Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.6% → 55.3%

Time: 33.9s

Alternatives: 5

Speedup: 151.0×

• could not determine a ground truth

  1. c0 = 1.5799590768404246e+209
  2. w = 6.417415336487265e-221
  3. h = -1.3835451913262504e-242
  4. D = -7.073986664411131e-293
  5. d = 3.502636434689754e+257
  6. M = -1605838292.073639

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: 55.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {D}^{2} \cdot \left(w \cdot h\right)\\ t_1 := \frac{{d}^{2}}{t\_0}\\ t_2 := \frac{c0}{2 \cdot w}\\ t_3 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\\ t_4 := t\_2 \cdot \left(t\_3 + \sqrt{t\_3 \cdot t\_3 - M \cdot M}\right)\\ \mathbf{if}\;t\_4 \leq -1 \cdot 10^{-316}:\\ \;\;\;\;\frac{\frac{c0}{w}}{2} \cdot \left(2 \cdot \frac{\frac{c0 \cdot {d}^{2}}{{D}^{2}}}{w \cdot h}\right)\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;t\_2 \cdot \frac{-{M}^{2}}{c0 \cdot \left(-\left(t\_1 + t\_1\right)\right)}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_2 \cdot \left(2 \cdot \left({d}^{2} \cdot \frac{c0}{t\_0}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (pow D 2.0) (* w h)))
        (t_1 (/ (pow d 2.0) t_0))
        (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)))))))
   (if (<= t_4 -1e-316)
     (*
      (/ (/ c0 w) 2.0)
      (* 2.0 (/ (/ (* c0 (pow d 2.0)) (pow D 2.0)) (* w h))))
     (if (<= t_4 0.0)
       (* t_2 (/ (- (pow M 2.0)) (* c0 (- (+ t_1 t_1)))))
       (if (<= t_4 INFINITY)
         (* t_2 (* 2.0 (* (pow d 2.0) (/ c0 t_0))))
         0.0)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = pow(D, 2.0) * (w * h);
	double t_1 = pow(d, 2.0) / t_0;
	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 tmp;
	if (t_4 <= -1e-316) {
		tmp = ((c0 / w) / 2.0) * (2.0 * (((c0 * pow(d, 2.0)) / pow(D, 2.0)) / (w * h)));
	} else if (t_4 <= 0.0) {
		tmp = t_2 * (-pow(M, 2.0) / (c0 * -(t_1 + t_1)));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_2 * (2.0 * (pow(d, 2.0) * (c0 / t_0)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = Math.pow(D, 2.0) * (w * h);
	double t_1 = Math.pow(d, 2.0) / t_0;
	double t_2 = c0 / (2.0 * w);
	double t_3 = (c0 * (d * d)) / ((D * D) * (w * h));
	double t_4 = t_2 * (t_3 + Math.sqrt(((t_3 * t_3) - (M * M))));
	double tmp;
	if (t_4 <= -1e-316) {
		tmp = ((c0 / w) / 2.0) * (2.0 * (((c0 * Math.pow(d, 2.0)) / Math.pow(D, 2.0)) / (w * h)));
	} else if (t_4 <= 0.0) {
		tmp = t_2 * (-Math.pow(M, 2.0) / (c0 * -(t_1 + t_1)));
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = t_2 * (2.0 * (Math.pow(d, 2.0) * (c0 / t_0)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = math.pow(D, 2.0) * (w * h)
	t_1 = math.pow(d, 2.0) / t_0
	t_2 = c0 / (2.0 * w)
	t_3 = (c0 * (d * d)) / ((D * D) * (w * h))
	t_4 = t_2 * (t_3 + math.sqrt(((t_3 * t_3) - (M * M))))
	tmp = 0
	if t_4 <= -1e-316:
		tmp = ((c0 / w) / 2.0) * (2.0 * (((c0 * math.pow(d, 2.0)) / math.pow(D, 2.0)) / (w * h)))
	elif t_4 <= 0.0:
		tmp = t_2 * (-math.pow(M, 2.0) / (c0 * -(t_1 + t_1)))
	elif t_4 <= math.inf:
		tmp = t_2 * (2.0 * (math.pow(d, 2.0) * (c0 / t_0)))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64((D ^ 2.0) * Float64(w * h))
	t_1 = Float64((d ^ 2.0) / t_0)
	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)))))
	tmp = 0.0
	if (t_4 <= -1e-316)
		tmp = Float64(Float64(Float64(c0 / w) / 2.0) * Float64(2.0 * Float64(Float64(Float64(c0 * (d ^ 2.0)) / (D ^ 2.0)) / Float64(w * h))));
	elseif (t_4 <= 0.0)
		tmp = Float64(t_2 * Float64(Float64(-(M ^ 2.0)) / Float64(c0 * Float64(-Float64(t_1 + t_1)))));
	elseif (t_4 <= Inf)
		tmp = Float64(t_2 * Float64(2.0 * Float64((d ^ 2.0) * Float64(c0 / t_0))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (D ^ 2.0) * (w * h);
	t_1 = (d ^ 2.0) / t_0;
	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))));
	tmp = 0.0;
	if (t_4 <= -1e-316)
		tmp = ((c0 / w) / 2.0) * (2.0 * (((c0 * (d ^ 2.0)) / (D ^ 2.0)) / (w * h)));
	elseif (t_4 <= 0.0)
		tmp = t_2 * (-(M ^ 2.0) / (c0 * -(t_1 + t_1)));
	elseif (t_4 <= Inf)
		tmp = t_2 * (2.0 * ((d ^ 2.0) * (c0 / t_0)));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[Power[D, 2.0], $MachinePrecision] * N[(w * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[d, 2.0], $MachinePrecision] / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(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]}, If[LessEqual[t$95$4, -1e-316], N[(N[(N[(c0 / w), $MachinePrecision] / 2.0), $MachinePrecision] * N[(2.0 * N[(N[(N[(c0 * N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[D, 2.0], $MachinePrecision]), $MachinePrecision] / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.0], N[(t$95$2 * N[((-N[Power[M, 2.0], $MachinePrecision]) / N[(c0 * (-N[(t$95$1 + t$95$1), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(t$95$2 * N[(2.0 * N[(N[Power[d, 2.0], $MachinePrecision] * N[(c0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]]]]]]

