Henrywood and Agarwal, Equation (13)

Percentage Accurate: 23.8% → 63.8%

Time: 32.5s

Alternatives: 10

Speedup: 151.0×

• could not determine a ground truth

  1. c0 = -2.601243995466845e+233
  2. w = 1.973094539180565e-268
  3. h = 1.0788056374716568e-232
  4. D = 9.262685047834647e-308
  5. d = 3.1538854574844056e+232
  6. M = 1.3617470856440585e-144

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: 23.8% 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: 63.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_1 := \left(t_0 + \sqrt{t_0 \cdot t_0 - M \cdot M}\right) \cdot \frac{c0}{2 \cdot w}\\ \mathbf{if}\;t_1 \leq \infty:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \frac{\frac{D \cdot \left(M \cdot M\right)}{\frac{d}{h}}}{d}\right)\\ \end{array} \end{array} \]

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	t_1 = (t_0 + sqrt(((t_0 * t_0) - (M * M)))) * (c0 / (2.0 * w));
	tmp = 0.0;
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = 0.25 * (D * (((D * (M * M)) / (d / h)) / d));
	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[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[(0.25 * N[(D * N[(N[(N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision] / N[(d / h), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 77.9%

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

   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. Taylor expanded in c0 around -inf 1.9%

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

      1. fma-def 1.9%

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

      2. times-frac 2.5%

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

      3. *-commutative 2.5%

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

      4. unpow2 2.5%

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

      5. unpow2 2.5%

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

      6. unpow2 2.5%

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

      7. associate-*r* 2.5%

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

   4. Simplified 30.1%

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

   5. Taylor expanded in c0 around 0 40.9%

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

      1. unpow2 40.9%

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

      2. unpow2 40.9%

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

      3. unpow2 40.9%

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

   7. Simplified 40.9%

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

      1. *-un-lft-identity 40.9%

         \[\leadsto 0.25 \cdot \color{blue}{\left(1 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}\right)} \]

      2. times-frac 52.1%

         \[\leadsto 0.25 \cdot \left(1 \cdot \color{blue}{\left(\frac{D \cdot D}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)}\right) \]

   9. Applied egg-rr 52.1%

      \[\leadsto 0.25 \cdot \color{blue}{\left(1 \cdot \left(\frac{D \cdot D}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\right)} \]
   10. Step-by-step derivation

       1. *-lft-identity 52.1%

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

       2. associate-/l* 55.6%

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

       3. unpow2 55.6%

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

       4. associate-/l* 56.8%

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

       5. unpow2 56.8%

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

   11. Simplified 56.8%

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

       1. associate-*l/ 58.0%

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

       2. associate-/r/ 57.3%

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

   13. Applied egg-rr 57.3%

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

       1. associate-/r/ 61.6%

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

       2. unpow2 61.6%

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

       3. associate-*l/ 61.9%

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

       4. *-commutative 61.9%

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

       5. associate-/l* 60.9%

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

       6. associate-*r/ 61.3%

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

       7. unpow2 61.3%

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

   15. Simplified 61.3%

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

4. Final simplification 67.1%

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

Alternative 2: 44.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ \mathbf{if}\;d \cdot d \leq 2 \cdot 10^{-249}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{d}\right)}{\frac{d}{D}}\\ \mathbf{elif}\;d \cdot d \leq 8.5 \cdot 10^{-7}:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\right)\\ \mathbf{elif}\;d \cdot d \leq 10^{+295}:\\ \;\;\;\;0.25 \cdot \left(\left(D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right) \cdot \frac{1}{d \cdot d}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \left({\left(\frac{d}{D}\right)}^{2} \cdot \frac{\frac{c0}{w}}{h}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w))))
   (if (<= (* d d) 2e-249)
     (* 0.25 (/ (* D (* (* M M) (/ h d))) (/ d D)))
     (if (<= (* d d) 8.5e-7)
       (* t_0 (* 2.0 (* (/ c0 (* w h)) (* (/ d D) (/ d D)))))
       (if (<= (* d d) 1e+295)
         (* 0.25 (* (* D (* D (* h (* M M)))) (/ 1.0 (* d d))))
         (* t_0 (* 2.0 (* (pow (/ d D) 2.0) (/ (/ c0 w) h)))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double tmp;
	if ((d * d) <= 2e-249) {
		tmp = 0.25 * ((D * ((M * M) * (h / d))) / (d / D));
	} else if ((d * d) <= 8.5e-7) {
		tmp = t_0 * (2.0 * ((c0 / (w * h)) * ((d / D) * (d / D))));
	} else if ((d * d) <= 1e+295) {
		tmp = 0.25 * ((D * (D * (h * (M * M)))) * (1.0 / (d * d)));
	} else {
		tmp = t_0 * (2.0 * (pow((d / D), 2.0) * ((c0 / w) / h)));
	}
	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) :: tmp
    t_0 = c0 / (2.0d0 * w)
    if ((d_1 * d_1) <= 2d-249) then
        tmp = 0.25d0 * ((d * ((m * m) * (h / d_1))) / (d_1 / d))
    else if ((d_1 * d_1) <= 8.5d-7) then
        tmp = t_0 * (2.0d0 * ((c0 / (w * h)) * ((d_1 / d) * (d_1 / d))))
    else if ((d_1 * d_1) <= 1d+295) then
        tmp = 0.25d0 * ((d * (d * (h * (m * m)))) * (1.0d0 / (d_1 * d_1)))
    else
        tmp = t_0 * (2.0d0 * (((d_1 / d) ** 2.0d0) * ((c0 / w) / h)))
    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 / (2.0 * w);
	double tmp;
	if ((d * d) <= 2e-249) {
		tmp = 0.25 * ((D * ((M * M) * (h / d))) / (d / D));
	} else if ((d * d) <= 8.5e-7) {
		tmp = t_0 * (2.0 * ((c0 / (w * h)) * ((d / D) * (d / D))));
	} else if ((d * d) <= 1e+295) {
		tmp = 0.25 * ((D * (D * (h * (M * M)))) * (1.0 / (d * d)));
	} else {
		tmp = t_0 * (2.0 * (Math.pow((d / D), 2.0) * ((c0 / w) / h)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	tmp = 0
	if (d * d) <= 2e-249:
		tmp = 0.25 * ((D * ((M * M) * (h / d))) / (d / D))
	elif (d * d) <= 8.5e-7:
		tmp = t_0 * (2.0 * ((c0 / (w * h)) * ((d / D) * (d / D))))
	elif (d * d) <= 1e+295:
		tmp = 0.25 * ((D * (D * (h * (M * M)))) * (1.0 / (d * d)))
	else:
		tmp = t_0 * (2.0 * (math.pow((d / D), 2.0) * ((c0 / w) / h)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	tmp = 0.0
	if (Float64(d * d) <= 2e-249)
		tmp = Float64(0.25 * Float64(Float64(D * Float64(Float64(M * M) * Float64(h / d))) / Float64(d / D)));
	elseif (Float64(d * d) <= 8.5e-7)
		tmp = Float64(t_0 * Float64(2.0 * Float64(Float64(c0 / Float64(w * h)) * Float64(Float64(d / D) * Float64(d / D)))));
	elseif (Float64(d * d) <= 1e+295)
		tmp = Float64(0.25 * Float64(Float64(D * Float64(D * Float64(h * Float64(M * M)))) * Float64(1.0 / Float64(d * d))));
	else
		tmp = Float64(t_0 * Float64(2.0 * Float64((Float64(d / D) ^ 2.0) * Float64(Float64(c0 / w) / h))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	tmp = 0.0;
	if ((d * d) <= 2e-249)
		tmp = 0.25 * ((D * ((M * M) * (h / d))) / (d / D));
	elseif ((d * d) <= 8.5e-7)
		tmp = t_0 * (2.0 * ((c0 / (w * h)) * ((d / D) * (d / D))));
	elseif ((d * d) <= 1e+295)
		tmp = 0.25 * ((D * (D * (h * (M * M)))) * (1.0 / (d * d)));
	else
		tmp = t_0 * (2.0 * (((d / D) ^ 2.0) * ((c0 / w) / h)));
	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]}, If[LessEqual[N[(d * d), $MachinePrecision], 2e-249], N[(0.25 * N[(N[(D * N[(N[(M * M), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(d * d), $MachinePrecision], 8.5e-7], N[(t$95$0 * N[(2.0 * N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(d * d), $MachinePrecision], 1e+295], N[(0.25 * N[(N[(D * N[(D * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(2.0 * N[(N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 d d) < 2.00000000000000011e-249

