 79.2%

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

      4. frac-times 79.3%

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

      5. unpow2 79.3%

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

      6. add-sqr-sqrt 47.7%

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

      7. sqrt-unprod 79.3%

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

      8. sqr-neg 79.3%

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

      9. sqrt-prod 31.6%

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

      10. add-sqr-sqrt 79.3%

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

   8. Applied egg-rr 79.3%

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

   10. Simplified 87.2%

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

   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))))) < -0.0

   1. Initial program 68.5%

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

      1. +-commutative 68.5%

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

      2. +-commutative 68.5%

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

      3. times-frac 48.6%

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

      4. fma-neg 48.6%

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

   3. Simplified 67.9%

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

   5. Taylor expanded in c0 around -inf 67.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) + 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right)} \]
   6. Step-by-step derivation

      1. fma-def 67.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(-1, c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right)} \]

      2. distribute-lft1-in 67.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}, 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      3. metadata-eval 67.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      4. associate-/r* 60.8%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \color{blue}{\frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      5. times-frac 60.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \color{blue}{\left(\frac{{D}^{2}}{c0} \cdot \frac{{M}^{2} \cdot \left(h \cdot w\right)}{{d}^{2}}\right)}\right) \]

      6. associate-*r* 60.6%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \left(\frac{{D}^{2}}{c0} \cdot \frac{\color{blue}{\left({M}^{2} \cdot h\right) \cdot w}}{{d}^{2}}\right)\right) \]

   7. Simplified 60.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \left(\frac{{D}^{2}}{c0} \cdot \frac{\left({M}^{2} \cdot h\right) \cdot w}{{d}^{2}}\right)\right)} \]

   8. Taylor expanded in c0 around 0 87.3%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. associate-/l* 87.2%

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

      2. associate-/r* 81.0%

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

   10. Simplified 81.0%

       \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2}}{\frac{\frac{{d}^{2}}{{M}^{2}}}{h}}} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\color{blue}{\sqrt{\frac{{d}^{2}}{{M}^{2}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}}{h}} \]

       2. sqrt-div 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{{M}^{2}}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       3. unpow2 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{{M}^{2}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       4. sqrt-prod 27.7%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{{M}^{2}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       5. add-sqr-sqrt 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{\color{blue}{d}}{\sqrt{{M}^{2}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       6. unpow2 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{\sqrt{\color{blue}{M \cdot M}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       7. sqrt-prod 34.4%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{\color{blue}{\sqrt{M} \cdot \sqrt{M}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       8. add-sqr-sqrt 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{\color{blue}{M}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       9. sqrt-div 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{{M}^{2}}}}}{h}} \]

       10. unpow2 80.9%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{{M}^{2}}}}{h}} \]

       11. sqrt-prod 27.7%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{{M}^{2}}}}{h}} \]

       12. add-sqr-sqrt 80.9%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{\color{blue}{d}}{\sqrt{{M}^{2}}}}{h}} \]

       13. unpow2 80.9%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{d}{\sqrt{\color{blue}{M \cdot M}}}}{h}} \]

       14. sqrt-prod 40.7%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{d}{\color{blue}{\sqrt{M} \cdot \sqrt{M}}}}{h}} \]

       15. add-sqr-sqrt 87.4%

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

   12. Applied egg-rr 87.4%

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

   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 73.8%

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

      1. +-commutative 73.8%

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

      2. +-commutative 73.8%

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

      3. times-frac 71.2%

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

      4. fma-neg 71.2%

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

   3. Simplified 71.2%

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

   6. Applied egg-rr 79.8%

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

      1. associate-/r* 79.9%

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

      2. associate-/r* 79.9%

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

      3. associate-/r* 82.0%

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

   8. Simplified 82.0%

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

      1. associate-*l/ 82.1%

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

   10. Applied egg-rr 82.0%

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

   11. Taylor expanded in d around inf 40.0%

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

   12. Taylor expanded in d around inf 80.1%

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

   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%

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

      1. +-commutative 0.0%

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

      2. +-commutative 0.0%

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

      3. times-frac 0.0%

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

      4. fma-neg 0.0%

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

   3. Simplified 3.9%

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

   5. Taylor expanded in c0 around -inf 1.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) + 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right)} \]
   6. Step-by-step derivation

      1. fma-def 1.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(-1, c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right)} \]

      2. distribute-lft1-in 1.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}, 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      3. metadata-eval 1.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      4. associate-/r* 1.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \color{blue}{\frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      5. times-frac 2.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \color{blue}{\left(\frac{{D}^{2}}{c0} \cdot \frac{{M}^{2} \cdot \left(h \cdot w\right)}{{d}^{2}}\right)}\right) \]

      6. associate-*r* 2.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \left(\frac{{D}^{2}}{c0} \cdot \frac{\color{blue}{\left({M}^{2} \cdot h\right) \cdot w}}{{d}^{2}}\right)\right) \]

   7. Simplified 2.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \left(\frac{{D}^{2}}{c0} \cdot \frac{\left({M}^{2} \cdot h\right) \cdot w}{{d}^{2}}\right)\right)} \]

   8. Taylor expanded in c0 around 0 44.8%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. associate-/l* 44.2%

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

      2. associate-/r* 44.8%

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

   10. Simplified 44.8%

       \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2}}{\frac{\frac{{d}^{2}}{{M}^{2}}}{h}}} \]
   11. Step-by-step derivation

       1. associate-/r/ 45.5%

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

       2. add-sqr-sqrt 45.5%

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

       3. pow2 45.5%

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

       4. sqrt-div 45.5%

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

       5. unpow2 45.5%

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

       6. sqrt-prod 26.7%

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

       7. add-sqr-sqrt 53.7%

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

       8. unpow2 53.7%

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

       9. sqrt-prod 31.5%

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

       10. add-sqr-sqrt 59.7%

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

   12. Applied egg-rr 59.7%

       \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{{\left(\frac{d}{M}\right)}^{2}} \cdot h\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq -\infty:\\ \;\;\;\;\frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \mathsf{fma}\left(\frac{c0}{w \cdot h}, {\left(\frac{d}{D}\right)}^{2}, M\right)\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq 0:\\ \;\;\;\;0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{d}{M}}{h}}\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(0.5 \cdot \left(\frac{D \cdot M}{d} \cdot \sqrt{\frac{w \cdot h}{c0}}\right) + \frac{d}{D} \cdot \sqrt{\frac{c0}{w \cdot h}}, \frac{d}{D} \cdot \sqrt{\frac{c0}{w \cdot h}}, \frac{{\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w}}{h}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(h \cdot \frac{{D}^{2}}{{\left(\frac{d}{M}\right)}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{w \cdot h}\\ t_1 := {\left(\frac{d}{D}\right)}^{2}\\ t_2 := t\_0 \cdot t\_1\\ t_3 := \frac{c0}{2 \cdot w}\\ t_4 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\\ t_5 := t\_3 \cdot \left(t\_4 + \sqrt{t\_4 \cdot t\_4 - M \cdot M}\right)\\ \mathbf{if}\;t\_5 \leq -\infty:\\ \;\;\;\;t\_3 \cdot \left(t\_2 + t\_0 \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\\ \mathbf{elif}\;t\_5 \leq 0:\\ \;\;\;\;0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{d}{M}}{h}}\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;t\_3 \cdot \left(t\_1 \cdot \frac{\frac{c0}{w}}{h} + t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(h \cdot \frac{{D}^{2}}{{\left(\frac{d}{M}\right)}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* w h)))
        (t_1 (pow (/ d D) 2.0))
        (t_2 (* t_0 t_1))
        (t_3 (/ c0 (* 2.0 w)))
        (t_4 (/ (* c0 (* d d)) (* (* D D) (* w h))))
        (t_5 (* t_3 (+ t_4 (sqrt (- (* t_4 t_4) (* M M)))))))
   (if (<= t_5 (- INFINITY))
     (* t_3 (+ t_2 (* t_0 (* (/ d D) (/ d D)))))
     (if (<= t_5 0.0)
       (* 0.25 (/ (pow D 2.0) (/ (* (/ d M) (/ d M)) h)))
       (if (<= t_5 INFINITY)
         (* t_3 (+ (* t_1 (/ (/ c0 w) h)) t_2))
         (* 0.25 (* h (/ (pow D 2.0) (pow (/ d M) 2.0)))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (w * h);
	double t_1 = pow((d / D), 2.0);
	double t_2 = t_0 * t_1;
	double t_3 = c0 / (2.0 * w);
	double t_4 = (c0 * (d * d)) / ((D * D) * (w * h));
	double t_5 = t_3 * (t_4 + sqrt(((t_4 * t_4) - (M * M))));
	double tmp;
	if (t_5 <= -((double) INFINITY)) {
		tmp = t_3 * (t_2 + (t_0 * ((d / D) * (d / D))));
	} else if (t_5 <= 0.0) {
		tmp = 0.25 * (pow(D, 2.0) / (((d / M) * (d / M)) / h));
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = t_3 * ((t_1 * ((c0 / w) / h)) + t_2);
	} else {
		tmp = 0.25 * (h * (pow(D, 2.0) / pow((d / M), 2.0)));
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (w * h);
	double t_1 = Math.pow((d / D), 2.0);
	double t_2 = t_0 * t_1;
	double t_3 = c0 / (2.0 * w);
	double t_4 = (c0 * (d * d)) / ((D * D) * (w * h));
	double t_5 = t_3 * (t_4 + Math.sqrt(((t_4 * t_4) - (M * M))));
	double tmp;
	if (t_5 <= -Double.POSITIVE_INFINITY) {
		tmp = t_3 * (t_2 + (t_0 * ((d / D) * (d / D))));
	} else if (t_5 <= 0.0) {
		tmp = 0.25 * (Math.pow(D, 2.0) / (((d / M) * (d / M)) / h));
	} else if (t_5 <= Double.POSITIVE_INFINITY) {
		tmp = t_3 * ((t_1 * ((c0 / w) / h)) + t_2);
	} else {
		tmp = 0.25 * (h * (Math.pow(D, 2.0) / Math.pow((d / M), 2.0)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (w * h)
	t_1 = math.pow((d / D), 2.0)
	t_2 = t_0 * t_1
	t_3 = c0 / (2.0 * w)
	t_4 = (c0 * (d * d)) / ((D * D) * (w * h))
	t_5 = t_3 * (t_4 + math.sqrt(((t_4 * t_4) - (M * M))))
	tmp = 0
	if t_5 <= -math.inf:
		tmp = t_3 * (t_2 + (t_0 * ((d / D) * (d / D))))
	elif t_5 <= 0.0:
		tmp = 0.25 * (math.pow(D, 2.0) / (((d / M) * (d / M)) / h))
	elif t_5 <= math.inf:
		tmp = t_3 * ((t_1 * ((c0 / w) / h)) + t_2)
	else:
		tmp = 0.25 * (h * (math.pow(D, 2.0) / math.pow((d / M), 2.0)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(w * h))
	t_1 = Float64(d / D) ^ 2.0
	t_2 = Float64(t_0 * t_1)
	t_3 = Float64(c0 / Float64(2.0 * w))
	t_4 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(D * D) * Float64(w * h)))
	t_5 = Float64(t_3 * Float64(t_4 + sqrt(Float64(Float64(t_4 * t_4) - Float64(M * M)))))
	tmp = 0.0
	if (t_5 <= Float64(-Inf))
		tmp = Float64(t_3 * Float64(t_2 + Float64(t_0 * Float64(Float64(d / D) * Float64(d / D)))));
	elseif (t_5 <= 0.0)
		tmp = Float64(0.25 * Float64((D ^ 2.0) / Float64(Float64(Float64(d / M) * Float64(d / M)) / h)));
	elseif (t_5 <= Inf)
		tmp = Float64(t_3 * Float64(Float64(t_1 * Float64(Float64(c0 / w) / h)) + t_2));
	else
		tmp = Float64(0.25 * Float64(h * Float64((D ^ 2.0) / (Float64(d / M) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (w * h);
	t_1 = (d / D) ^ 2.0;
	t_2 = t_0 * t_1;
	t_3 = c0 / (2.0 * w);
	t_4 = (c0 * (d * d)) / ((D * D) * (w * h));
	t_5 = t_3 * (t_4 + sqrt(((t_4 * t_4) - (M * M))));
	tmp = 0.0;
	if (t_5 <= -Inf)
		tmp = t_3 * (t_2 + (t_0 * ((d / D) * (d / D))));
	elseif (t_5 <= 0.0)
		tmp = 0.25 * ((D ^ 2.0) / (((d / M) * (d / M)) / h));
	elseif (t_5 <= Inf)
		tmp = t_3 * ((t_1 * ((c0 / w) / h)) + t_2);
	else
		tmp = 0.25 * (h * ((D ^ 2.0) / ((d / M) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$3 * N[(t$95$4 + N[Sqrt[N[(N[(t$95$4 * t$95$4), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, (-Infinity)], N[(t$95$3 * N[(t$95$2 + N[(t$95$0 * N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 0.0], N[(0.25 * N[(N[Power[D, 2.0], $MachinePrecision] / N[(N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, Infinity], N[(t$95$3 * N[(N[(t$95$1 * N[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(h * N[(N[Power[D, 2.0], $MachinePrecision] / N[Power[N[(d / M), $MachinePrecision], 2.0], $MachinePrecision]), $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))))) < -inf.0