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))))) < -9.999999837e-317

   1. Initial program 75.5%

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

   3. Simplified 72.0%

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

      1. times-frac 72.0%

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

   6. Applied egg-rr 72.0%

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

   7. Taylor expanded in c0 around inf 84.6%

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

      1. associate-/r* 87.4%

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

   9. Simplified 87.4%

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

   if -9.999999837e-317 < (*.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 63.9%

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

      1. +-commutative 63.9%

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

      2. +-commutative 63.9%

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

      3. times-frac 27.4%

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

      4. fma-neg 27.4%

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

   3. Simplified 63.6%

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

   6. Applied egg-rr 63.6%

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

      1. associate--r- 75.5%

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

      2. +-inverses 75.5%

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

      3. *-commutative 75.5%

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

   8. Simplified 75.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{0 + {M}^{2}}{\frac{c0}{h \cdot w} \cdot {\left(\frac{d}{D}\right)}^{2} - \sqrt{{\left(\frac{c0}{h \cdot w}\right)}^{2} \cdot {\left(\frac{d}{D}\right)}^{4} - {M}^{2}}}} \]

   9. Taylor expanded in c0 around -inf 87.1%

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

       1. associate-*r/ 87.1%

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

       2. neg-mul-1 87.1%

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

       3. sub-neg 87.1%

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

       4. mul-1-neg 87.1%

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

       5. distribute-neg-out 87.1%

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

   11. Simplified 87.1%

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

   if -0.0 < (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 80.8%

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

      1. +-commutative 80.8%

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

      2. +-commutative 80.8%

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

      3. times-frac 75.9%

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

      4. fma-neg 75.9%

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

   3. Simplified 78.0%

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

   5. Taylor expanded in c0 around inf 88.0%

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

      1. associate-/l* 87.9%

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

      2. associate-*r* 85.7%

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

      3. *-commutative 85.7%

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

      4. *-commutative 85.7%

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

      5. associate-/r/ 88.0%

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

      6. *-commutative 88.0%

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

      7. *-commutative 88.0%

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

      8. associate-*r* 90.1%

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

   7. Simplified 90.1%

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

   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)))))