   1. Initial program 12.5%

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

   2. Taylor expanded in c0 around -inf 2.7%

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

      1. fma-def 2.7%

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

      2. times-frac 2.7%

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

      3. *-commutative 2.7%

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

      4. unpow2 2.7%

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

      5. unpow2 2.7%

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

      6. unpow2 2.7%

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

      7. associate-*r* 2.7%

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

   4. Simplified 12.9%

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

   5. Taylor expanded in c0 around 0 17.5%

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

      1. unpow2 17.5%

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

      2. unpow2 17.5%

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

      3. unpow2 17.5%

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

   7. Simplified 17.5%

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

      1. *-un-lft-identity 17.5%

         \[\leadsto 0.25 \cdot \color{blue}{\left(1 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}\right)} \]

      2. times-frac 41.1%

         \[\leadsto 0.25 \cdot \left(1 \cdot \color{blue}{\left(\frac{D \cdot D}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)}\right) \]

   9. Applied egg-rr 41.1%

      \[\leadsto 0.25 \cdot \color{blue}{\left(1 \cdot \left(\frac{D \cdot D}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\right)} \]
   10. Step-by-step derivation

       1. *-lft-identity 41.1%

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

       2. associate-/l* 56.7%

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

       3. unpow2 56.7%

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

       4. associate-/l* 58.9%

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

       5. unpow2 58.9%

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

   11. Simplified 58.9%

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

       1. associate-*l/ 58.3%

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

       2. associate-/r/ 60.3%

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

   13. Applied egg-rr 60.3%

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

   if 2.00000000000000011e-249 < (*.f64 d d) < 8.50000000000000014e-7

   1. Initial program 48.9%

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

      1. times-frac 45.8%

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

      2. fma-def 45.7%

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

      3. times-frac 45.7%

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

      4. difference-of-squares 45.7%

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

   3. Simplified 48.9%

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

      1. fma-udef 49.0%

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

      2. pow2 49.0%

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

      3. associate-*l* 49.0%

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

      4. div-inv 49.0%

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

      5. clear-num 48.9%

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

      6. associate-*r/ 45.7%

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

      7. *-commutative 45.7%

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

   5. Applied egg-rr 51.8%

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

      1. distribute-lft-in 51.6%

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

      2. *-commutative 51.6%

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

      3. associate-*l/ 51.6%

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

      4. *-commutative 51.6%

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

      5. associate-*r/ 51.6%

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

      6. associate-/l* 51.7%

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

      7. associate-*l/ 51.7%

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

   7. Applied egg-rr 51.7%

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

      1. distribute-lft-out 51.9%

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

      2. times-frac 51.8%

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

      3. associate-/r/ 51.8%

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

      4. associate-*l/ 51.8%

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

      5. associate-*r/ 51.8%

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

      6. times-frac 52.0%

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

   9. Simplified 52.0%

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

   10. Taylor expanded in c0 around inf 49.2%

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

       1. times-frac 52.3%

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

       2. unpow2 52.3%

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

       3. unpow2 52.3%

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

       4. times-frac 58.0%

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

       5. unpow2 58.0%

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

   12. Simplified 58.0%

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

       1. unpow2 43.3%

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

   14. Applied egg-rr 58.0%

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

   if 8.50000000000000014e-7 < (*.f64 d d) < 9.9999999999999998e294

   1. Initial program 26.8%

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

   2. Taylor expanded in c0 around -inf 9.4%

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

      1. fma-def 9.4%

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

      2. times-frac 10.8%

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

      3. *-commutative 10.8%

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

      4. unpow2 10.8%

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

      5. unpow2 10.8%

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

      6. unpow2 10.8%

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

      7. associate-*r* 10.8%

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

   4. Simplified 35.9%

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

   5. Taylor expanded in c0 around 0 53.1%

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

      1. unpow2 53.1%

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

      2. unpow2 53.1%

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

      3. unpow2 53.1%

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

   7. Simplified 53.1%

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

      1. div-inv 53.1%

         \[\leadsto 0.25 \cdot \color{blue}{\left(\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{1}{d \cdot d}\right)} \]

      2. associate-*l* 56.3%

         \[\leadsto 0.25 \cdot \left(\color{blue}{\left(D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)} \cdot \frac{1}{d \cdot d}\right) \]

   9. Applied egg-rr 56.3%

      \[\leadsto 0.25 \cdot \color{blue}{\left(\left(D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right) \cdot \frac{1}{d \cdot d}\right)} \]

   if 9.9999999999999998e294 < (*.f64 d d)

   1. Initial program 26.9%

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

      1. times-frac 27.8%

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

      2. fma-def 27.8%

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

      3. times-frac 27.8%

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

      4. difference-of-squares 36.5%

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

   3. Simplified 36.5%

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

      1. fma-udef 36.5%

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

      2. pow2 36.5%

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

      3. associate-*l* 36.5%

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

      4. div-inv 36.5%

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

      5. clear-num 36.5%

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

      6. associate-*r/ 36.5%

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

      7. *-commutative 36.5%

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

   5. Applied egg-rr 43.3%

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

      1. distribute-lft-in 43.3%

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

      2. *-commutative 43.3%

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

      3. associate-*l/ 42.4%

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

      4. *-commutative 42.4%

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

      5. associate-*r/ 40.5%

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

      6. associate-/l* 42.4%

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

      7. associate-*l/ 42.5%

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

   7. Applied egg-rr 42.5%

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

      1. distribute-lft-out 42.5%

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

      2. times-frac 41.5%

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

      3. associate-/r/ 41.5%

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

      4. associate-*l/ 40.5%

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

      5. associate-*r/ 41.5%

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

      6. times-frac 41.5%

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

   9. Simplified 41.5%

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

   10. Taylor expanded in c0 around inf 36.2%

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

       1. times-frac 37.2%

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

       2. unpow2 37.2%

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

       3. unpow2 37.2%

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

       4. times-frac 49.6%

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

       5. unpow2 49.6%

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

   12. Simplified 49.6%

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

       1. associate-*r/ 48.8%

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

   14. Applied egg-rr 48.8%

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

       1. associate-*r/ 49.6%

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

       2. associate-/r* 49.7%

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

   16. Simplified 49.7%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \cdot d \leq 2 \cdot 10^{-249}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{d}\right)}{\frac{d}{D}}\\ \mathbf{elif}\;d \cdot d \leq 8.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\right)\\ \mathbf{elif}\;d \cdot d \leq 10^{+295}:\\ \;\;\;\;0.25 \cdot \left(\left(D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right) \cdot \frac{1}{d \cdot d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left({\left(\frac{d}{D}\right)}^{2} \cdot \frac{\frac{c0}{w}}{h}\right)\right)\\ \end{array} \]