   1. Initial program 74.9%

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

      1. +-commutative 74.9%

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

      2. +-commutative 74.9%

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

      3. times-frac 74.9%

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

      4. fma-neg 74.9%

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

   3. Simplified 74.9%

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

      1. frac-times 74.9%

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

   6. Applied egg-rr 74.9%

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

   7. Taylor expanded in c0 around inf 79.8%

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

      1. *-commutative 79.8%

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

      2. *-commutative 79.8%

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

      3. times-frac 79.8%

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

      4. *-commutative 79.8%

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

      5. unpow2 79.8%

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

      6. associate-/r* 79.8%

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

      7. unpow2 79.8%

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

      8. associate-*l/ 79.8%

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

      9. associate-*r/ 79.8%

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

      10. unpow2 79.8%

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

   9. Simplified 79.8%

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

       1. frac-times 74.9%

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

   11. Applied egg-rr 87.2%

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

   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))))) < -0.0

   1. Initial program 68.5%

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

      1. +-commutative 68.5%

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

      2. +-commutative 68.5%

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

      3. times-frac 48.6%

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

      4. fma-neg 48.6%

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

   3. Simplified 67.9%

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

   5. Taylor expanded in c0 around -inf 67.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) + 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right)} \]
   6. Step-by-step derivation

      1. fma-def 67.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(-1, c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right)} \]

      2. distribute-lft1-in 67.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}, 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      3. metadata-eval 67.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      4. associate-/r* 60.8%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \color{blue}{\frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      5. times-frac 60.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \color{blue}{\left(\frac{{D}^{2}}{c0} \cdot \frac{{M}^{2} \cdot \left(h \cdot w\right)}{{d}^{2}}\right)}\right) \]

      6. associate-*r* 60.6%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \left(\frac{{D}^{2}}{c0} \cdot \frac{\color{blue}{\left({M}^{2} \cdot h\right) \cdot w}}{{d}^{2}}\right)\right) \]

   7. Simplified 60.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \left(\frac{{D}^{2}}{c0} \cdot \frac{\left({M}^{2} \cdot h\right) \cdot w}{{d}^{2}}\right)\right)} \]

   8. Taylor expanded in c0 around 0 87.3%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. associate-/l* 87.2%

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

      2. associate-/r* 81.0%

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

   10. Simplified 81.0%

       \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2}}{\frac{\frac{{d}^{2}}{{M}^{2}}}{h}}} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\color{blue}{\sqrt{\frac{{d}^{2}}{{M}^{2}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}}{h}} \]

       2. sqrt-div 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{{M}^{2}}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       3. unpow2 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{{M}^{2}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       4. sqrt-prod 27.7%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{{M}^{2}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       5. add-sqr-sqrt 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{\color{blue}{d}}{\sqrt{{M}^{2}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       6. unpow2 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{\sqrt{\color{blue}{M \cdot M}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       7. sqrt-prod 34.4%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{\color{blue}{\sqrt{M} \cdot \sqrt{M}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       8. add-sqr-sqrt 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{\color{blue}{M}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       9. sqrt-div 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{{M}^{2}}}}}{h}} \]

       10. unpow2 80.9%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{{M}^{2}}}}{h}} \]

       11. sqrt-prod 27.7%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{{M}^{2}}}}{h}} \]

       12. add-sqr-sqrt 80.9%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{\color{blue}{d}}{\sqrt{{M}^{2}}}}{h}} \]

       13. unpow2 80.9%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{d}{\sqrt{\color{blue}{M \cdot M}}}}{h}} \]

       14. sqrt-prod 40.7%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{d}{\color{blue}{\sqrt{M} \cdot \sqrt{M}}}}{h}} \]

       15. add-sqr-sqrt 87.4%

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

   12. Applied egg-rr 87.4%

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

   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 73.8%

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

      1. +-commutative 73.8%

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

      2. +-commutative 73.8%

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

      3. times-frac 71.2%

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

      4. fma-neg 71.2%

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

   3. Simplified 71.2%

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

      1. frac-times 71.1%

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

   6. Applied egg-rr 71.1%

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

   7. Taylor expanded in c0 around inf 71.2%

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

      1. *-commutative 71.2%

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

      2. *-commutative 71.2%

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

      3. times-frac 71.2%

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

      4. *-commutative 71.2%

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

      5. unpow2 71.2%

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

      6. associate-/r* 71.2%

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

      7. unpow2 71.2%

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

      8. associate-*l/ 71.2%

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

      9. associate-*r/ 71.2%

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

      10. unpow2 71.2%

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

   9. Simplified 71.2%

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

       1. expm1-log1p-u 70.6%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\right)\right)} + \frac{c0}{h \cdot w} \cdot {\left(\frac{d}{D}\right)}^{2}\right) \]

       2. expm1-udef 70.6%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\right)} - 1\right)} + \frac{c0}{h \cdot w} \cdot {\left(\frac{d}{D}\right)}^{2}\right) \]

       3. frac-times 76.2%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(e^{\mathsf{log1p}\left(\frac{c0}{w \cdot h} \cdot \color{blue}{\left(\frac{d}{D} \cdot \frac{d}{D}\right)}\right)} - 1\right) + \frac{c0}{h \cdot w} \cdot {\left(\frac{d}{D}\right)}^{2}\right) \]

       4. pow2 76.2%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(e^{\mathsf{log1p}\left(\frac{c0}{w \cdot h} \cdot \color{blue}{{\left(\frac{d}{D}\right)}^{2}}\right)} - 1\right) + \frac{c0}{h \cdot w} \cdot {\left(\frac{d}{D}\right)}^{2}\right) \]

   11. Applied egg-rr 76.2%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\right)} - 1\right)} + \frac{c0}{h \cdot w} \cdot {\left(\frac{d}{D}\right)}^{2}\right) \]
   12. Step-by-step derivation

       1. expm1-def 79.0%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\right)\right)} + \frac{c0}{h \cdot w} \cdot {\left(\frac{d}{D}\right)}^{2}\right) \]

       2. expm1-log1p 79.9%

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

       3. associate-/r* 80.0%

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

   13. Simplified 80.0%

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

   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%

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

      1. +-commutative 0.0%

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

      2. +-commutative 0.0%

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

      3. times-frac 0.0%

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

      4. fma-neg 0.0%

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

   3. Simplified 3.9%

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

   5. Taylor expanded in c0 around -inf 1.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) + 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right)} \]
   6. Step-by-step derivation

      1. fma-def 1.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(-1, c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right)} \]

      2. distribute-lft1-in 1.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}, 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      3. metadata-eval 1.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      4. associate-/r* 1.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \color{blue}{\frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      5. times-frac 2.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \color{blue}{\left(\frac{{D}^{2}}{c0} \cdot \frac{{M}^{2} \cdot \left(h \cdot w\right)}{{d}^{2}}\right)}\right) \]

      6. associate-*r* 2.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \left(\frac{{D}^{2}}{c0} \cdot \frac{\color{blue}{\left({M}^{2} \cdot h\right) \cdot w}}{{d}^{2}}\right)\right) \]

   7. Simplified 2.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \left(\frac{{D}^{2}}{c0} \cdot \frac{\left({M}^{2} \cdot h\right) \cdot w}{{d}^{2}}\right)\right)} \]

   8. Taylor expanded in c0 around 0 44.8%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. associate-/l* 44.2%

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

      2. associate-/r* 44.8%

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

   10. Simplified 44.8%

       \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2}}{\frac{\frac{{d}^{2}}{{M}^{2}}}{h}}} \]
   11. Step-by-step derivation

       1. associate-/r/ 45.5%

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

       2. add-sqr-sqrt 45.5%

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

       3. pow2 45.5%

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

       4. sqrt-div 45.5%

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

       5. unpow2 45.5%

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

       6. sqrt-prod 26.7%

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

       7. add-sqr-sqrt 53.7%

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

       8. unpow2 53.7%

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

       9. sqrt-prod 31.5%

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

       10. add-sqr-sqrt 59.7%

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

   12. Applied egg-rr 59.7%

       \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{{\left(\frac{d}{M}\right)}^{2}} \cdot h\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 68.9%