   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 0.6%

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

      4. fma-neg 0.6%

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

   3. Simplified 1.8%

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

   5. Taylor expanded in c0 around -inf 2.4%

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

      1. mul-1-neg 2.4%

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

      2. distribute-lft-in 1.2%

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

   7. Simplified 39.3%

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

   8. Taylor expanded in c0 around 0 48.4%

      \[\leadsto \color{blue}{0} \]
3. Recombined 4 regimes into one program.

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq -1 \cdot 10^{-316}:\\ \;\;\;\;\frac{\frac{c0}{w}}{2} \cdot \left(2 \cdot \frac{\frac{c0 \cdot {d}^{2}}{{D}^{2}}}{w \cdot h}\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq 0:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \frac{-{M}^{2}}{c0 \cdot \left(-\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)\right)}\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left({d}^{2} \cdot \frac{c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 54.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\\ t_2 := t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-316}:\\ \;\;\;\;\frac{\frac{c0}{w}}{2} \cdot \left(2 \cdot \frac{\frac{c0 \cdot {d}^{2}}{{D}^{2}}}{w \cdot h}\right)\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(0.5, \frac{\frac{{D}^{2}}{c0} \cdot \left(\left(w \cdot h\right) \cdot {M}^{2}\right)}{{d}^{2}}, 0\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_0 \cdot \left(2 \cdot \left({d}^{2} \cdot \frac{c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w)))
        (t_1 (/ (* c0 (* d d)) (* (* D D) (* w h))))
        (t_2 (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))))
   (if (<= t_2 -1e-316)
     (*
      (/ (/ c0 w) 2.0)
      (* 2.0 (/ (/ (* c0 (pow d 2.0)) (pow D 2.0)) (* w h))))
     (if (<= t_2 0.0)
       (*
        t_0
        (fma
         0.5
         (/ (* (/ (pow D 2.0) c0) (* (* w h) (pow M 2.0))) (pow d 2.0))
         0.0))
       (if (<= t_2 INFINITY)
         (* t_0 (* 2.0 (* (pow d 2.0) (/ c0 (* (pow D 2.0) (* w h))))))
         0.0)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((D * D) * (w * h));
	double t_2 = t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= -1e-316) {
		tmp = ((c0 / w) / 2.0) * (2.0 * (((c0 * pow(d, 2.0)) / pow(D, 2.0)) / (w * h)));
	} else if (t_2 <= 0.0) {
		tmp = t_0 * fma(0.5, (((pow(D, 2.0) / c0) * ((w * h) * pow(M, 2.0))) / pow(d, 2.0)), 0.0);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_0 * (2.0 * (pow(d, 2.0) * (c0 / (pow(D, 2.0) * (w * h)))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(D * D) * Float64(w * h)))
	t_2 = Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M)))))
	tmp = 0.0
	if (t_2 <= -1e-316)
		tmp = Float64(Float64(Float64(c0 / w) / 2.0) * Float64(2.0 * Float64(Float64(Float64(c0 * (d ^ 2.0)) / (D ^ 2.0)) / Float64(w * h))));
	elseif (t_2 <= 0.0)
		tmp = Float64(t_0 * fma(0.5, Float64(Float64(Float64((D ^ 2.0) / c0) * Float64(Float64(w * h) * (M ^ 2.0))) / (d ^ 2.0)), 0.0));
	elseif (t_2 <= Inf)
		tmp = Float64(t_0 * Float64(2.0 * Float64((d ^ 2.0) * Float64(c0 / Float64((D ^ 2.0) * Float64(w * h))))));
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-316], N[(N[(N[(c0 / w), $MachinePrecision] / 2.0), $MachinePrecision] * N[(2.0 * N[(N[(N[(c0 * N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[D, 2.0], $MachinePrecision]), $MachinePrecision] / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(t$95$0 * N[(0.5 * N[(N[(N[(N[Power[D, 2.0], $MachinePrecision] / c0), $MachinePrecision] * N[(N[(w * h), $MachinePrecision] * N[Power[M, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(t$95$0 * N[(2.0 * N[(N[Power[d, 2.0], $MachinePrecision] * N[(c0 / N[(N[Power[D, 2.0], $MachinePrecision] * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]]]]