Alternative 3: 44.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{d}{D}\right)}^{2}\\ \mathbf{if}\;d \cdot d \leq 2 \cdot 10^{-249}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{d}\right)}{\frac{d}{D}}\\ \mathbf{elif}\;d \cdot d \leq 8.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{c0 \cdot \left(2 \cdot \left(t_0 \cdot \frac{c0}{w \cdot h}\right)\right)}{2 \cdot w}\\ \mathbf{elif}\;d \cdot d \leq 10^{+295}:\\ \;\;\;\;0.25 \cdot \left(\left(D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right) \cdot \frac{1}{d \cdot d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(t_0 \cdot \frac{\frac{c0}{w}}{h}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (pow (/ d D) 2.0)))
   (if (<= (* d d) 2e-249)
     (* 0.25 (/ (* D (* (* M M) (/ h d))) (/ d D)))
     (if (<= (* d d) 8.5e-7)
       (/ (* c0 (* 2.0 (* t_0 (/ c0 (* w h))))) (* 2.0 w))
       (if (<= (* d d) 1e+295)
         (* 0.25 (* (* D (* D (* h (* M M)))) (/ 1.0 (* d d))))
         (* (/ c0 (* 2.0 w)) (* 2.0 (* t_0 (/ (/ c0 w) h)))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = pow((d / D), 2.0);
	double tmp;
	if ((d * d) <= 2e-249) {
		tmp = 0.25 * ((D * ((M * M) * (h / d))) / (d / D));
	} else if ((d * d) <= 8.5e-7) {
		tmp = (c0 * (2.0 * (t_0 * (c0 / (w * h))))) / (2.0 * w);
	} else if ((d * d) <= 1e+295) {
		tmp = 0.25 * ((D * (D * (h * (M * M)))) * (1.0 / (d * d)));
	} else {
		tmp = (c0 / (2.0 * w)) * (2.0 * (t_0 * ((c0 / w) / h)));
	}
	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) :: tmp
    t_0 = (d_1 / d) ** 2.0d0
    if ((d_1 * d_1) <= 2d-249) then
        tmp = 0.25d0 * ((d * ((m * m) * (h / d_1))) / (d_1 / d))
    else if ((d_1 * d_1) <= 8.5d-7) then
        tmp = (c0 * (2.0d0 * (t_0 * (c0 / (w * h))))) / (2.0d0 * w)
    else if ((d_1 * d_1) <= 1d+295) then
        tmp = 0.25d0 * ((d * (d * (h * (m * m)))) * (1.0d0 / (d_1 * d_1)))
    else
        tmp = (c0 / (2.0d0 * w)) * (2.0d0 * (t_0 * ((c0 / w) / h)))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = Math.pow((d / D), 2.0);
	double tmp;
	if ((d * d) <= 2e-249) {
		tmp = 0.25 * ((D * ((M * M) * (h / d))) / (d / D));
	} else if ((d * d) <= 8.5e-7) {
		tmp = (c0 * (2.0 * (t_0 * (c0 / (w * h))))) / (2.0 * w);
	} else if ((d * d) <= 1e+295) {
		tmp = 0.25 * ((D * (D * (h * (M * M)))) * (1.0 / (d * d)));
	} else {
		tmp = (c0 / (2.0 * w)) * (2.0 * (t_0 * ((c0 / w) / h)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = math.pow((d / D), 2.0)
	tmp = 0
	if (d * d) <= 2e-249:
		tmp = 0.25 * ((D * ((M * M) * (h / d))) / (d / D))
	elif (d * d) <= 8.5e-7:
		tmp = (c0 * (2.0 * (t_0 * (c0 / (w * h))))) / (2.0 * w)
	elif (d * d) <= 1e+295:
		tmp = 0.25 * ((D * (D * (h * (M * M)))) * (1.0 / (d * d)))
	else:
		tmp = (c0 / (2.0 * w)) * (2.0 * (t_0 * ((c0 / w) / h)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(d / D) ^ 2.0
	tmp = 0.0
	if (Float64(d * d) <= 2e-249)
		tmp = Float64(0.25 * Float64(Float64(D * Float64(Float64(M * M) * Float64(h / d))) / Float64(d / D)));
	elseif (Float64(d * d) <= 8.5e-7)
		tmp = Float64(Float64(c0 * Float64(2.0 * Float64(t_0 * Float64(c0 / Float64(w * h))))) / Float64(2.0 * w));
	elseif (Float64(d * d) <= 1e+295)
		tmp = Float64(0.25 * Float64(Float64(D * Float64(D * Float64(h * Float64(M * M)))) * Float64(1.0 / Float64(d * d))));
	else
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(2.0 * Float64(t_0 * Float64(Float64(c0 / w) / h))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (d / D) ^ 2.0;
	tmp = 0.0;
	if ((d * d) <= 2e-249)
		tmp = 0.25 * ((D * ((M * M) * (h / d))) / (d / D));
	elseif ((d * d) <= 8.5e-7)
		tmp = (c0 * (2.0 * (t_0 * (c0 / (w * h))))) / (2.0 * w);
	elseif ((d * d) <= 1e+295)
		tmp = 0.25 * ((D * (D * (h * (M * M)))) * (1.0 / (d * d)));
	else
		tmp = (c0 / (2.0 * w)) * (2.0 * (t_0 * ((c0 / w) / h)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(d * d), $MachinePrecision], 2e-249], N[(0.25 * N[(N[(D * N[(N[(M * M), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(d * d), $MachinePrecision], 8.5e-7], N[(N[(c0 * N[(2.0 * N[(t$95$0 * N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(d * d), $MachinePrecision], 1e+295], N[(0.25 * N[(N[(D * N[(D * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(t$95$0 * N[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 d d) < 2.00000000000000011e-249

   1. Initial program 12.5%

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

   2. Taylor expanded in c0 around -inf 2.7%

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

      1. fma-def 2.7%

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

      2. times-frac 2.7%

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

      3. *-commutative 2.7%

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

      4. unpow2 2.7%

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

      5. unpow2 2.7%

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

      6. unpow2 2.7%

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

      7. associate-*r* 2.7%

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

   4. Simplified 12.9%

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

   5. Taylor expanded in c0 around 0 17.5%

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

      1. unpow2 17.5%

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

      2. unpow2 17.5%

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

      3. unpow2 17.5%

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

   7. Simplified 17.5%

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

      1. *-un-lft-identity 17.5%

         \[\leadsto 0.25 \cdot \color{blue}{\left(1 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}\right)} \]

      2. times-frac 41.1%

         \[\leadsto 0.25 \cdot \left(1 \cdot \color{blue}{\left(\frac{D \cdot D}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)}\right) \]

   9. Applied egg-rr 41.1%

      \[\leadsto 0.25 \cdot \color{blue}{\left(1 \cdot \left(\frac{D \cdot D}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\right)} \]
   10. Step-by-step derivation

       1. *-lft-identity 41.1%

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

       2. associate-/l* 56.7%

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

       3. unpow2 56.7%

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

       4. associate-/l* 58.9%

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

       5. unpow2 58.9%

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

   11. Simplified 58.9%

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

       1. associate-*l/ 58.3%

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

       2. associate-/r/ 60.3%

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

   13. Applied egg-rr 60.3%

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

   if 2.00000000000000011e-249 < (*.f64 d d) < 8.50000000000000014e-7

   1. Initial program 48.9%

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

      1. times-frac 45.8%

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

      2. fma-def 45.7%

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

      3. times-frac 45.7%

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

      4. difference-of-squares 45.7%

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

   3. Simplified 48.9%

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

      1. fma-udef 49.0%

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

      2. pow2 49.0%

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

      3. associate-*l* 49.0%

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

      4. div-inv 49.0%

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

      5. clear-num 48.9%

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

      6. associate-*r/ 45.7%

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

      7. *-commutative 45.7%

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

   5. Applied egg-rr 51.8%

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

      1. distribute-lft-in 51.6%

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

      2. *-commutative 51.6%

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

      3. associate-*l/ 51.6%

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

      4. *-commutative 51.6%

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

      5. associate-*r/ 51.6%

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

      6. associate-/l* 51.7%

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

      7. associate-*l/ 51.7%

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

   7. Applied egg-rr 51.7%

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

      1. distribute-lft-out 51.9%

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

      2. times-frac 51.8%

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

      3. associate-/r/ 51.8%

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

      4. associate-*l/ 51.8%

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

      5. associate-*r/ 51.8%

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

      6. times-frac 52.0%

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

   9. Simplified 52.0%

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

   10. Taylor expanded in c0 around inf 49.2%

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

       1. times-frac 52.3%

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

       2. unpow2 52.3%

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

       3. unpow2 52.3%

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

       4. times-frac 58.0%

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

       5. unpow2 58.0%

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

   12. Simplified 58.0%

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

       1. associate-*l/ 58.1%

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

       2. *-commutative 58.1%

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

   14. Applied egg-rr 58.1%

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

   if 8.50000000000000014e-7 < (*.f64 d d) < 9.9999999999999998e294

   1. Initial program 26.8%

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

   2. Taylor expanded in c0 around -inf 9.4%

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

      1. fma-def 9.4%

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

      2. times-frac 10.8%

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

      3. *-commutative 10.8%

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

      4. unpow2 10.8%

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

      5. unpow2 10.8%

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

      6. unpow2 10.8%

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

      7. associate-*r* 10.8%

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

   4. Simplified 35.9%

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

   5. Taylor expanded in c0 around 0 53.1%

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

      1. unpow2 53.1%

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

      2. unpow2 53.1%

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

      3. unpow2 53.1%

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

   7. Simplified 53.1%

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

      1. div-inv 53.1%

         \[\leadsto 0.25 \cdot \color{blue}{\left(\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{1}{d \cdot d}\right)} \]

      2. associate-*l* 56.3%

         \[\leadsto 0.25 \cdot \left(\color{blue}{\left(D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)} \cdot \frac{1}{d \cdot d}\right) \]