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

Alternative 7: 62.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{c0}{w}}{h}\\ t_1 := \frac{c0}{w \cdot h}\\ t_2 := \frac{c0}{2 \cdot w}\\ t_3 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\\ t_4 := t\_2 \cdot \left(t\_3 + \sqrt{t\_3 \cdot t\_3 - M \cdot M}\right)\\ t_5 := {\left(\frac{d}{D}\right)}^{2}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(t\_0, \frac{d}{D} \cdot \frac{d}{D}, \mathsf{fma}\left(t\_1, t\_5, M\right)\right)\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{d}{M}}{h}}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_2 \cdot \left(t\_5 \cdot t\_0 + t\_1 \cdot t\_5\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(h \cdot \frac{{D}^{2}}{{\left(\frac{d}{M}\right)}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (/ c0 w) h))
        (t_1 (/ c0 (* w h)))
        (t_2 (/ c0 (* 2.0 w)))
        (t_3 (/ (* c0 (* d d)) (* (* D D) (* w h))))
        (t_4 (* t_2 (+ t_3 (sqrt (- (* t_3 t_3) (* M M))))))
        (t_5 (pow (/ d D) 2.0)))
   (if (<= t_4 (- INFINITY))
     (* (/ (/ c0 w) 2.0) (fma t_0 (* (/ d D) (/ d D)) (fma t_1 t_5 M)))
     (if (<= t_4 0.0)
       (* 0.25 (/ (pow D 2.0) (/ (* (/ d M) (/ d M)) h)))
       (if (<= t_4 INFINITY)
         (* t_2 (+ (* t_5 t_0) (* t_1 t_5)))
         (* 0.25 (* h (/ (pow D 2.0) (pow (/ d M) 2.0)))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 / w) / h;
	double t_1 = c0 / (w * h);
	double t_2 = c0 / (2.0 * w);
	double t_3 = (c0 * (d * d)) / ((D * D) * (w * h));
	double t_4 = t_2 * (t_3 + sqrt(((t_3 * t_3) - (M * M))));
	double t_5 = pow((d / D), 2.0);
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = ((c0 / w) / 2.0) * fma(t_0, ((d / D) * (d / D)), fma(t_1, t_5, M));
	} else if (t_4 <= 0.0) {
		tmp = 0.25 * (pow(D, 2.0) / (((d / M) * (d / M)) / h));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_2 * ((t_5 * t_0) + (t_1 * t_5));
	} else {
		tmp = 0.25 * (h * (pow(D, 2.0) / pow((d / M), 2.0)));
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 / w) / h)
	t_1 = Float64(c0 / Float64(w * h))
	t_2 = Float64(c0 / Float64(2.0 * w))
	t_3 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(D * D) * Float64(w * h)))
	t_4 = Float64(t_2 * Float64(t_3 + sqrt(Float64(Float64(t_3 * t_3) - Float64(M * M)))))
	t_5 = Float64(d / D) ^ 2.0
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(c0 / w) / 2.0) * fma(t_0, Float64(Float64(d / D) * Float64(d / D)), fma(t_1, t_5, M)));
	elseif (t_4 <= 0.0)
		tmp = Float64(0.25 * Float64((D ^ 2.0) / Float64(Float64(Float64(d / M) * Float64(d / M)) / h)));
	elseif (t_4 <= Inf)
		tmp = Float64(t_2 * Float64(Float64(t_5 * t_0) + Float64(t_1 * t_5)));
	else
		tmp = Float64(0.25 * Float64(h * Float64((D ^ 2.0) / (Float64(d / M) ^ 2.0))));
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / 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 * d), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(w * h), $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]}, Block[{t$95$5 = N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(N[(N[(c0 / w), $MachinePrecision] / 2.0), $MachinePrecision] * N[(t$95$0 * N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * t$95$5 + M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.0], N[(0.25 * N[(N[Power[D, 2.0], $MachinePrecision] / N[(N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(t$95$2 * N[(N[(t$95$5 * t$95$0), $MachinePrecision] + N[(t$95$1 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(h * N[(N[Power[D, 2.0], $MachinePrecision] / N[Power[N[(d / M), $MachinePrecision], 2.0], $MachinePrecision]), $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))))) < -inf.0

   1. Initial program 74.9%

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

   3. Simplified 74.5%

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

      1. times-frac 74.4%

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

   6. Applied egg-rr 74.4%

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

      1. sqrt-prod 79.2%

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

      2. pow2 79.2%

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

      3. associate-/l/ 79.2%

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

      4. frac-times 79.3%

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

      5. unpow2 79.3%

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

      6. add-sqr-sqrt 47.7%

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

      7. sqrt-unprod 79.3%

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

      8. sqr-neg 79.3%

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

      9. sqrt-prod 31.6%

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

      10. add-sqr-sqrt 79.3%

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

   8. Applied egg-rr 79.3%

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

   10. Simplified 87.2%

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

   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))))) < -0.0

   1. Initial program 68.5%

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

      1. +-commutative 68.5%

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

      2. +-commutative 68.5%

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

      3. times-frac 48.6%

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

      4. fma-neg 48.6%

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

   3. Simplified 67.9%

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

   5. Taylor expanded in c0 around -inf 67.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) + 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right)} \]
   6. Step-by-step derivation

      1. fma-def 67.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(-1, c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right)} \]

      2. distribute-lft1-in 67.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}, 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      3. metadata-eval 67.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      4. associate-/r* 60.8%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \color{blue}{\frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      5. times-frac 60.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \color{blue}{\left(\frac{{D}^{2}}{c0} \cdot \frac{{M}^{2} \cdot \left(h \cdot w\right)}{{d}^{2}}\right)}\right) \]

      6. associate-*r* 60.6%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \left(\frac{{D}^{2}}{c0} \cdot \frac{\color{blue}{\left({M}^{2} \cdot h\right) \cdot w}}{{d}^{2}}\right)\right) \]

   7. Simplified 60.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \left(\frac{{D}^{2}}{c0} \cdot \frac{\left({M}^{2} \cdot h\right) \cdot w}{{d}^{2}}\right)\right)} \]

   8. Taylor expanded in c0 around 0 87.3%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. associate-/l* 87.2%

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

      2. associate-/r* 81.0%

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

   10. Simplified 81.0%

       \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2}}{\frac{\frac{{d}^{2}}{{M}^{2}}}{h}}} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\color{blue}{\sqrt{\frac{{d}^{2}}{{M}^{2}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}}{h}} \]

       2. sqrt-div 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{{M}^{2}}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       3. unpow2 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{{M}^{2}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       4. sqrt-prod 27.7%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{{M}^{2}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       5. add-sqr-sqrt 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{\color{blue}{d}}{\sqrt{{M}^{2}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       6. unpow2 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{\sqrt{\color{blue}{M \cdot M}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       7. sqrt-prod 34.4%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{\color{blue}{\sqrt{M} \cdot \sqrt{M}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       8. add-sqr-sqrt 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{\color{blue}{M}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       9. sqrt-div 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{{M}^{2}}}}}{h}} \]

       10. unpow2 80.9%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{{M}^{2}}}}{h}} \]

       11. sqrt-prod 27.7%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{{M}^{2}}}}{h}} \]

       12. add-sqr-sqrt 80.9%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{\color{blue}{d}}{\sqrt{{M}^{2}}}}{h}} \]

       13. unpow2 80.9%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{d}{\sqrt{\color{blue}{M \cdot M}}}}{h}} \]

       14. sqrt-prod 40.7%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{d}{\color{blue}{\sqrt{M} \cdot \sqrt{M}}}}{h}} \]

       15. add-sqr-sqrt 87.4%

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

   12. Applied egg-rr 87.4%

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

   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 73.8%

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

      1. +-commutative 73.8%

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

      2. +-commutative 73.8%

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

      3. times-frac 71.2%

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

      4. fma-neg 71.2%

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

   3. Simplified 71.2%

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

      1. frac-times 71.1%

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

   6. Applied egg-rr 71.1%

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

   7. Taylor expanded in c0 around inf 71.2%

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

      1. *-commutative 71.2%

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

      2. *-commutative 71.2%

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

      3. times-frac 71.2%

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

      4. *-commutative 71.2%

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

      5. unpow2 71.2%

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

      6. associate-/r* 71.2%

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

      7. unpow2 71.2%

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

      8. associate-*l/ 71.2%

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

      9. associate-*r/ 71.2%

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

      10. unpow2 71.2%

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

   9. Simplified 71.2%

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

       1. expm1-log1p-u 70.6%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\right)\right)} + \frac{c0}{h \cdot w} \cdot {\left(\frac{d}{D}\right)}^{2}\right) \]

       2. expm1-udef 70.6%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\right)} - 1\right)} + \frac{c0}{h \cdot w} \cdot {\left(\frac{d}{D}\right)}^{2}\right) \]

       3. frac-times 76.2%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(e^{\mathsf{log1p}\left(\frac{c0}{w \cdot h} \cdot \color{blue}{\left(\frac{d}{D} \cdot \frac{d}{D}\right)}\right)} - 1\right) + \frac{c0}{h \cdot w} \cdot {\left(\frac{d}{D}\right)}^{2}\right) \]

       4. pow2 76.2%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(e^{\mathsf{log1p}\left(\frac{c0}{w \cdot h} \cdot \color{blue}{{\left(\frac{d}{D}\right)}^{2}}\right)} - 1\right) + \frac{c0}{h \cdot w} \cdot {\left(\frac{d}{D}\right)}^{2}\right) \]

   11. Applied egg-rr 76.2%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\right)} - 1\right)} + \frac{c0}{h \cdot w} \cdot {\left(\frac{d}{D}\right)}^{2}\right) \]
   12. Step-by-step derivation

       1. expm1-def 79.0%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\right)\right)} + \frac{c0}{h \cdot w} \cdot {\left(\frac{d}{D}\right)}^{2}\right) \]

       2. expm1-log1p 79.9%

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

       3. associate-/r* 80.0%

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

   13. Simplified 80.0%

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

   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%

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

      1. +-commutative 0.0%

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

      2. +-commutative 0.0%

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

      3. times-frac 0.0%

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

      4. fma-neg 0.0%

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

   3. Simplified 3.9%

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

   5. Taylor expanded in c0 around -inf 1.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) + 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right)} \]
   6. Step-by-step derivation

      1. fma-def 1.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(-1, c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right)} \]

      2. distribute-lft1-in 1.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}, 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      3. metadata-eval 1.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      4. associate-/r* 1.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \color{blue}{\frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      5. times-frac 2.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \color{blue}{\left(\frac{{D}^{2}}{c0} \cdot \frac{{M}^{2} \cdot \left(h \cdot w\right)}{{d}^{2}}\right)}\right) \]

      6. associate-*r* 2.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \left(\frac{{D}^{2}}{c0} \cdot \frac{\color{blue}{\left({M}^{2} \cdot h\right) \cdot w}}{{d}^{2}}\right)\right) \]

   7. Simplified 2.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \left(\frac{{D}^{2}}{c0} \cdot \frac{\left({M}^{2} \cdot h\right) \cdot w}{{d}^{2}}\right)\right)} \]

   8. Taylor expanded in c0 around 0 44.8%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. associate-/l* 44.2%

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

      2. associate-/r* 44.8%

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

   10. Simplified 44.8%

       \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2}}{\frac{\frac{{d}^{2}}{{M}^{2}}}{h}}} \]
   11. Step-by-step derivation