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))))) < -9.999999837e-317

   1. Initial program 75.5%

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

   3. Simplified 72.0%

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

      1. times-frac 72.0%

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

   6. Applied egg-rr 72.0%

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

   7. Taylor expanded in c0 around inf 84.6%

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

      1. associate-/r* 87.4%

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

   9. Simplified 87.4%

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

   if -9.999999837e-317 < (*.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 63.9%

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

      1. +-commutative 63.9%

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

      2. +-commutative 63.9%

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

      3. times-frac 27.4%

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

      4. fma-neg 27.4%

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

   3. Simplified 63.6%

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

   5. Taylor expanded in c0 around -inf 62.7%

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

   7. Simplified 75.2%

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

   if -0.0 < (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 80.8%

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

      1. +-commutative 80.8%

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

      2. +-commutative 80.8%

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

      3. times-frac 75.9%

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

      4. fma-neg 75.9%

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

   3. Simplified 78.0%

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

   5. Taylor expanded in c0 around inf 88.0%

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

      1. associate-/l* 87.9%

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

      2. associate-*r* 85.7%

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

      3. *-commutative 85.7%

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

      4. *-commutative 85.7%

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

      5. associate-/r/ 88.0%

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

      6. *-commutative 88.0%

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

      7. *-commutative 88.0%

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

      8. associate-*r* 90.1%

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

   7. Simplified 90.1%

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

   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)))))

   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 0.6%

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

      4. fma-neg 0.6%

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

   3. Simplified 1.8%

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

   5. Taylor expanded in c0 around -inf 2.4%

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

      1. mul-1-neg 2.4%

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

      2. distribute-lft-in 1.2%

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

   7. Simplified 39.3%

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

   8. Taylor expanded in c0 around 0 48.4%

      \[\leadsto \color{blue}{0} \]
3. Recombined 4 regimes into one program.

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq -1 \cdot 10^{-316}:\\ \;\;\;\;\frac{\frac{c0}{w}}{2} \cdot \left(2 \cdot \frac{\frac{c0 \cdot {d}^{2}}{{D}^{2}}}{w \cdot h}\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq 0:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(0.5, \frac{\frac{{D}^{2}}{c0} \cdot \left(\left(w \cdot h\right) \cdot {M}^{2}\right)}{{d}^{2}}, 0\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left({d}^{2} \cdot \frac{c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 54.0% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(D * D) * Float64(w * h)))
	tmp = 0.0
	if (Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_0 * Float64(2.0 * Float64((d ^ 2.0) * Float64(c0 / Float64((D ^ 2.0) * 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 / (2.0 * w);
	t_1 = (c0 * (d * d)) / ((D * D) * (w * h));
	tmp = 0.0;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= Inf)
		tmp = t_0 * (2.0 * ((d ^ 2.0) * (c0 / ((D ^ 2.0) * (w * h)))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 77.2%

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

      1. +-commutative 77.2%

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

      2. +-commutative 77.2%

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

      3. times-frac 69.8%

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

      4. fma-neg 69.8%

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

   3. Simplified 74.4%

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

   5. Taylor expanded in c0 around inf 78.7%

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

      1. associate-/l* 76.4%

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

      2. associate-*r* 75.2%

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

      3. *-commutative 75.2%

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

      4. *-commutative 75.2%

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

      5. associate-/r/ 78.7%

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

      6. *-commutative 78.7%

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

      7. *-commutative 78.7%

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

      8. associate-*r* 79.8%

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

   7. Simplified 79.8%

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

   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)))))