   9. Applied egg-rr 56.3%

      \[\leadsto 0.25 \cdot \color{blue}{\left(\left(D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right) \cdot \frac{1}{d \cdot d}\right)} \]

   if 9.9999999999999998e294 < (*.f64 d d)

   1. Initial program 26.9%

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

      1. times-frac 27.8%

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

      2. fma-def 27.8%

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

      3. times-frac 27.8%

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

      4. difference-of-squares 36.5%

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

   3. Simplified 36.5%

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

      1. fma-udef 36.5%

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

      2. pow2 36.5%

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

      3. associate-*l* 36.5%

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

      4. div-inv 36.5%

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

      5. clear-num 36.5%

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

      6. associate-*r/ 36.5%

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

      7. *-commutative 36.5%

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

   5. Applied egg-rr 43.3%

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

      1. distribute-lft-in 43.3%

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

      2. *-commutative 43.3%

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

      3. associate-*l/ 42.4%

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

      4. *-commutative 42.4%

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

      5. associate-*r/ 40.5%

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

      6. associate-/l* 42.4%

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

      7. associate-*l/ 42.5%

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

   7. Applied egg-rr 42.5%

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

      1. distribute-lft-out 42.5%

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

      2. times-frac 41.5%

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

      3. associate-/r/ 41.5%

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

      4. associate-*l/ 40.5%

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

      5. associate-*r/ 41.5%

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

      6. times-frac 41.5%

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

   9. Simplified 41.5%

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

   10. Taylor expanded in c0 around inf 36.2%

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

       1. times-frac 37.2%

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

       2. unpow2 37.2%

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

       3. unpow2 37.2%

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

       4. times-frac 49.6%

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

       5. unpow2 49.6%

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

   12. Simplified 49.6%

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

       1. associate-*r/ 48.8%

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

   14. Applied egg-rr 48.8%

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

       1. associate-*r/ 49.6%

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

       2. associate-/r* 49.7%

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

   16. Simplified 49.7%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \cdot d \leq 2 \cdot 10^{-249}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{d}\right)}{\frac{d}{D}}\\ \mathbf{elif}\;d \cdot d \leq 8.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{c0 \cdot \left(2 \cdot \left({\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h}\right)\right)}{2 \cdot w}\\ \mathbf{elif}\;d \cdot d \leq 10^{+295}:\\ \;\;\;\;0.25 \cdot \left(\left(D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right) \cdot \frac{1}{d \cdot d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left({\left(\frac{d}{D}\right)}^{2} \cdot \frac{\frac{c0}{w}}{h}\right)\right)\\ \end{array} \]

Alternative 4: 44.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{d}{D}\right)}^{2}\\ t_1 := \frac{c0}{2 \cdot w}\\ \mathbf{if}\;d \cdot d \leq 2 \cdot 10^{-249}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{d}\right)}{\frac{d}{D}}\\ \mathbf{elif}\;d \cdot d \leq 8.5 \cdot 10^{-7}:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \left(t_0 \cdot \left(c0 \cdot \frac{1}{w \cdot h}\right)\right)\right)\\ \mathbf{elif}\;d \cdot d \leq 10^{+295}:\\ \;\;\;\;0.25 \cdot \left(\left(D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right) \cdot \frac{1}{d \cdot d}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \left(t_0 \cdot \frac{\frac{c0}{w}}{h}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (pow (/ d D) 2.0)) (t_1 (/ c0 (* 2.0 w))))
   (if (<= (* d d) 2e-249)
     (* 0.25 (/ (* D (* (* M M) (/ h d))) (/ d D)))
     (if (<= (* d d) 8.5e-7)
       (* t_1 (* 2.0 (* t_0 (* c0 (/ 1.0 (* w h))))))
       (if (<= (* d d) 1e+295)
         (* 0.25 (* (* D (* D (* h (* M M)))) (/ 1.0 (* d d))))
         (* t_1 (* 2.0 (* t_0 (/ (/ c0 w) h)))))))))

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 tmp;
	if ((d * d) <= 2e-249) {
		tmp = 0.25 * ((D * ((M * M) * (h / d))) / (d / D));
	} else if ((d * d) <= 8.5e-7) {
		tmp = t_1 * (2.0 * (t_0 * (c0 * (1.0 / (w * h)))));
	} else if ((d * d) <= 1e+295) {
		tmp = 0.25 * ((D * (D * (h * (M * M)))) * (1.0 / (d * d)));
	} else {
		tmp = t_1 * (2.0 * (t_0 * ((c0 / w) / h)));
	}
	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) :: tmp
    t_0 = (d_1 / d) ** 2.0d0
    t_1 = c0 / (2.0d0 * w)
    if ((d_1 * d_1) <= 2d-249) then
        tmp = 0.25d0 * ((d * ((m * m) * (h / d_1))) / (d_1 / d))
    else if ((d_1 * d_1) <= 8.5d-7) then
        tmp = t_1 * (2.0d0 * (t_0 * (c0 * (1.0d0 / (w * h)))))
    else if ((d_1 * d_1) <= 1d+295) then
        tmp = 0.25d0 * ((d * (d * (h * (m * m)))) * (1.0d0 / (d_1 * d_1)))
    else
        tmp = t_1 * (2.0d0 * (t_0 * ((c0 / w) / h)))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = Math.pow((d / D), 2.0);
	double t_1 = c0 / (2.0 * w);
	double tmp;
	if ((d * d) <= 2e-249) {
		tmp = 0.25 * ((D * ((M * M) * (h / d))) / (d / D));
	} else if ((d * d) <= 8.5e-7) {
		tmp = t_1 * (2.0 * (t_0 * (c0 * (1.0 / (w * h)))));
	} else if ((d * d) <= 1e+295) {
		tmp = 0.25 * ((D * (D * (h * (M * M)))) * (1.0 / (d * d)));
	} else {
		tmp = t_1 * (2.0 * (t_0 * ((c0 / w) / h)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = math.pow((d / D), 2.0)
	t_1 = c0 / (2.0 * w)
	tmp = 0
	if (d * d) <= 2e-249:
		tmp = 0.25 * ((D * ((M * M) * (h / d))) / (d / D))
	elif (d * d) <= 8.5e-7:
		tmp = t_1 * (2.0 * (t_0 * (c0 * (1.0 / (w * h)))))
	elif (d * d) <= 1e+295:
		tmp = 0.25 * ((D * (D * (h * (M * M)))) * (1.0 / (d * d)))
	else:
		tmp = t_1 * (2.0 * (t_0 * ((c0 / w) / h)))
	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))
	tmp = 0.0
	if (Float64(d * d) <= 2e-249)
		tmp = Float64(0.25 * Float64(Float64(D * Float64(Float64(M * M) * Float64(h / d))) / Float64(d / D)));
	elseif (Float64(d * d) <= 8.5e-7)
		tmp = Float64(t_1 * Float64(2.0 * Float64(t_0 * Float64(c0 * Float64(1.0 / Float64(w * h))))));
	elseif (Float64(d * d) <= 1e+295)
		tmp = Float64(0.25 * Float64(Float64(D * Float64(D * Float64(h * Float64(M * M)))) * Float64(1.0 / Float64(d * d))));
	else
		tmp = Float64(t_1 * Float64(2.0 * Float64(t_0 * Float64(Float64(c0 / w) / h))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (d / D) ^ 2.0;
	t_1 = c0 / (2.0 * w);
	tmp = 0.0;
	if ((d * d) <= 2e-249)
		tmp = 0.25 * ((D * ((M * M) * (h / d))) / (d / D));
	elseif ((d * d) <= 8.5e-7)
		tmp = t_1 * (2.0 * (t_0 * (c0 * (1.0 / (w * h)))));
	elseif ((d * d) <= 1e+295)
		tmp = 0.25 * ((D * (D * (h * (M * M)))) * (1.0 / (d * d)));
	else
		tmp = t_1 * (2.0 * (t_0 * ((c0 / w) / h)));
	end
	tmp_2 = 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]}, If[LessEqual[N[(d * d), $MachinePrecision], 2e-249], N[(0.25 * N[(N[(D * N[(N[(M * M), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(d * d), $MachinePrecision], 8.5e-7], N[(t$95$1 * N[(2.0 * N[(t$95$0 * N[(c0 * N[(1.0 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(d * d), $MachinePrecision], 1e+295], N[(0.25 * N[(N[(D * N[(D * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(2.0 * N[(t$95$0 * N[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 d d) < 2.00000000000000011e-249