       1. associate-/r/ 45.5%

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

       2. add-sqr-sqrt 45.5%

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

       3. pow2 45.5%

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

       4. sqrt-div 45.5%

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

       5. unpow2 45.5%

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

       6. sqrt-prod 26.7%

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

       7. add-sqr-sqrt 53.7%

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

       8. unpow2 53.7%

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

       9. sqrt-prod 31.5%

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

       10. add-sqr-sqrt 59.7%

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

   12. Applied egg-rr 59.7%

       \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{{\left(\frac{d}{M}\right)}^{2}} \cdot h\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 68.9%

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

Alternative 8: 62.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{w \cdot h}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\\ t_3 := t\_1 \cdot \left(t\_2 + \sqrt{t\_2 \cdot t\_2 - M \cdot M}\right)\\ t_4 := t\_1 \cdot \left(t\_0 \cdot {\left(\frac{d}{D}\right)}^{2} + t\_0 \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{d}{M}}{h}}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(h \cdot \frac{{D}^{2}}{{\left(\frac{d}{M}\right)}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* w h)))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (/ (* c0 (* d d)) (* (* D D) (* w h))))
        (t_3 (* t_1 (+ t_2 (sqrt (- (* t_2 t_2) (* M M))))))
        (t_4
         (* t_1 (+ (* t_0 (pow (/ d D) 2.0)) (* t_0 (* (/ d D) (/ d D)))))))
   (if (<= t_3 (- INFINITY))
     t_4
     (if (<= t_3 0.0)
       (* 0.25 (/ (pow D 2.0) (/ (* (/ d M) (/ d M)) h)))
       (if (<= t_3 INFINITY)
         t_4
         (* 0.25 (* h (/ (pow D 2.0) (pow (/ d M) 2.0)))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (w * h);
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((D * D) * (w * h));
	double t_3 = t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))));
	double t_4 = t_1 * ((t_0 * pow((d / D), 2.0)) + (t_0 * ((d / D) * (d / D))));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_4;
	} else if (t_3 <= 0.0) {
		tmp = 0.25 * (pow(D, 2.0) / (((d / M) * (d / M)) / h));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = 0.25 * (h * (pow(D, 2.0) / pow((d / M), 2.0)));
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (w * h);
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((D * D) * (w * h));
	double t_3 = t_1 * (t_2 + Math.sqrt(((t_2 * t_2) - (M * M))));
	double t_4 = t_1 * ((t_0 * Math.pow((d / D), 2.0)) + (t_0 * ((d / D) * (d / D))));
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = t_4;
	} else if (t_3 <= 0.0) {
		tmp = 0.25 * (Math.pow(D, 2.0) / (((d / M) * (d / M)) / h));
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = t_4;
	} else {
		tmp = 0.25 * (h * (Math.pow(D, 2.0) / Math.pow((d / M), 2.0)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (w * h)
	t_1 = c0 / (2.0 * w)
	t_2 = (c0 * (d * d)) / ((D * D) * (w * h))
	t_3 = t_1 * (t_2 + math.sqrt(((t_2 * t_2) - (M * M))))
	t_4 = t_1 * ((t_0 * math.pow((d / D), 2.0)) + (t_0 * ((d / D) * (d / D))))
	tmp = 0
	if t_3 <= -math.inf:
		tmp = t_4
	elif t_3 <= 0.0:
		tmp = 0.25 * (math.pow(D, 2.0) / (((d / M) * (d / M)) / h))
	elif t_3 <= math.inf:
		tmp = t_4
	else:
		tmp = 0.25 * (h * (math.pow(D, 2.0) / math.pow((d / M), 2.0)))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (w * h);
	t_1 = c0 / (2.0 * w);
	t_2 = (c0 * (d * d)) / ((D * D) * (w * h));
	t_3 = t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))));
	t_4 = t_1 * ((t_0 * ((d / D) ^ 2.0)) + (t_0 * ((d / D) * (d / D))));
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = t_4;
	elseif (t_3 <= 0.0)
		tmp = 0.25 * ((D ^ 2.0) / (((d / M) * (d / M)) / h));
	elseif (t_3 <= Inf)
		tmp = t_4;
	else
		tmp = 0.25 * (h * ((D ^ 2.0) / ((d / M) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 * N[(N[(t$95$0 * N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$4, If[LessEqual[t$95$3, 0.0], N[(0.25 * N[(N[Power[D, 2.0], $MachinePrecision] / N[(N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$4, N[(0.25 * N[(h * N[(N[Power[D, 2.0], $MachinePrecision] / N[Power[N[(d / M), $MachinePrecision], 2.0], $MachinePrecision]), $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))))) < -inf.0 or -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 74.3%

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

      1. +-commutative 74.3%

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

      2. +-commutative 74.3%

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

      3. times-frac 72.9%

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

      4. fma-neg 72.9%

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

   3. Simplified 72.9%

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

      1. frac-times 72.9%

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

   6. Applied egg-rr 72.9%

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

   7. Taylor expanded in c0 around inf 75.2%

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

      1. *-commutative 75.2%

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

      2. *-commutative 75.2%

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

      3. times-frac 75.2%

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

      4. *-commutative 75.2%

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

      5. unpow2 75.2%

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

      6. associate-/r* 75.2%

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

      7. unpow2 75.2%

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

      8. associate-*l/ 75.2%

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

      9. associate-*r/ 75.2%

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

      10. unpow2 75.2%

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

   9. Simplified 75.2%

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

       1. frac-times 72.9%

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

   11. Applied egg-rr 83.3%

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

   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))))) < -0.0

   1. Initial program 68.5%

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

      1. +-commutative 68.5%

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

      2. +-commutative 68.5%

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

      3. times-frac 48.6%

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

      4. fma-neg 48.6%

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

   3. Simplified 67.9%

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

   5. Taylor expanded in c0 around -inf 67.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) + 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right)} \]
   6. Step-by-step derivation

      1. fma-def 67.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(-1, c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right)} \]

      2. distribute-lft1-in 67.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}, 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      3. metadata-eval 67.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      4. associate-/r* 60.8%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \color{blue}{\frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      5. times-frac 60.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \color{blue}{\left(\frac{{D}^{2}}{c0} \cdot \frac{{M}^{2} \cdot \left(h \cdot w\right)}{{d}^{2}}\right)}\right) \]

      6. associate-*r* 60.6%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \left(\frac{{D}^{2}}{c0} \cdot \frac{\color{blue}{\left({M}^{2} \cdot h\right) \cdot w}}{{d}^{2}}\right)\right) \]

   7. Simplified 60.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \left(\frac{{D}^{2}}{c0} \cdot \frac{\left({M}^{2} \cdot h\right) \cdot w}{{d}^{2}}\right)\right)} \]

   8. Taylor expanded in c0 around 0 87.3%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. associate-/l* 87.2%

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

      2. associate-/r* 81.0%

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

   10. Simplified 81.0%

       \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2}}{\frac{\frac{{d}^{2}}{{M}^{2}}}{h}}} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\color{blue}{\sqrt{\frac{{d}^{2}}{{M}^{2}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}}{h}} \]

       2. sqrt-div 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{{M}^{2}}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       3. unpow2 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{{M}^{2}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       4. sqrt-prod 27.7%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{{M}^{2}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       5. add-sqr-sqrt 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{\color{blue}{d}}{\sqrt{{M}^{2}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       6. unpow2 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{\sqrt{\color{blue}{M \cdot M}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       7. sqrt-prod 34.4%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{\color{blue}{\sqrt{M} \cdot \sqrt{M}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       8. add-sqr-sqrt 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{\color{blue}{M}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       9. sqrt-div 80.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{{M}^{2}}}}}{h}} \]

       10. unpow2 80.9%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{{M}^{2}}}}{h}} \]

       11. sqrt-prod 27.7%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{{M}^{2}}}}{h}} \]

       12. add-sqr-sqrt 80.9%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{\color{blue}{d}}{\sqrt{{M}^{2}}}}{h}} \]

       13. unpow2 80.9%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{d}{\sqrt{\color{blue}{M \cdot M}}}}{h}} \]

       14. sqrt-prod 40.7%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{d}{\color{blue}{\sqrt{M} \cdot \sqrt{M}}}}{h}} \]

       15. add-sqr-sqrt 87.4%

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

   12. Applied egg-rr 87.4%

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

   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%

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

      1. +-commutative 0.0%

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

      2. +-commutative 0.0%

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

      3. times-frac 0.0%

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

      4. fma-neg 0.0%

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

   3. Simplified 3.9%

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

   5. Taylor expanded in c0 around -inf 1.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) + 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right)} \]
   6. Step-by-step derivation

      1. fma-def 1.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(-1, c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right)} \]

      2. distribute-lft1-in 1.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}, 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      3. metadata-eval 1.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      4. associate-/r* 1.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \color{blue}{\frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      5. times-frac 2.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \color{blue}{\left(\frac{{D}^{2}}{c0} \cdot \frac{{M}^{2} \cdot \left(h \cdot w\right)}{{d}^{2}}\right)}\right) \]

      6. associate-*r* 2.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \left(\frac{{D}^{2}}{c0} \cdot \frac{\color{blue}{\left({M}^{2} \cdot h\right) \cdot w}}{{d}^{2}}\right)\right) \]

   7. Simplified 2.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \left(\frac{{D}^{2}}{c0} \cdot \frac{\left({M}^{2} \cdot h\right) \cdot w}{{d}^{2}}\right)\right)} \]

   8. Taylor expanded in c0 around 0 44.8%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. associate-/l* 44.2%

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

      2. associate-/r* 44.8%

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

   10. Simplified 44.8%

       \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2}}{\frac{\frac{{d}^{2}}{{M}^{2}}}{h}}} \]
   11. Step-by-step derivation

       1. associate-/r/ 45.5%

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

       2. add-sqr-sqrt 45.5%

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

       3. pow2 45.5%

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

       4. sqrt-div 45.5%

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

       5. unpow2 45.5%

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

       6. sqrt-prod 26.7%

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

       7. add-sqr-sqrt 53.7%

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

       8. unpow2 53.7%

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

       9. sqrt-prod 31.5%

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

       10. add-sqr-sqrt 59.7%

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

   12. Applied egg-rr 59.7%

       \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{{\left(\frac{d}{M}\right)}^{2}} \cdot h\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.8%