   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 0.6%

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

      4. fma-neg 0.6%

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

   3. Simplified 1.8%

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

   5. Taylor expanded in c0 around -inf 2.4%

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

      1. mul-1-neg 2.4%

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

      2. distribute-lft-in 1.2%

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

   7. Simplified 39.3%

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

   8. Taylor expanded in c0 around 0 48.4%

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

4. Final simplification 58.7%

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

Alternative 4: 54.2% accurate, 0.5× speedup

Math

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

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)))))
   (if (<= (* t_1 (+ t_2 (sqrt (- (* t_2 t_2) (* M M))))) INFINITY)
     (* t_1 (+ (* (* (/ d D) (/ d D)) t_0) (* (pow (/ d D) 2.0) t_0)))
     0.0)))

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 77.2%

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

      1. +-commutative 77.2%

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

      2. +-commutative 77.2%

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

      3. times-frac 69.8%

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

      4. fma-neg 69.8%

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

   3. Simplified 74.4%

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

   5. Taylor expanded in c0 around inf 74.9%

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

      1. *-commutative 74.9%

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

      2. associate-/l/ 75.9%

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

      3. associate-*l/ 77.0%

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

      4. associate-/l* 76.7%

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

      5. *-commutative 76.7%

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

   7. Simplified 76.7%

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

      1. expm1-log1p-u 76.5%

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

      2. clear-num 76.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \frac{\frac{c0}{h \cdot w}}{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{{d}^{2}}{{D}^{2}}}}\right)\right)}\right) \]

      3. unpow2 76.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \frac{\frac{c0}{h \cdot w}}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{\color{blue}{d \cdot d}}{{D}^{2}}}\right)\right)}\right) \]

      4. unpow2 76.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \frac{\frac{c0}{h \cdot w}}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{d \cdot d}{\color{blue}{D \cdot D}}}\right)\right)}\right) \]

      5. frac-times 76.7%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \frac{\frac{c0}{h \cdot w}}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\color{blue}{\frac{d}{D} \cdot \frac{d}{D}}}\right)\right)}\right) \]

      6. pow2 76.7%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \frac{\frac{c0}{h \cdot w}}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\color{blue}{{\left(\frac{d}{D}\right)}^{2}}}\right)\right)}\right) \]

      7. pow-flip 76.7%

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

      8. metadata-eval 76.7%

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

   9. Applied egg-rr 76.7%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \frac{\frac{c0}{h \cdot w}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{d}{D}\right)}^{-2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. frac-times 78.7%

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

   11. Applied egg-rr 78.7%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \color{blue}{\left(\frac{d}{D} \cdot \frac{d}{D}\right)} + \frac{\frac{c0}{h \cdot w}}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{d}{D}\right)}^{-2}\right)\right)}\right) \]
   12. Step-by-step derivation

       1. expm1-log1p-u 79.0%

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

       2. div-inv 79.0%

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

       3. pow-flip 79.3%

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

       4. metadata-eval 79.3%

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

       5. associate-/l/ 79.3%

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

       6. *-commutative 79.3%

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

       7. associate-/l/ 79.3%

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

   13. Applied egg-rr 79.3%

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

   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 0.6%

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

      4. fma-neg 0.6%

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

   3. Simplified 1.8%

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

   5. Taylor expanded in c0 around -inf 2.4%

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

      1. mul-1-neg 2.4%

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

      2. distribute-lft-in 1.2%

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

   7. Simplified 39.3%

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

   8. Taylor expanded in c0 around 0 48.4%

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

4. Final simplification 58.6%

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

Alternative 5: 33.1% 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 25.3%

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

   1. +-commutative 25.3%

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

   2. +-commutative 25.3%

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

   3. times-frac 23.3%

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

   4. fma-neg 23.3%

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

3. Simplified 25.6%

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

5. Taylor expanded in c0 around -inf 4.4%

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

   1. mul-1-neg 4.4%

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

   2. distribute-lft-in 3.6%

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

7. Simplified 29.8%

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

8. Taylor expanded in c0 around 0 36.1%

   \[\leadsto \color{blue}{0} \]

9. Final simplification 36.1%

   \[\leadsto 0 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024033 
(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))))))