   1. Initial program 12.5%

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

   2. Taylor expanded in c0 around -inf 2.7%

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

      1. fma-def 2.7%

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

      2. times-frac 2.7%

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

      3. *-commutative 2.7%

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

      4. unpow2 2.7%

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

      5. unpow2 2.7%

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

      6. unpow2 2.7%

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

      7. associate-*r* 2.7%

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

   4. Simplified 12.9%

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

   5. Taylor expanded in c0 around 0 17.5%

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

      1. unpow2 17.5%

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

      2. unpow2 17.5%

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

      3. unpow2 17.5%

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

   7. Simplified 17.5%

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

      1. *-un-lft-identity 17.5%

         \[\leadsto 0.25 \cdot \color{blue}{\left(1 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}\right)} \]

      2. times-frac 41.1%

         \[\leadsto 0.25 \cdot \left(1 \cdot \color{blue}{\left(\frac{D \cdot D}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)}\right) \]

   9. Applied egg-rr 41.1%

      \[\leadsto 0.25 \cdot \color{blue}{\left(1 \cdot \left(\frac{D \cdot D}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\right)} \]
   10. Step-by-step derivation

       1. *-lft-identity 41.1%

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

       2. associate-/l* 56.7%

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

       3. unpow2 56.7%

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

       4. associate-/l* 58.9%

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

       5. unpow2 58.9%

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

   11. Simplified 58.9%

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

       1. associate-*l/ 58.3%

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

       2. associate-/r/ 60.3%

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

   13. Applied egg-rr 60.3%

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

   if 2.00000000000000011e-249 < (*.f64 d d) < 8.50000000000000014e-7

   1. Initial program 48.9%

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

      1. times-frac 45.8%

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

      2. fma-def 45.7%

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

      3. times-frac 45.7%

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

      4. difference-of-squares 45.7%

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

   3. Simplified 48.9%

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

      1. fma-udef 49.0%

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

      2. pow2 49.0%

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

      3. associate-*l* 49.0%

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

      4. div-inv 49.0%

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

      5. clear-num 48.9%

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

      6. associate-*r/ 45.7%

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

      7. *-commutative 45.7%

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

   5. Applied egg-rr 51.8%

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

      1. distribute-lft-in 51.6%

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

      2. *-commutative 51.6%

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

      3. associate-*l/ 51.6%

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

      4. *-commutative 51.6%

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

      5. associate-*r/ 51.6%

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

      6. associate-/l* 51.7%

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

      7. associate-*l/ 51.7%

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

   7. Applied egg-rr 51.7%

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

      1. distribute-lft-out 51.9%

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

      2. times-frac 51.8%

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

      3. associate-/r/ 51.8%

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

      4. associate-*l/ 51.8%

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

      5. associate-*r/ 51.8%

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

      6. times-frac 52.0%

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

   9. Simplified 52.0%

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

   10. Taylor expanded in c0 around inf 49.2%

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

       1. times-frac 52.3%

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

       2. unpow2 52.3%

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

       3. unpow2 52.3%

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

       4. times-frac 58.0%

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

       5. unpow2 58.0%

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

   12. Simplified 58.0%

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

       1. div-inv 58.2%

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

   14. Applied egg-rr 58.2%

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

   if 8.50000000000000014e-7 < (*.f64 d d) < 9.9999999999999998e294

   1. Initial program 26.8%

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

   2. Taylor expanded in c0 around -inf 9.4%

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

      1. fma-def 9.4%

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

      2. times-frac 10.8%

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

      3. *-commutative 10.8%

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

      4. unpow2 10.8%

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

      5. unpow2 10.8%

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

      6. unpow2 10.8%

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

      7. associate-*r* 10.8%

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

   4. Simplified 35.9%

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

   5. Taylor expanded in c0 around 0 53.1%

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

      1. unpow2 53.1%

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

      2. unpow2 53.1%

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

      3. unpow2 53.1%

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

   7. Simplified 53.1%

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

      1. div-inv 53.1%

         \[\leadsto 0.25 \cdot \color{blue}{\left(\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{1}{d \cdot d}\right)} \]

      2. associate-*l* 56.3%

         \[\leadsto 0.25 \cdot \left(\color{blue}{\left(D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)} \cdot \frac{1}{d \cdot d}\right) \]

   9. Applied egg-rr 56.3%

      \[\leadsto 0.25 \cdot \color{blue}{\left(\left(D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right) \cdot \frac{1}{d \cdot d}\right)} \]

   if 9.9999999999999998e294 < (*.f64 d d)

   1. Initial program 26.9%

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

      1. times-frac 27.8%

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

      2. fma-def 27.8%

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

      3. times-frac 27.8%

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

      4. difference-of-squares 36.5%

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

   3. Simplified 36.5%

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

      1. fma-udef 36.5%

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

      2. pow2 36.5%

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

      3. associate-*l* 36.5%

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

      4. div-inv 36.5%

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

      5. clear-num 36.5%

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

      6. associate-*r/ 36.5%

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

      7. *-commutative 36.5%

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

   5. Applied egg-rr 43.3%

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

      1. distribute-lft-in 43.3%

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

      2. *-commutative 43.3%

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

      3. associate-*l/ 42.4%

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

      4. *-commutative 42.4%

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

      5. associate-*r/ 40.5%

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

      6. associate-/l* 42.4%

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

      7. associate-*l/ 42.5%

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

   7. Applied egg-rr 42.5%

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

      1. distribute-lft-out 42.5%

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

      2. times-frac 41.5%

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

      3. associate-/r/ 41.5%

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

      4. associate-*l/ 40.5%

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

      5. associate-*r/ 41.5%

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

      6. times-frac 41.5%

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

   9. Simplified 41.5%

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

   10. Taylor expanded in c0 around inf 36.2%

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

       1. times-frac 37.2%

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

       2. unpow2 37.2%

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

       3. unpow2 37.2%

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

       4. times-frac 49.6%

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

       5. unpow2 49.6%

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

   12. Simplified 49.6%

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

       1. associate-*r/ 48.8%

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

   14. Applied egg-rr 48.8%

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

       1. associate-*r/ 49.6%

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

       2. associate-/r* 49.7%

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

   16. Simplified 49.7%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \cdot d \leq 2 \cdot 10^{-249}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{d}\right)}{\frac{d}{D}}\\ \mathbf{elif}\;d \cdot d \leq 8.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left({\left(\frac{d}{D}\right)}^{2} \cdot \left(c0 \cdot \frac{1}{w \cdot h}\right)\right)\right)\\ \mathbf{elif}\;d \cdot d \leq 10^{+295}:\\ \;\;\;\;0.25 \cdot \left(\left(D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right) \cdot \frac{1}{d \cdot d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left({\left(\frac{d}{D}\right)}^{2} \cdot \frac{\frac{c0}{w}}{h}\right)\right)\\ \end{array} \]