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

Alternative 9: 61.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\\ t_1 := \frac{c0}{w \cdot h}\\ t_2 := \frac{c0}{2 \cdot w}\\ t_3 := t\_2 \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right)\\ \mathbf{if}\;t\_3 \leq -\infty \lor \neg \left(t\_3 \leq 0\right) \land t\_3 \leq \infty:\\ \;\;\;\;t\_2 \cdot \left(t\_1 \cdot {\left(\frac{d}{D}\right)}^{2} + t\_1 \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{d}{M}}{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* D D) (* w h))))
        (t_1 (/ c0 (* w h)))
        (t_2 (/ c0 (* 2.0 w)))
        (t_3 (* t_2 (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))))
   (if (or (<= t_3 (- INFINITY)) (and (not (<= t_3 0.0)) (<= t_3 INFINITY)))
     (* t_2 (+ (* t_1 (pow (/ d D) 2.0)) (* t_1 (* (/ d D) (/ d D)))))
     (* 0.25 (/ (pow D 2.0) (/ (* (/ d M) (/ d M)) h))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((D * D) * (w * h));
	double t_1 = c0 / (w * h);
	double t_2 = c0 / (2.0 * w);
	double t_3 = t_2 * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	double tmp;
	if ((t_3 <= -((double) INFINITY)) || (!(t_3 <= 0.0) && (t_3 <= ((double) INFINITY)))) {
		tmp = t_2 * ((t_1 * pow((d / D), 2.0)) + (t_1 * ((d / D) * (d / D))));
	} else {
		tmp = 0.25 * (pow(D, 2.0) / (((d / M) * (d / M)) / h));
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((D * D) * (w * h));
	double t_1 = c0 / (w * h);
	double t_2 = c0 / (2.0 * w);
	double t_3 = t_2 * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
	double tmp;
	if ((t_3 <= -Double.POSITIVE_INFINITY) || (!(t_3 <= 0.0) && (t_3 <= Double.POSITIVE_INFINITY))) {
		tmp = t_2 * ((t_1 * Math.pow((d / D), 2.0)) + (t_1 * ((d / D) * (d / D))));
	} else {
		tmp = 0.25 * (Math.pow(D, 2.0) / (((d / M) * (d / M)) / h));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((D * D) * (w * h))
	t_1 = c0 / (w * h)
	t_2 = c0 / (2.0 * w)
	t_3 = t_2 * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))
	tmp = 0
	if (t_3 <= -math.inf) or (not (t_3 <= 0.0) and (t_3 <= math.inf)):
		tmp = t_2 * ((t_1 * math.pow((d / D), 2.0)) + (t_1 * ((d / D) * (d / D))))
	else:
		tmp = 0.25 * (math.pow(D, 2.0) / (((d / M) * (d / M)) / h))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(D * D) * Float64(w * h)))
	t_1 = Float64(c0 / Float64(w * h))
	t_2 = Float64(c0 / Float64(2.0 * w))
	t_3 = Float64(t_2 * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))))
	tmp = 0.0
	if ((t_3 <= Float64(-Inf)) || (!(t_3 <= 0.0) && (t_3 <= Inf)))
		tmp = Float64(t_2 * Float64(Float64(t_1 * (Float64(d / D) ^ 2.0)) + Float64(t_1 * Float64(Float64(d / D) * Float64(d / D)))));
	else
		tmp = Float64(0.25 * Float64((D ^ 2.0) / Float64(Float64(Float64(d / M) * Float64(d / M)) / h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((D * D) * (w * h));
	t_1 = c0 / (w * h);
	t_2 = c0 / (2.0 * w);
	t_3 = t_2 * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	tmp = 0.0;
	if ((t_3 <= -Inf) || (~((t_3 <= 0.0)) && (t_3 <= Inf)))
		tmp = t_2 * ((t_1 * ((d / D) ^ 2.0)) + (t_1 * ((d / D) * (d / D))));
	else
		tmp = 0.25 * ((D ^ 2.0) / (((d / M) * (d / M)) / h));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$3, (-Infinity)], And[N[Not[LessEqual[t$95$3, 0.0]], $MachinePrecision], LessEqual[t$95$3, Infinity]]], N[(t$95$2 * N[(N[(t$95$1 * N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[Power[D, 2.0], $MachinePrecision] / N[(N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision] / h), $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 or -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 74.3%

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

      1. +-commutative 74.3%

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

      2. +-commutative 74.3%

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

      3. times-frac 72.9%

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

      4. fma-neg 72.9%

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

   3. Simplified 72.9%

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

      1. frac-times 72.9%

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

   6. Applied egg-rr 72.9%

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

   7. Taylor expanded in c0 around inf 75.2%

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

      1. *-commutative 75.2%

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

      2. *-commutative 75.2%

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

      3. times-frac 75.2%

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

      4. *-commutative 75.2%

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

      5. unpow2 75.2%

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

      6. associate-/r* 75.2%

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

      7. unpow2 75.2%

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

      8. associate-*l/ 75.2%

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

      9. associate-*r/ 75.2%

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

      10. unpow2 75.2%

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

   9. Simplified 75.2%

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

       1. frac-times 72.9%

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

   11. Applied egg-rr 83.3%

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

   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))))) < -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 5.9%

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

      1. +-commutative 5.9%

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

      2. +-commutative 5.9%

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

      3. times-frac 4.2%

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

      4. fma-neg 4.2%

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

   3. Simplified 9.4%

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

   5. Taylor expanded in c0 around -inf 7.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) + 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right)} \]
   6. Step-by-step derivation

      1. fma-def 7.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(-1, c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right)} \]

      2. distribute-lft1-in 7.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}, 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      3. metadata-eval 7.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      4. associate-/r* 6.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \color{blue}{\frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      5. times-frac 7.2%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \color{blue}{\left(\frac{{D}^{2}}{c0} \cdot \frac{{M}^{2} \cdot \left(h \cdot w\right)}{{d}^{2}}\right)}\right) \]

      6. associate-*r* 7.2%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \left(\frac{{D}^{2}}{c0} \cdot \frac{\color{blue}{\left({M}^{2} \cdot h\right) \cdot w}}{{d}^{2}}\right)\right) \]

   7. Simplified 7.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \left(\frac{{D}^{2}}{c0} \cdot \frac{\left({M}^{2} \cdot h\right) \cdot w}{{d}^{2}}\right)\right)} \]

   8. Taylor expanded in c0 around 0 48.5%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. associate-/l* 47.9%

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

      2. associate-/r* 48.0%

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

   10. Simplified 48.0%

       \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2}}{\frac{\frac{{d}^{2}}{{M}^{2}}}{h}}} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 47.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\color{blue}{\sqrt{\frac{{d}^{2}}{{M}^{2}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}}{h}} \]

       2. sqrt-div 48.0%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{{M}^{2}}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       3. unpow2 48.0%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{{M}^{2}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       4. sqrt-prod 23.0%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{{M}^{2}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       5. add-sqr-sqrt 46.2%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{\color{blue}{d}}{\sqrt{{M}^{2}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       6. unpow2 46.2%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{\sqrt{\color{blue}{M \cdot M}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       7. sqrt-prod 21.0%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{\color{blue}{\sqrt{M} \cdot \sqrt{M}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       8. add-sqr-sqrt 44.8%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{\color{blue}{M}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       9. sqrt-div 44.8%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{{M}^{2}}}}}{h}} \]

       10. unpow2 44.8%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{{M}^{2}}}}{h}} \]

       11. sqrt-prod 21.7%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{{M}^{2}}}}{h}} \]

       12. add-sqr-sqrt 49.1%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{\color{blue}{d}}{\sqrt{{M}^{2}}}}{h}} \]

       13. unpow2 49.1%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{d}{\sqrt{\color{blue}{M \cdot M}}}}{h}} \]

       14. sqrt-prod 29.5%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{d}{\color{blue}{\sqrt{M} \cdot \sqrt{M}}}}{h}} \]

       15. add-sqr-sqrt 59.2%

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

   12. Applied egg-rr 59.2%

       \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\color{blue}{\frac{d}{M} \cdot \frac{d}{M}}}{h}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.9%

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

Alternative 10: 37.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := d \cdot \frac{d}{D}\\ t_1 := \frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\\ t_2 := \frac{c0}{2 \cdot w}\\ \mathbf{if}\;D \leq 1.55 \cdot 10^{-204}:\\ \;\;\;\;0\\ \mathbf{elif}\;D \leq 3.4 \cdot 10^{-142}:\\ \;\;\;\;t\_2 \cdot \left(t\_1 + \frac{\frac{c0}{w} \cdot t\_0}{h \cdot D}\right)\\ \mathbf{elif}\;D \leq 5.9 \cdot 10^{+71}:\\ \;\;\;\;0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{d}{M}}{h}}\\ \mathbf{elif}\;D \leq 6.5 \cdot 10^{+141}:\\ \;\;\;\;t\_2 \cdot \left(t\_1 + \frac{t\_0}{D \cdot \frac{w \cdot h}{c0}}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* d (/ d D)))
        (t_1 (* (/ c0 (* w h)) (/ (* d d) (* D D))))
        (t_2 (/ c0 (* 2.0 w))))
   (if (<= D 1.55e-204)
     0.0
     (if (<= D 3.4e-142)
       (* t_2 (+ t_1 (/ (* (/ c0 w) t_0) (* h D))))
       (if (<= D 5.9e+71)
         (* 0.25 (/ (pow D 2.0) (/ (* (/ d M) (/ d M)) h)))
         (if (<= D 6.5e+141)
           (* t_2 (+ t_1 (/ t_0 (* D (/ (* w h) c0)))))
           0.0))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = d * (d / D);
	double t_1 = (c0 / (w * h)) * ((d * d) / (D * D));
	double t_2 = c0 / (2.0 * w);
	double tmp;
	if (D <= 1.55e-204) {
		tmp = 0.0;
	} else if (D <= 3.4e-142) {
		tmp = t_2 * (t_1 + (((c0 / w) * t_0) / (h * D)));
	} else if (D <= 5.9e+71) {
		tmp = 0.25 * (pow(D, 2.0) / (((d / M) * (d / M)) / h));
	} else if (D <= 6.5e+141) {
		tmp = t_2 * (t_1 + (t_0 / (D * ((w * h) / c0))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = d_1 * (d_1 / d)
    t_1 = (c0 / (w * h)) * ((d_1 * d_1) / (d * d))
    t_2 = c0 / (2.0d0 * w)
    if (d <= 1.55d-204) then
        tmp = 0.0d0
    else if (d <= 3.4d-142) then
        tmp = t_2 * (t_1 + (((c0 / w) * t_0) / (h * d)))
    else if (d <= 5.9d+71) then
        tmp = 0.25d0 * ((d ** 2.0d0) / (((d_1 / m) * (d_1 / m)) / h))
    else if (d <= 6.5d+141) then
        tmp = t_2 * (t_1 + (t_0 / (d * ((w * h) / c0))))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = d * (d / D);
	double t_1 = (c0 / (w * h)) * ((d * d) / (D * D));
	double t_2 = c0 / (2.0 * w);
	double tmp;
	if (D <= 1.55e-204) {
		tmp = 0.0;
	} else if (D <= 3.4e-142) {
		tmp = t_2 * (t_1 + (((c0 / w) * t_0) / (h * D)));
	} else if (D <= 5.9e+71) {
		tmp = 0.25 * (Math.pow(D, 2.0) / (((d / M) * (d / M)) / h));
	} else if (D <= 6.5e+141) {
		tmp = t_2 * (t_1 + (t_0 / (D * ((w * h) / c0))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = d * (d / D)
	t_1 = (c0 / (w * h)) * ((d * d) / (D * D))
	t_2 = c0 / (2.0 * w)
	tmp = 0
	if D <= 1.55e-204:
		tmp = 0.0
	elif D <= 3.4e-142:
		tmp = t_2 * (t_1 + (((c0 / w) * t_0) / (h * D)))
	elif D <= 5.9e+71:
		tmp = 0.25 * (math.pow(D, 2.0) / (((d / M) * (d / M)) / h))
	elif D <= 6.5e+141:
		tmp = t_2 * (t_1 + (t_0 / (D * ((w * h) / c0))))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(d * Float64(d / D))
	t_1 = Float64(Float64(c0 / Float64(w * h)) * Float64(Float64(d * d) / Float64(D * D)))
	t_2 = Float64(c0 / Float64(2.0 * w))
	tmp = 0.0
	if (D <= 1.55e-204)
		tmp = 0.0;
	elseif (D <= 3.4e-142)
		tmp = Float64(t_2 * Float64(t_1 + Float64(Float64(Float64(c0 / w) * t_0) / Float64(h * D))));
	elseif (D <= 5.9e+71)
		tmp = Float64(0.25 * Float64((D ^ 2.0) / Float64(Float64(Float64(d / M) * Float64(d / M)) / h)));
	elseif (D <= 6.5e+141)
		tmp = Float64(t_2 * Float64(t_1 + Float64(t_0 / Float64(D * Float64(Float64(w * h) / c0)))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = d * (d / D);
	t_1 = (c0 / (w * h)) * ((d * d) / (D * D));
	t_2 = c0 / (2.0 * w);
	tmp = 0.0;
	if (D <= 1.55e-204)
		tmp = 0.0;
	elseif (D <= 3.4e-142)
		tmp = t_2 * (t_1 + (((c0 / w) * t_0) / (h * D)));
	elseif (D <= 5.9e+71)
		tmp = 0.25 * ((D ^ 2.0) / (((d / M) * (d / M)) / h));
	elseif (D <= 6.5e+141)
		tmp = t_2 * (t_1 + (t_0 / (D * ((w * h) / c0))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(d * N[(d / D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[D, 1.55e-204], 0.0, If[LessEqual[D, 3.4e-142], N[(t$95$2 * N[(t$95$1 + N[(N[(N[(c0 / w), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[D, 5.9e+71], N[(0.25 * N[(N[Power[D, 2.0], $MachinePrecision] / N[(N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[D, 6.5e+141], N[(t$95$2 * N[(t$95$1 + N[(t$95$0 / N[(D * N[(N[(w * h), $MachinePrecision] / c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if D < 1.55e-204 or 6.50000000000000053e141 < D

   1. Initial program 26.7%

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

      1. +-commutative 26.7%

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

      2. +-commutative 26.7%

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

      3. times-frac 25.4%

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

      4. fma-neg 25.4%

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

   3. Simplified 28.1%

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

   5. Taylor expanded in c0 around -inf 4.0%

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

      1. mul-1-neg 4.0%

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

      2. distribute-lft-in 3.3%

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

   7. Simplified 33.0%

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

   8. Taylor expanded in c0 around 0 35.8%

      \[\leadsto \color{blue}{0} \]

   if 1.55e-204 < D < 3.40000000000000029e-142

   1. Initial program 36.8%

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

      1. +-commutative 36.8%

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

      2. +-commutative 36.8%

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

      3. times-frac 36.7%

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

      4. fma-neg 36.7%

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

   3. Simplified 36.7%

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

      1. frac-times 36.7%

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

   6. Applied egg-rr 36.7%

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

   7. Taylor expanded in c0 around inf 47.5%

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

      1. *-commutative 47.5%

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

      2. *-commutative 47.5%

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

      3. times-frac 47.6%

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

      4. *-commutative 47.6%

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

      5. unpow2 47.6%

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

      6. associate-/r* 47.7%

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

      7. unpow2 47.7%

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

      8. associate-*l/ 47.6%

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

      9. associate-*r/ 47.6%

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

      10. unpow2 47.6%

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

   9. Simplified 47.6%

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

       1. pow2 47.6%

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

       2. associate-/l/ 47.6%

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

       3. associate-*r/ 47.7%

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

       4. frac-times 47.7%

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

   11. Applied egg-rr 47.7%

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

   if 3.40000000000000029e-142 < D < 5.9000000000000002e71

   1. Initial program 27.0%

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

      1. +-commutative 27.0%

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

      2. +-commutative 27.0%

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

      3. times-frac 23.5%

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

      4. fma-neg 23.5%

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

   3. Simplified 30.5%

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

   5. Taylor expanded in c0 around -inf 10.9%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) + 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right)} \]
   6. Step-by-step derivation

      1. fma-def 10.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(-1, c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right)} \]

      2. distribute-lft1-in 10.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}, 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      3. metadata-eval 10.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      4. associate-/r* 9.2%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \color{blue}{\frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      5. times-frac 9.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \color{blue}{\left(\frac{{D}^{2}}{c0} \cdot \frac{{M}^{2} \cdot \left(h \cdot w\right)}{{d}^{2}}\right)}\right) \]

      6. associate-*r* 9.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \left(\frac{{D}^{2}}{c0} \cdot \frac{\color{blue}{\left({M}^{2} \cdot h\right) \cdot w}}{{d}^{2}}\right)\right) \]

   7. Simplified 9.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\frac{{d}^{2}}{{D}^{2}}}{h \cdot w}\right), 0.5 \cdot \left(\frac{{D}^{2}}{c0} \cdot \frac{\left({M}^{2} \cdot h\right) \cdot w}{{d}^{2}}\right)\right)} \]

   8. Taylor expanded in c0 around 0 53.7%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. associate-/l* 53.7%

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

      2. associate-/r* 52.0%

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

   10. Simplified 52.0%

       \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2}}{\frac{\frac{{d}^{2}}{{M}^{2}}}{h}}} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 52.0%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\color{blue}{\sqrt{\frac{{d}^{2}}{{M}^{2}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}}{h}} \]

       2. sqrt-div 52.0%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{{M}^{2}}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       3. unpow2 52.0%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{{M}^{2}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       4. sqrt-prod 25.3%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{{M}^{2}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       5. add-sqr-sqrt 51.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{\color{blue}{d}}{\sqrt{{M}^{2}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       6. unpow2 51.9%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{\sqrt{\color{blue}{M \cdot M}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       7. sqrt-prod 19.8%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{\color{blue}{\sqrt{M} \cdot \sqrt{M}}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       8. add-sqr-sqrt 45.2%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{\color{blue}{M}} \cdot \sqrt{\frac{{d}^{2}}{{M}^{2}}}}{h}} \]

       9. sqrt-div 45.2%

          \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{{M}^{2}}}}}{h}} \]

       10. unpow2 45.2%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{{M}^{2}}}}{h}} \]

       11. sqrt-prod 18.6%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{{M}^{2}}}}{h}} \]

       12. add-sqr-sqrt 49.0%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{\color{blue}{d}}{\sqrt{{M}^{2}}}}{h}} \]

       13. unpow2 49.0%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{d}{\sqrt{\color{blue}{M \cdot M}}}}{h}} \]

       14. sqrt-prod 30.4%

           \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{d}{\color{blue}{\sqrt{M} \cdot \sqrt{M}}}}{h}} \]

       15. add-sqr-sqrt 62.6%

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

   12. Applied egg-rr 62.6%

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

   if 5.9000000000000002e71 < D < 6.50000000000000053e141

   1. Initial program 29.2%

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

      1. +-commutative 29.2%

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

      2. +-commutative 29.2%

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

      3. times-frac 28.7%

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

      4. fma-neg 28.7%

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

   3. Simplified 34.1%

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

      1. frac-times 34.1%

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

   6. Applied egg-rr 34.1%

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

   7. Taylor expanded in c0 around inf 45.9%

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

      1. *-commutative 45.9%

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

      2. *-commutative 45.9%

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

      3. times-frac 46.0%

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

      4. *-commutative 46.0%

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

      5. unpow2 46.0%

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

      6. associate-/r* 46.0%

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

      7. unpow2 46.0%

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

      8. associate-*l/ 46.0%

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

      9. associate-*r/ 46.0%

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

      10. unpow2 46.0%

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

   9. Simplified 46.0%

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

       1. pow2 46.0%

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

       2. associate-/l/ 46.1%

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

       3. *-commutative 46.1%

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

       4. associate-*r/ 46.1%

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

       5. associate-/l/ 46.0%

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

       6. clear-num 46.1%

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

       7. frac-times 46.0%

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

   11. Applied egg-rr 46.0%

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

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 1.55 \cdot 10^{-204}:\\ \;\;\;\;0\\ \mathbf{elif}\;D \leq 3.4 \cdot 10^{-142}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \frac{\frac{c0}{w} \cdot \left(d \cdot \frac{d}{D}\right)}{h \cdot D}\right)\\ \mathbf{elif}\;D \leq 5.9 \cdot 10^{+71}:\\ \;\;\;\;0.25 \cdot \frac{{D}^{2}}{\frac{\frac{d}{M} \cdot \frac{d}{M}}{h}}\\ \mathbf{elif}\;D \leq 6.5 \cdot 10^{+141}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \frac{d \cdot \frac{d}{D}}{D \cdot \frac{w \cdot h}{c0}}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 36.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -4.4 \cdot 10^{+104}:\\ \;\;\;\;0\\ \mathbf{elif}\;w \leq -4.2 \cdot 10^{-106} \lor \neg \left(w \leq -2.5 \cdot 10^{-266}\right) \land w \leq 1.9 \cdot 10^{-96}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \frac{c0 \cdot \left(d \cdot \frac{d}{D}\right)}{D \cdot \left(w \cdot h\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= w -4.4e+104)
   0.0
   (if (or (<= w -4.2e-106) (and (not (<= w -2.5e-266)) (<= w 1.9e-96)))
     (*
      (/ c0 (* 2.0 w))
      (+
       (* (/ c0 (* w h)) (/ (* d d) (* D D)))
       (/ (* c0 (* d (/ d D))) (* D (* w h)))))
     0.0)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (w <= -4.4e+104) {
		tmp = 0.0;
	} else if ((w <= -4.2e-106) || (!(w <= -2.5e-266) && (w <= 1.9e-96))) {
		tmp = (c0 / (2.0 * w)) * (((c0 / (w * h)) * ((d * d) / (D * D))) + ((c0 * (d * (d / D))) / (D * (w * h))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