Alternative 5: 43.7% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \cdot d \leq 2 \cdot 10^{-249}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{d}\right)}{\frac{d}{D}}\\ \mathbf{elif}\;d \cdot d \leq 8.5 \cdot 10^{-7} \lor \neg \left(d \cdot d \leq 10^{+295}\right):\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right) \cdot \frac{1}{d \cdot d}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= (* d d) 2e-249)
   (* 0.25 (/ (* D (* (* M M) (/ h d))) (/ d D)))
   (if (or (<= (* d d) 8.5e-7) (not (<= (* d d) 1e+295)))
     (* (/ c0 (* 2.0 w)) (* 2.0 (* (/ c0 (* w h)) (* (/ d D) (/ d D)))))
     (* 0.25 (* (* D (* D (* h (* M M)))) (/ 1.0 (* d d)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((d * d) <= 2e-249) {
		tmp = 0.25 * ((D * ((M * M) * (h / d))) / (d / D));
	} else if (((d * d) <= 8.5e-7) || !((d * d) <= 1e+295)) {
		tmp = (c0 / (2.0 * w)) * (2.0 * ((c0 / (w * h)) * ((d / D) * (d / D))));
	} else {
		tmp = 0.25 * ((D * (D * (h * (M * M)))) * (1.0 / (d * d)));
	}
	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 ((d_1 * d_1) <= 2d-249) then
        tmp = 0.25d0 * ((d * ((m * m) * (h / d_1))) / (d_1 / d))
    else if (((d_1 * d_1) <= 8.5d-7) .or. (.not. ((d_1 * d_1) <= 1d+295))) then
        tmp = (c0 / (2.0d0 * w)) * (2.0d0 * ((c0 / (w * h)) * ((d_1 / d) * (d_1 / d))))
    else
        tmp = 0.25d0 * ((d * (d * (h * (m * m)))) * (1.0d0 / (d_1 * d_1)))
    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 ((d * d) <= 2e-249) {
		tmp = 0.25 * ((D * ((M * M) * (h / d))) / (d / D));
	} else if (((d * d) <= 8.5e-7) || !((d * d) <= 1e+295)) {
		tmp = (c0 / (2.0 * w)) * (2.0 * ((c0 / (w * h)) * ((d / D) * (d / D))));
	} else {
		tmp = 0.25 * ((D * (D * (h * (M * M)))) * (1.0 / (d * d)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (d * d) <= 2e-249:
		tmp = 0.25 * ((D * ((M * M) * (h / d))) / (d / D))
	elif ((d * d) <= 8.5e-7) or not ((d * d) <= 1e+295):
		tmp = (c0 / (2.0 * w)) * (2.0 * ((c0 / (w * h)) * ((d / D) * (d / D))))
	else:
		tmp = 0.25 * ((D * (D * (h * (M * M)))) * (1.0 / (d * d)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (Float64(d * d) <= 2e-249)
		tmp = Float64(0.25 * Float64(Float64(D * Float64(Float64(M * M) * Float64(h / d))) / Float64(d / D)));
	elseif ((Float64(d * d) <= 8.5e-7) || !(Float64(d * d) <= 1e+295))
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(2.0 * Float64(Float64(c0 / Float64(w * h)) * Float64(Float64(d / D) * Float64(d / D)))));
	else
		tmp = Float64(0.25 * Float64(Float64(D * Float64(D * Float64(h * Float64(M * M)))) * Float64(1.0 / Float64(d * d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((d * d) <= 2e-249)
		tmp = 0.25 * ((D * ((M * M) * (h / d))) / (d / D));
	elseif (((d * d) <= 8.5e-7) || ~(((d * d) <= 1e+295)))
		tmp = (c0 / (2.0 * w)) * (2.0 * ((c0 / (w * h)) * ((d / D) * (d / D))));
	else
		tmp = 0.25 * ((D * (D * (h * (M * M)))) * (1.0 / (d * d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[N[(d * d), $MachinePrecision], 2e-249], N[(0.25 * N[(N[(D * N[(N[(M * M), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(d * d), $MachinePrecision], 8.5e-7], N[Not[LessEqual[N[(d * d), $MachinePrecision], 1e+295]], $MachinePrecision]], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(D * N[(D * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 d d) < 2.00000000000000011e-249

   1. Initial program 12.5%

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

   2. Taylor expanded in c0 around -inf 2.7%

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

      1. fma-def 2.7%

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

      2. times-frac 2.7%

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

      3. *-commutative 2.7%

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

      4. unpow2 2.7%

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

      5. unpow2 2.7%

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

      6. unpow2 2.7%

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

      7. associate-*r* 2.7%

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

   4. Simplified 12.9%

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

   5. Taylor expanded in c0 around 0 17.5%

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

      1. unpow2 17.5%

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

      2. unpow2 17.5%

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

      3. unpow2 17.5%

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

   7. Simplified 17.5%

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

      1. *-un-lft-identity 17.5%

         \[\leadsto 0.25 \cdot \color{blue}{\left(1 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}\right)} \]

      2. times-frac 41.1%

         \[\leadsto 0.25 \cdot \left(1 \cdot \color{blue}{\left(\frac{D \cdot D}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)}\right) \]

   9. Applied egg-rr 41.1%

      \[\leadsto 0.25 \cdot \color{blue}{\left(1 \cdot \left(\frac{D \cdot D}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\right)} \]
   10. Step-by-step derivation

       1. *-lft-identity 41.1%

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

       2. associate-/l* 56.7%

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

       3. unpow2 56.7%

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

       4. associate-/l* 58.9%

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

       5. unpow2 58.9%

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

   11. Simplified 58.9%

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

       1. associate-*l/ 58.3%

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

       2. associate-/r/ 60.3%

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

   13. Applied egg-rr 60.3%

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

   if 2.00000000000000011e-249 < (*.f64 d d) < 8.50000000000000014e-7 or 9.9999999999999998e294 < (*.f64 d d)

   1. Initial program 32.2%

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

      1. times-frac 32.1%

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

      2. fma-def 32.1%

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

      3. times-frac 32.1%

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

      4. difference-of-squares 38.7%

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

   3. Simplified 39.5%

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

      1. fma-udef 39.5%

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

      2. pow2 39.5%

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

      3. associate-*l* 39.5%

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

      4. div-inv 39.5%

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

      5. clear-num 39.5%

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

      6. associate-*r/ 38.7%

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

      7. *-commutative 38.7%

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

   5. Applied egg-rr 45.3%

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

      1. distribute-lft-in 45.3%

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

      2. *-commutative 45.3%

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

      3. associate-*l/ 44.6%

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

      4. *-commutative 44.6%

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

      5. associate-*r/ 43.2%

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

      6. associate-/l* 44.6%

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

      7. associate-*l/ 44.7%

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

   7. Applied egg-rr 44.7%

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

      1. distribute-lft-out 44.7%

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

      2. times-frac 44.0%

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

      3. associate-/r/ 44.0%

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

      4. associate-*l/ 43.2%

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

      5. associate-*r/ 44.0%

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

      6. times-frac 44.0%

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

   9. Simplified 44.0%

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

   10. Taylor expanded in c0 around inf 39.3%

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

       1. times-frac 40.8%

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

       2. unpow2 40.8%

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

       3. unpow2 40.8%

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

       4. times-frac 51.7%

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

       5. unpow2 51.7%

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

   12. Simplified 51.7%

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

       1. unpow2 39.4%

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

   14. Applied egg-rr 51.7%

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

   if 8.50000000000000014e-7 < (*.f64 d d) < 9.9999999999999998e294

   1. Initial program 26.8%

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

   2. Taylor expanded in c0 around -inf 9.4%

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

      1. fma-def 9.4%

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

      2. times-frac 10.8%

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

      3. *-commutative 10.8%

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

      4. unpow2 10.8%

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

      5. unpow2 10.8%

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

      6. unpow2 10.8%

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

      7. associate-*r* 10.8%

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

   4. Simplified 35.9%

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

   5. Taylor expanded in c0 around 0 53.1%

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

      1. unpow2 53.1%

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

      2. unpow2 53.1%

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

      3. unpow2 53.1%

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

   7. Simplified 53.1%

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

      1. div-inv 53.1%

         \[\leadsto 0.25 \cdot \color{blue}{\left(\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{1}{d \cdot d}\right)} \]

      2. associate-*l* 56.3%

         \[\leadsto 0.25 \cdot \left(\color{blue}{\left(D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)} \cdot \frac{1}{d \cdot d}\right) \]

   9. Applied egg-rr 56.3%

      \[\leadsto 0.25 \cdot \color{blue}{\left(\left(D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right) \cdot \frac{1}{d \cdot d}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \cdot d \leq 2 \cdot 10^{-249}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{d}\right)}{\frac{d}{D}}\\ \mathbf{elif}\;d \cdot d \leq 8.5 \cdot 10^{-7} \lor \neg \left(d \cdot d \leq 10^{+295}\right):\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right) \cdot \frac{1}{d \cdot d}\right)\\ \end{array} \]

Alternative 6: 44.0% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.25 \cdot \left(D \cdot \frac{\frac{D \cdot \left(M \cdot M\right)}{\frac{d}{h}}}{d}\right)\\ t_1 := \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot c0}{h \cdot \left(w \cdot w\right)}\\ \mathbf{if}\;D \leq -1.4 \cdot 10^{+133}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;D \leq -2.6 \cdot 10^{-89}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{d \cdot \left(c0 \cdot d\right)}{D \cdot D}}{w \cdot h}\right)\\ \mathbf{elif}\;D \leq -2.1 \cdot 10^{-273}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;D \leq 1.28 \cdot 10^{-287}:\\ \;\;\;\;\frac{d \cdot d}{D \cdot D} \cdot \left(\frac{c0}{h} \cdot \frac{\frac{c0}{w}}{w}\right)\\ \mathbf{elif}\;D \leq 2.2 \cdot 10^{+73}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* 0.25 (* D (/ (/ (* D (* M M)) (/ d h)) d))))
        (t_1 (* (* (/ d D) (/ d D)) (/ (* c0 c0) (* h (* w w))))))
   (if (<= D -1.4e+133)
     t_1
     (if (<= D -2.6e-89)
       (* (/ c0 (* 2.0 w)) (* 2.0 (/ (/ (* d (* c0 d)) (* D D)) (* w h))))
       (if (<= D -2.1e-273)
         t_0
         (if (<= D 1.28e-287)
           (* (/ (* d d) (* D D)) (* (/ c0 h) (/ (/ c0 w) w)))
           (if (<= D 2.2e+73) t_0 t_1)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = 0.25 * (D * (((D * (M * M)) / (d / h)) / d));
	double t_1 = ((d / D) * (d / D)) * ((c0 * c0) / (h * (w * w)));
	double tmp;
	if (D <= -1.4e+133) {
		tmp = t_1;
	} else if (D <= -2.6e-89) {
		tmp = (c0 / (2.0 * w)) * (2.0 * (((d * (c0 * d)) / (D * D)) / (w * h)));
	} else if (D <= -2.1e-273) {
		tmp = t_0;
	} else if (D <= 1.28e-287) {
		tmp = ((d * d) / (D * D)) * ((c0 / h) * ((c0 / w) / w));
	} else if (D <= 2.2e+73) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: tmp
    t_0 = 0.25d0 * (d * (((d * (m * m)) / (d_1 / h)) / d_1))
    t_1 = ((d_1 / d) * (d_1 / d)) * ((c0 * c0) / (h * (w * w)))
    if (d <= (-1.4d+133)) then
        tmp = t_1
    else if (d <= (-2.6d-89)) then
        tmp = (c0 / (2.0d0 * w)) * (2.0d0 * (((d_1 * (c0 * d_1)) / (d * d)) / (w * h)))
    else if (d <= (-2.1d-273)) then
        tmp = t_0
    else if (d <= 1.28d-287) then
        tmp = ((d_1 * d_1) / (d * d)) * ((c0 / h) * ((c0 / w) / w))
    else if (d <= 2.2d+73) then
        tmp = t_0
    else
        tmp = t_1
    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 = 0.25 * (D * (((D * (M * M)) / (d / h)) / d));
	double t_1 = ((d / D) * (d / D)) * ((c0 * c0) / (h * (w * w)));
	double tmp;
	if (D <= -1.4e+133) {
		tmp = t_1;
	} else if (D <= -2.6e-89) {
		tmp = (c0 / (2.0 * w)) * (2.0 * (((d * (c0 * d)) / (D * D)) / (w * h)));
	} else if (D <= -2.1e-273) {
		tmp = t_0;
	} else if (D <= 1.28e-287) {
		tmp = ((d * d) / (D * D)) * ((c0 / h) * ((c0 / w) / w));
	} else if (D <= 2.2e+73) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = 0.25 * (D * (((D * (M * M)) / (d / h)) / d))
	t_1 = ((d / D) * (d / D)) * ((c0 * c0) / (h * (w * w)))
	tmp = 0
	if D <= -1.4e+133:
		tmp = t_1
	elif D <= -2.6e-89:
		tmp = (c0 / (2.0 * w)) * (2.0 * (((d * (c0 * d)) / (D * D)) / (w * h)))
	elif D <= -2.1e-273:
		tmp = t_0
	elif D <= 1.28e-287:
		tmp = ((d * d) / (D * D)) * ((c0 / h) * ((c0 / w) / w))
	elif D <= 2.2e+73:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(0.25 * Float64(D * Float64(Float64(Float64(D * Float64(M * M)) / Float64(d / h)) / d)))
	t_1 = Float64(Float64(Float64(d / D) * Float64(d / D)) * Float64(Float64(c0 * c0) / Float64(h * Float64(w * w))))
	tmp = 0.0
	if (D <= -1.4e+133)
		tmp = t_1;
	elseif (D <= -2.6e-89)
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(2.0 * Float64(Float64(Float64(d * Float64(c0 * d)) / Float64(D * D)) / Float64(w * h))));
	elseif (D <= -2.1e-273)
		tmp = t_0;
	elseif (D <= 1.28e-287)
		tmp = Float64(Float64(Float64(d * d) / Float64(D * D)) * Float64(Float64(c0 / h) * Float64(Float64(c0 / w) / w)));
	elseif (D <= 2.2e+73)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = 0.25 * (D * (((D * (M * M)) / (d / h)) / d));
	t_1 = ((d / D) * (d / D)) * ((c0 * c0) / (h * (w * w)));
	tmp = 0.0;
	if (D <= -1.4e+133)
		tmp = t_1;
	elseif (D <= -2.6e-89)
		tmp = (c0 / (2.0 * w)) * (2.0 * (((d * (c0 * d)) / (D * D)) / (w * h)));
	elseif (D <= -2.1e-273)
		tmp = t_0;
	elseif (D <= 1.28e-287)
		tmp = ((d * d) / (D * D)) * ((c0 / h) * ((c0 / w) / w));
	elseif (D <= 2.2e+73)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(0.25 * N[(D * N[(N[(N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision] / N[(d / h), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] * N[(N[(c0 * c0), $MachinePrecision] / N[(h * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[D, -1.4e+133], t$95$1, If[LessEqual[D, -2.6e-89], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[(N[(d * N[(c0 * d), $MachinePrecision]), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision] / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[D, -2.1e-273], t$95$0, If[LessEqual[D, 1.28e-287], N[(N[(N[(d * d), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(c0 / h), $MachinePrecision] * N[(N[(c0 / w), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[D, 2.2e+73], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if D < -1.40000000000000008e133 or 2.2e73 < D

   1. Initial program 22.9%

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

      1. times-frac 25.3%

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

      2. fma-def 22.9%

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

      3. times-frac 23.0%

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

      4. difference-of-squares 25.5%

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

   3. Simplified 25.7%

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

      1. fma-udef 28.1%

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

      2. pow2 28.1%

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

      3. associate-*l* 25.6%

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

      4. div-inv 23.2%

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

      5. clear-num 23.2%

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

      6. associate-*r/ 25.3%

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

      7. *-commutative 25.3%

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

   5. Applied egg-rr 41.1%

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

      1. distribute-lft-in 41.1%

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

      2. *-commutative 41.1%

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

      3. associate-*l/ 38.7%

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

      4. *-commutative 38.7%

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

      5. associate-*r/ 36.1%

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

      6. associate-/l* 36.3%

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

      7. associate-*l/ 36.3%

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

   7. Applied egg-rr 36.3%

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

      1. distribute-lft-out 36.4%

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

      2. times-frac 36.2%

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

      3. associate-/r/ 33.9%

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

      4. associate-*l/ 33.6%

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

      5. associate-*r/ 33.9%

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

      6. times-frac 38.8%

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

   9. Simplified 38.8%

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

   10. Taylor expanded in c0 around inf 25.4%

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

       1. times-frac 28.2%

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

       2. unpow2 28.2%

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

       3. unpow2 28.2%

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

       4. times-frac 58.6%

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

       5. unpow2 58.6%

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

       6. unpow2 58.6%

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

       7. *-commutative 58.6%

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

       8. unpow2 58.6%

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

   12. Simplified 58.6%

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

       1. unpow2 58.6%

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

   14. Applied egg-rr 58.6%

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

   if -1.40000000000000008e133 < D < -2.5999999999999999e-89

   1. Initial program 41.3%

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

      1. times-frac 39.2%

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

      2. fma-def 39.2%

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

      3. times-frac 39.2%

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

      4. difference-of-squares 46.0%

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

   3. Simplified 43.9%

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

      1. fma-udef 43.9%

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

      2. pow2 43.9%

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

      3. associate-*l* 43.9%

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

      4. div-inv 43.9%

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

      5. clear-num 43.9%

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

      6. associate-*r/ 46.0%

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

      7. *-commutative 46.0%

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

   5. Applied egg-rr 48.4%

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

   6. Taylor expanded in c0 around inf 55.2%

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

      1. associate-/r* 55.4%

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

      2. unpow2 55.4%

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

      3. associate-*l* 57.7%

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

      4. unpow2 57.7%

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

   8. Simplified 57.7%

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

   if -2.5999999999999999e-89 < D < -2.1000000000000002e-273 or 1.28e-287 < D < 2.2e73