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) :: tmp
    if (w <= (-4.4d+104)) then
        tmp = 0.0d0
    else if ((w <= (-4.2d-106)) .or. (.not. (w <= (-2.5d-266))) .and. (w <= 1.9d-96)) then
        tmp = (c0 / (2.0d0 * w)) * (((c0 / (w * h)) * ((d_1 * d_1) / (d * d))) + ((c0 * (d_1 * (d_1 / d))) / (d * (w * h))))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (w <= -4.4e+104) {
		tmp = 0.0;
	} else if ((w <= -4.2e-106) || (!(w <= -2.5e-266) && (w <= 1.9e-96))) {
		tmp = (c0 / (2.0 * w)) * (((c0 / (w * h)) * ((d * d) / (D * D))) + ((c0 * (d * (d / D))) / (D * (w * h))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if w <= -4.4e+104:
		tmp = 0.0
	elif (w <= -4.2e-106) or (not (w <= -2.5e-266) and (w <= 1.9e-96)):
		tmp = (c0 / (2.0 * w)) * (((c0 / (w * h)) * ((d * d) / (D * D))) + ((c0 * (d * (d / D))) / (D * (w * h))))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (w <= -4.4e+104)
		tmp = 0.0;
	elseif ((w <= -4.2e-106) || (!(w <= -2.5e-266) && (w <= 1.9e-96)))
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(Float64(Float64(c0 / Float64(w * h)) * Float64(Float64(d * d) / Float64(D * D))) + Float64(Float64(c0 * Float64(d * Float64(d / D))) / Float64(D * Float64(w * h)))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (w <= -4.4e+104)
		tmp = 0.0;
	elseif ((w <= -4.2e-106) || (~((w <= -2.5e-266)) && (w <= 1.9e-96)))
		tmp = (c0 / (2.0 * w)) * (((c0 / (w * h)) * ((d * d) / (D * D))) + ((c0 * (d * (d / D))) / (D * (w * h))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[w, -4.4e+104], 0.0, If[Or[LessEqual[w, -4.2e-106], And[N[Not[LessEqual[w, -2.5e-266]], $MachinePrecision], LessEqual[w, 1.9e-96]]], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(c0 * N[(d * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(D * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if w < -4.40000000000000001e104 or -4.20000000000000007e-106 < w < -2.49999999999999996e-266 or 1.9e-96 < w

   1. Initial program 19.9%

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

      1. +-commutative 19.9%

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

      2. +-commutative 19.9%

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

      3. times-frac 18.5%

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

      4. fma-neg 18.5%

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

   3. Simplified 21.3%

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

   5. Taylor expanded in c0 around -inf 7.9%

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

      1. mul-1-neg 7.9%

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

      2. distribute-lft-in 7.1%

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

   7. Simplified 44.6%

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

   8. Taylor expanded in c0 around 0 48.8%

      \[\leadsto \color{blue}{0} \]

   if -4.40000000000000001e104 < w < -4.20000000000000007e-106 or -2.49999999999999996e-266 < w < 1.9e-96

   1. Initial program 37.9%

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

      1. +-commutative 37.9%

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

      2. +-commutative 37.9%

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

      3. times-frac 36.1%

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

      4. fma-neg 36.1%

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

   3. Simplified 40.7%

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

      1. frac-times 39.8%

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

   6. Applied egg-rr 39.8%

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

   7. Taylor expanded in c0 around inf 39.5%

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

      1. *-commutative 39.5%

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

      2. *-commutative 39.5%

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

      3. times-frac 41.7%

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

      4. *-commutative 41.7%

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

      5. unpow2 41.7%

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

      6. associate-/r* 42.6%

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

      7. unpow2 42.6%

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

      8. associate-*l/ 42.6%

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

      9. associate-*r/ 42.6%

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

      10. unpow2 42.6%

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

   9. Simplified 42.6%

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

       1. pow2 42.6%

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

       2. associate-/l/ 42.6%

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

       3. *-commutative 42.6%

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

       4. associate-*r/ 42.6%

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

       5. associate-/l/ 42.6%

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

       6. frac-times 42.5%

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

   11. Applied egg-rr 42.5%

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

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -4.4 \cdot 10^{+104}:\\ \;\;\;\;0\\ \mathbf{elif}\;w \leq -4.2 \cdot 10^{-106} \lor \neg \left(w \leq -2.5 \cdot 10^{-266}\right) \land w \leq 1.9 \cdot 10^{-96}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \frac{c0 \cdot \left(d \cdot \frac{d}{D}\right)}{D \cdot \left(w \cdot h\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 36.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := d \cdot \frac{d}{D}\\ \mathbf{if}\;w \leq -8 \cdot 10^{+104}:\\ \;\;\;\;0\\ \mathbf{elif}\;w \leq -4 \cdot 10^{-104}:\\ \;\;\;\;t\_1 \cdot \left(t\_0 + \frac{t\_2 \cdot \frac{c0}{h}}{w \cdot D}\right)\\ \mathbf{elif}\;w \leq -9.4 \cdot 10^{-268}:\\ \;\;\;\;0\\ \mathbf{elif}\;w \leq 3 \cdot 10^{-96}:\\ \;\;\;\;t\_1 \cdot \left(t\_0 + \frac{c0 \cdot t\_2}{D \cdot \left(w \cdot h\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ c0 (* w h)) (/ (* d d) (* D D))))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (* d (/ d D))))
   (if (<= w -8e+104)
     0.0
     (if (<= w -4e-104)
       (* t_1 (+ t_0 (/ (* t_2 (/ c0 h)) (* w D))))
       (if (<= w -9.4e-268)
         0.0
         (if (<= w 3e-96)
           (* t_1 (+ t_0 (/ (* c0 t_2) (* D (* w h)))))
           0.0))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 / (w * h)) * ((d * d) / (D * D));
	double t_1 = c0 / (2.0 * w);
	double t_2 = d * (d / D);
	double tmp;
	if (w <= -8e+104) {
		tmp = 0.0;
	} else if (w <= -4e-104) {
		tmp = t_1 * (t_0 + ((t_2 * (c0 / h)) / (w * D)));
	} else if (w <= -9.4e-268) {
		tmp = 0.0;
	} else if (w <= 3e-96) {
		tmp = t_1 * (t_0 + ((c0 * t_2) / (D * (w * h))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (c0 / (w * h)) * ((d_1 * d_1) / (d * d))
    t_1 = c0 / (2.0d0 * w)
    t_2 = d_1 * (d_1 / d)
    if (w <= (-8d+104)) then
        tmp = 0.0d0
    else if (w <= (-4d-104)) then
        tmp = t_1 * (t_0 + ((t_2 * (c0 / h)) / (w * d)))
    else if (w <= (-9.4d-268)) then
        tmp = 0.0d0
    else if (w <= 3d-96) then
        tmp = t_1 * (t_0 + ((c0 * t_2) / (d * (w * h))))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 / (w * h)) * ((d * d) / (D * D));
	double t_1 = c0 / (2.0 * w);
	double t_2 = d * (d / D);
	double tmp;
	if (w <= -8e+104) {
		tmp = 0.0;
	} else if (w <= -4e-104) {
		tmp = t_1 * (t_0 + ((t_2 * (c0 / h)) / (w * D)));
	} else if (w <= -9.4e-268) {
		tmp = 0.0;
	} else if (w <= 3e-96) {
		tmp = t_1 * (t_0 + ((c0 * t_2) / (D * (w * h))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 / (w * h)) * ((d * d) / (D * D))
	t_1 = c0 / (2.0 * w)
	t_2 = d * (d / D)
	tmp = 0
	if w <= -8e+104:
		tmp = 0.0
	elif w <= -4e-104:
		tmp = t_1 * (t_0 + ((t_2 * (c0 / h)) / (w * D)))
	elif w <= -9.4e-268:
		tmp = 0.0
	elif w <= 3e-96:
		tmp = t_1 * (t_0 + ((c0 * t_2) / (D * (w * h))))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 / Float64(w * h)) * Float64(Float64(d * d) / Float64(D * D)))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(d * Float64(d / D))
	tmp = 0.0
	if (w <= -8e+104)
		tmp = 0.0;
	elseif (w <= -4e-104)
		tmp = Float64(t_1 * Float64(t_0 + Float64(Float64(t_2 * Float64(c0 / h)) / Float64(w * D))));
	elseif (w <= -9.4e-268)
		tmp = 0.0;
	elseif (w <= 3e-96)
		tmp = Float64(t_1 * Float64(t_0 + Float64(Float64(c0 * t_2) / Float64(D * Float64(w * h)))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 / (w * h)) * ((d * d) / (D * D));
	t_1 = c0 / (2.0 * w);
	t_2 = d * (d / D);
	tmp = 0.0;
	if (w <= -8e+104)
		tmp = 0.0;
	elseif (w <= -4e-104)
		tmp = t_1 * (t_0 + ((t_2 * (c0 / h)) / (w * D)));
	elseif (w <= -9.4e-268)
		tmp = 0.0;
	elseif (w <= 3e-96)
		tmp = t_1 * (t_0 + ((c0 * t_2) / (D * (w * h))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(d * N[(d / D), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, -8e+104], 0.0, If[LessEqual[w, -4e-104], N[(t$95$1 * N[(t$95$0 + N[(N[(t$95$2 * N[(c0 / h), $MachinePrecision]), $MachinePrecision] / N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, -9.4e-268], 0.0, If[LessEqual[w, 3e-96], N[(t$95$1 * N[(t$95$0 + N[(N[(c0 * t$95$2), $MachinePrecision] / N[(D * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if w < -8e104 or -3.99999999999999971e-104 < w < -9.39999999999999947e-268 or 3e-96 < w

   1. Initial program 19.9%

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

      1. +-commutative 19.9%

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

      2. +-commutative 19.9%

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

      3. times-frac 18.5%

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

      4. fma-neg 18.5%

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

   3. Simplified 21.3%

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

   5. Taylor expanded in c0 around -inf 7.9%

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

      1. mul-1-neg 7.9%

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

      2. distribute-lft-in 7.1%

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

   7. Simplified 44.6%

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

   8. Taylor expanded in c0 around 0 48.8%

      \[\leadsto \color{blue}{0} \]

   if -8e104 < w < -3.99999999999999971e-104

   1. Initial program 47.6%

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

      1. +-commutative 47.6%

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

      2. +-commutative 47.6%

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

      3. times-frac 44.4%

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

      4. fma-neg 44.4%

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

   3. Simplified 47.4%

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

      1. frac-times 44.4%

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

   6. Applied egg-rr 44.4%

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

   7. Taylor expanded in c0 around inf 42.0%

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

      1. *-commutative 42.0%

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

      2. *-commutative 42.0%

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

      3. times-frac 41.9%

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

      4. *-commutative 41.9%

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

      5. unpow2 41.9%

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

      6. associate-/r* 41.9%

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

      7. unpow2 41.9%

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

      8. associate-*l/ 41.9%

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

      9. associate-*r/ 41.9%

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

      10. unpow2 41.9%

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

   9. Simplified 41.9%

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

       1. pow2 41.9%

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

       2. associate-/r* 41.9%

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

       3. associate-*r/ 41.9%

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

       4. frac-times 42.0%

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

   11. Applied egg-rr 42.0%

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

   if -9.39999999999999947e-268 < w < 3e-96

   1. Initial program 34.1%

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

      1. +-commutative 34.1%

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

      2. +-commutative 34.1%

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

      3. times-frac 32.8%

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

      4. fma-neg 32.8%

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

   3. Simplified 38.0%

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

      1. frac-times 38.0%

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

   6. Applied egg-rr 38.0%

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

   7. Taylor expanded in c0 around inf 38.5%

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

      1. *-commutative 38.5%

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

      2. *-commutative 38.5%

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

      3. times-frac 41.7%

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

      4. *-commutative 41.7%

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

      5. unpow2 41.7%

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

      6. associate-/r* 42.8%

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

      7. unpow2 42.8%

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

      8. associate-*l/ 42.9%

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

      9. associate-*r/ 42.8%

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

      10. unpow2 42.8%

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

   9. Simplified 42.8%

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

       1. pow2 42.8%

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

       2. associate-/l/ 42.8%

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

       3. *-commutative 42.8%

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

       4. associate-*r/ 42.9%

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

       5. associate-/l/ 42.9%

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

       6. frac-times 42.7%

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

   11. Applied egg-rr 42.7%

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

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -8 \cdot 10^{+104}:\\ \;\;\;\;0\\ \mathbf{elif}\;w \leq -4 \cdot 10^{-104}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \frac{\left(d \cdot \frac{d}{D}\right) \cdot \frac{c0}{h}}{w \cdot D}\right)\\ \mathbf{elif}\;w \leq -9.4 \cdot 10^{-268}:\\ \;\;\;\;0\\ \mathbf{elif}\;w \leq 3 \cdot 10^{-96}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \frac{c0 \cdot \left(d \cdot \frac{d}{D}\right)}{D \cdot \left(w \cdot h\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 36.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := d \cdot \frac{d}{D}\\ \mathbf{if}\;w \leq -4.5 \cdot 10^{+104}:\\ \;\;\;\;0\\ \mathbf{elif}\;w \leq -6.2 \cdot 10^{-106}:\\ \;\;\;\;t\_1 \cdot \left(t\_0 + \frac{t\_2 \cdot \frac{c0}{h}}{w \cdot D}\right)\\ \mathbf{elif}\;w \leq -7.6 \cdot 10^{-268}:\\ \;\;\;\;0\\ \mathbf{elif}\;w \leq 2.65 \cdot 10^{-96}:\\ \;\;\;\;t\_1 \cdot \left(t\_0 + \frac{t\_2}{D \cdot \frac{w \cdot h}{c0}}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ c0 (* w h)) (/ (* d d) (* D D))))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (* d (/ d D))))
   (if (<= w -4.5e+104)
     0.0
     (if (<= w -6.2e-106)
       (* t_1 (+ t_0 (/ (* t_2 (/ c0 h)) (* w D))))
       (if (<= w -7.6e-268)
         0.0
         (if (<= w 2.65e-96)
           (* t_1 (+ t_0 (/ t_2 (* D (/ (* w h) c0)))))
           0.0))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 / (w * h)) * ((d * d) / (D * D));
	double t_1 = c0 / (2.0 * w);
	double t_2 = d * (d / D);
	double tmp;
	if (w <= -4.5e+104) {
		tmp = 0.0;
	} else if (w <= -6.2e-106) {
		tmp = t_1 * (t_0 + ((t_2 * (c0 / h)) / (w * D)));
	} else if (w <= -7.6e-268) {
		tmp = 0.0;
	} else if (w <= 2.65e-96) {
		tmp = t_1 * (t_0 + (t_2 / (D * ((w * h) / c0))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (c0 / (w * h)) * ((d_1 * d_1) / (d * d))
    t_1 = c0 / (2.0d0 * w)
    t_2 = d_1 * (d_1 / d)
    if (w <= (-4.5d+104)) then
        tmp = 0.0d0
    else if (w <= (-6.2d-106)) then
        tmp = t_1 * (t_0 + ((t_2 * (c0 / h)) / (w * d)))
    else if (w <= (-7.6d-268)) then
        tmp = 0.0d0
    else if (w <= 2.65d-96) then
        tmp = t_1 * (t_0 + (t_2 / (d * ((w * h) / c0))))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 / (w * h)) * ((d * d) / (D * D));
	double t_1 = c0 / (2.0 * w);
	double t_2 = d * (d / D);
	double tmp;
	if (w <= -4.5e+104) {
		tmp = 0.0;
	} else if (w <= -6.2e-106) {
		tmp = t_1 * (t_0 + ((t_2 * (c0 / h)) / (w * D)));
	} else if (w <= -7.6e-268) {
		tmp = 0.0;
	} else if (w <= 2.65e-96) {
		tmp = t_1 * (t_0 + (t_2 / (D * ((w * h) / c0))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 / (w * h)) * ((d * d) / (D * D))
	t_1 = c0 / (2.0 * w)
	t_2 = d * (d / D)
	tmp = 0
	if w <= -4.5e+104:
		tmp = 0.0
	elif w <= -6.2e-106:
		tmp = t_1 * (t_0 + ((t_2 * (c0 / h)) / (w * D)))
	elif w <= -7.6e-268:
		tmp = 0.0
	elif w <= 2.65e-96:
		tmp = t_1 * (t_0 + (t_2 / (D * ((w * h) / c0))))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 / Float64(w * h)) * Float64(Float64(d * d) / Float64(D * D)))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(d * Float64(d / D))
	tmp = 0.0
	if (w <= -4.5e+104)
		tmp = 0.0;
	elseif (w <= -6.2e-106)
		tmp = Float64(t_1 * Float64(t_0 + Float64(Float64(t_2 * Float64(c0 / h)) / Float64(w * D))));
	elseif (w <= -7.6e-268)
		tmp = 0.0;
	elseif (w <= 2.65e-96)
		tmp = Float64(t_1 * Float64(t_0 + Float64(t_2 / Float64(D * Float64(Float64(w * h) / c0)))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 / (w * h)) * ((d * d) / (D * D));
	t_1 = c0 / (2.0 * w);
	t_2 = d * (d / D);
	tmp = 0.0;
	if (w <= -4.5e+104)
		tmp = 0.0;
	elseif (w <= -6.2e-106)
		tmp = t_1 * (t_0 + ((t_2 * (c0 / h)) / (w * D)));
	elseif (w <= -7.6e-268)
		tmp = 0.0;
	elseif (w <= 2.65e-96)
		tmp = t_1 * (t_0 + (t_2 / (D * ((w * h) / c0))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(d * N[(d / D), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, -4.5e+104], 0.0, If[LessEqual[w, -6.2e-106], N[(t$95$1 * N[(t$95$0 + N[(N[(t$95$2 * N[(c0 / h), $MachinePrecision]), $MachinePrecision] / N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, -7.6e-268], 0.0, If[LessEqual[w, 2.65e-96], N[(t$95$1 * N[(t$95$0 + N[(t$95$2 / N[(D * N[(N[(w * h), $MachinePrecision] / c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if w < -4.4999999999999998e104 or -6.19999999999999971e-106 < w < -7.6000000000000005e-268 or 2.6500000000000001e-96 < w

   1. Initial program 19.9%

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

      1. +-commutative 19.9%

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

      2. +-commutative 19.9%

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

      3. times-frac 18.5%

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

      4. fma-neg 18.5%

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

   3. Simplified 21.3%

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

   5. Taylor expanded in c0 around -inf 7.9%

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

      1. mul-1-neg 7.9%

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

      2. distribute-lft-in 7.1%

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

   7. Simplified 44.6%

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

   8. Taylor expanded in c0 around 0 48.8%

      \[\leadsto \color{blue}{0} \]

   if -4.4999999999999998e104 < w < -6.19999999999999971e-106

   1. Initial program 47.6%

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

      1. +-commutative 47.6%

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

      2. +-commutative 47.6%

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

      3. times-frac 44.4%

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

      4. fma-neg 44.4%

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

   3. Simplified 47.4%

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

      1. frac-times 44.4%

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

   6. Applied egg-rr 44.4%

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

   7. Taylor expanded in c0 around inf 42.0%

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

      1. *-commutative 42.0%

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

      2. *-commutative 42.0%

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

      3. times-frac 41.9%

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

      4. *-commutative 41.9%

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

      5. unpow2 41.9%

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

      6. associate-/r* 41.9%

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

      7. unpow2 41.9%

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

      8. associate-*l/ 41.9%

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

      9. associate-*r/ 41.9%

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

      10. unpow2 41.9%

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

   9. Simplified 41.9%

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

       1. pow2 41.9%

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

       2. associate-/r* 41.9%

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

       3. associate-*r/ 41.9%

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

       4. frac-times 42.0%

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

   11. Applied egg-rr 42.0%

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

   if -7.6000000000000005e-268 < w < 2.6500000000000001e-96

   1. Initial program 34.1%

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

      1. +-commutative 34.1%

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

      2. +-commutative 34.1%

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

      3. times-frac 32.8%

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

      4. fma-neg 32.8%

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

   3. Simplified 38.0%

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

      1. frac-times 38.0%

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

   6. Applied egg-rr 38.0%

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

   7. Taylor expanded in c0 around inf 38.5%

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

      1. *-commutative 38.5%

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

      2. *-commutative 38.5%

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

      3. times-frac 41.7%

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

      4. *-commutative 41.7%

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

      5. unpow2 41.7%

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

      6. associate-/r* 42.8%

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

      7. unpow2 42.8%

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

      8. associate-*l/ 42.9%

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

      9. associate-*r/ 42.8%

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

      10. unpow2 42.8%

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

   9. Simplified 42.8%

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

       1. pow2 42.8%

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

       2. associate-/l/ 42.8%

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

       3. *-commutative 42.8%

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

       4. associate-*r/ 42.9%

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

       5. associate-/l/ 42.9%

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

       6. clear-num 42.9%

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

       7. frac-times 42.9%

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

   11. Applied egg-rr 42.9%

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

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -4.5 \cdot 10^{+104}:\\ \;\;\;\;0\\ \mathbf{elif}\;w \leq -6.2 \cdot 10^{-106}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \frac{\left(d \cdot \frac{d}{D}\right) \cdot \frac{c0}{h}}{w \cdot D}\right)\\ \mathbf{elif}\;w \leq -7.6 \cdot 10^{-268}:\\ \;\;\;\;0\\ \mathbf{elif}\;w \leq 2.65 \cdot 10^{-96}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \frac{d \cdot \frac{d}{D}}{D \cdot \frac{w \cdot h}{c0}}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 33.4% accurate, 151.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (c0 w h D d M) :precision binary64 0.0)

C

double code(double c0, double w, double h, double D, double d, double M) {
	return 0.0;
}

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
    code = 0.0d0
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return 0.0;
}

Python

def code(c0, w, h, D, d, M):
	return 0.0

Julia

function code(c0, w, h, D, d, M)
	return 0.0
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = 0.0;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := 0.0

Derivation

1. Initial program 27.8%

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

   1. +-commutative 27.8%

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

   2. +-commutative 27.8%

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

   3. times-frac 26.2%

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

   4. fma-neg 26.2%

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

3. Simplified 29.8%

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

5. Taylor expanded in c0 around -inf 5.3%

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

   1. mul-1-neg 5.3%

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

   2. distribute-lft-in 4.9%

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

7. Simplified 32.1%

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

8. Taylor expanded in c0 around 0 35.8%

   \[\leadsto \color{blue}{0} \]

9. Final simplification 35.8%

   \[\leadsto 0 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024040 
(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))))))