   1. Initial program 21.1%

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

   2. Taylor expanded in c0 around -inf 2.2%

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

      1. fma-def 2.2%

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

      2. times-frac 2.2%

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

      3. *-commutative 2.2%

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

      4. unpow2 2.2%

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

      5. unpow2 2.2%

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

      6. unpow2 2.2%

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

      7. associate-*r* 2.2%

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

   4. Simplified 25.4%

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

   5. Taylor expanded in c0 around 0 35.5%

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

      1. unpow2 35.5%

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

      2. unpow2 35.5%

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

      3. unpow2 35.5%

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

   7. Simplified 35.5%

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

      1. *-un-lft-identity 35.5%

         \[\leadsto 0.25 \cdot \color{blue}{\left(1 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}\right)} \]

      2. times-frac 41.4%

         \[\leadsto 0.25 \cdot \left(1 \cdot \color{blue}{\left(\frac{D \cdot D}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)}\right) \]

   9. Applied egg-rr 41.4%

      \[\leadsto 0.25 \cdot \color{blue}{\left(1 \cdot \left(\frac{D \cdot D}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\right)} \]
   10. Step-by-step derivation

       1. *-lft-identity 41.4%

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

       2. associate-/l* 45.8%

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

       3. unpow2 45.8%

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

       4. associate-/l* 47.2%

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

       5. unpow2 47.2%

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

   11. Simplified 47.2%

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

       1. associate-*l/ 48.5%

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

       2. associate-/r/ 49.7%

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

   13. Applied egg-rr 49.7%

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

       1. associate-/r/ 54.3%

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

       2. unpow2 54.3%

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

       3. associate-*l/ 53.3%

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

       4. *-commutative 53.3%

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

       5. associate-/l* 53.7%

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

       6. associate-*r/ 52.8%

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

       7. unpow2 52.8%

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

   15. Simplified 52.8%

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

   if -2.1000000000000002e-273 < D < 1.28e-287

   1. Initial program 50.0%

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

      1. times-frac 50.0%

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

      2. fma-def 50.0%

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

      3. times-frac 50.0%

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

      4. difference-of-squares 50.0%

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

   3. Simplified 54.5%

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

      1. fma-udef 54.5%

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

      2. pow2 54.5%

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

      3. associate-*l* 54.5%

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

      4. div-inv 54.5%

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

      5. clear-num 54.5%

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

      6. associate-*r/ 50.0%

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

      7. *-commutative 50.0%

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

   5. Applied egg-rr 59.1%

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

      1. distribute-lft-in 59.1%

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

      2. *-commutative 59.1%

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

      3. associate-*l/ 59.1%

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

      4. *-commutative 59.1%

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

      5. associate-*r/ 54.5%

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

      6. associate-/l* 59.1%

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

      7. associate-*l/ 59.1%

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

   7. Applied egg-rr 59.1%

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

      1. distribute-lft-out 59.1%

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

      2. times-frac 59.1%

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

      3. associate-/r/ 59.1%

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

      4. associate-*l/ 54.5%

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

      5. associate-*r/ 59.1%

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

      6. times-frac 59.3%

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

   9. Simplified 59.3%

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

   10. Taylor expanded in c0 around inf 50.4%

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

       1. times-frac 54.9%

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

       2. unpow2 54.9%

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

       3. unpow2 54.9%

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

       4. times-frac 59.5%

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

       5. unpow2 59.5%

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

   12. Simplified 59.5%

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

   13. Taylor expanded in c0 around 0 41.0%

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

       1. times-frac 45.5%

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

       2. unpow2 45.5%

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

       3. unpow2 45.5%

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

       4. unpow2 45.5%

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

       5. *-commutative 45.5%

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

       6. times-frac 45.8%

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

       7. unpow2 45.8%

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

       8. associate-/r* 50.4%

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

   15. Simplified 50.4%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq -1.4 \cdot 10^{+133}:\\ \;\;\;\;\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot c0}{h \cdot \left(w \cdot w\right)}\\ \mathbf{elif}\;D \leq -2.6 \cdot 10^{-89}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{d \cdot \left(c0 \cdot d\right)}{D \cdot D}}{w \cdot h}\right)\\ \mathbf{elif}\;D \leq -2.1 \cdot 10^{-273}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \frac{\frac{D \cdot \left(M \cdot M\right)}{\frac{d}{h}}}{d}\right)\\ \mathbf{elif}\;D \leq 1.28 \cdot 10^{-287}:\\ \;\;\;\;\frac{d \cdot d}{D \cdot D} \cdot \left(\frac{c0}{h} \cdot \frac{\frac{c0}{w}}{w}\right)\\ \mathbf{elif}\;D \leq 2.2 \cdot 10^{+73}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \frac{\frac{D \cdot \left(M \cdot M\right)}{\frac{d}{h}}}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot c0}{h \cdot \left(w \cdot w\right)}\\ \end{array} \]

Alternative 7: 45.5% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \cdot D \leq 5 \cdot 10^{+140}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \frac{\frac{D \cdot \left(M \cdot M\right)}{\frac{d}{h}}}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot c0}{h \cdot \left(w \cdot w\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= (* D D) 5e+140)
   (* 0.25 (* D (/ (/ (* D (* M M)) (/ d h)) d)))
   (* (* (/ d D) (/ d D)) (/ (* c0 c0) (* h (* w w))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((D * D) <= 5e+140) {
		tmp = 0.25 * (D * (((D * (M * M)) / (d / h)) / d));
	} else {
		tmp = ((d / D) * (d / D)) * ((c0 * c0) / (h * (w * w)));
	}
	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 ((d * d) <= 5d+140) then
        tmp = 0.25d0 * (d * (((d * (m * m)) / (d_1 / h)) / d_1))
    else
        tmp = ((d_1 / d) * (d_1 / d)) * ((c0 * c0) / (h * (w * w)))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((D * D) <= 5e+140) {
		tmp = 0.25 * (D * (((D * (M * M)) / (d / h)) / d));
	} else {
		tmp = ((d / D) * (d / D)) * ((c0 * c0) / (h * (w * w)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (D * D) <= 5e+140:
		tmp = 0.25 * (D * (((D * (M * M)) / (d / h)) / d))
	else:
		tmp = ((d / D) * (d / D)) * ((c0 * c0) / (h * (w * w)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (Float64(D * D) <= 5e+140)
		tmp = Float64(0.25 * Float64(D * Float64(Float64(Float64(D * Float64(M * M)) / Float64(d / h)) / d)));
	else
		tmp = Float64(Float64(Float64(d / D) * Float64(d / D)) * Float64(Float64(c0 * c0) / Float64(h * Float64(w * w))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((D * D) <= 5e+140)
		tmp = 0.25 * (D * (((D * (M * M)) / (d / h)) / d));
	else
		tmp = ((d / D) * (d / D)) * ((c0 * c0) / (h * (w * w)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[N[(D * D), $MachinePrecision], 5e+140], N[(0.25 * N[(D * N[(N[(N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision] / N[(d / h), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] * N[(N[(c0 * c0), $MachinePrecision] / N[(h * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 D D) < 5.00000000000000008e140

   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. Taylor expanded in c0 around -inf 2.6%

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

      1. fma-def 2.6%

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

      2. times-frac 3.1%

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

      3. *-commutative 3.1%

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

      4. unpow2 3.1%

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

      5. unpow2 3.1%

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

      6. unpow2 3.1%

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

      7. associate-*r* 3.1%

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

   4. Simplified 25.7%

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

   5. Taylor expanded in c0 around 0 33.6%

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

      1. unpow2 33.6%

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

      2. unpow2 33.6%

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

      3. unpow2 33.6%

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

   7. Simplified 33.6%

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

      1. *-un-lft-identity 33.6%

         \[\leadsto 0.25 \cdot \color{blue}{\left(1 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}\right)} \]

      2. times-frac 40.2%

         \[\leadsto 0.25 \cdot \left(1 \cdot \color{blue}{\left(\frac{D \cdot D}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)}\right) \]

   9. Applied egg-rr 40.2%

      \[\leadsto 0.25 \cdot \color{blue}{\left(1 \cdot \left(\frac{D \cdot D}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\right)} \]
   10. Step-by-step derivation

       1. *-lft-identity 40.2%

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

       2. associate-/l* 43.4%

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

       3. unpow2 43.4%

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

       4. associate-/l* 43.9%

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

       5. unpow2 43.9%

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

   11. Simplified 43.9%

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

       1. associate-*l/ 45.4%

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

       2. associate-/r/ 45.7%

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

   13. Applied egg-rr 45.7%

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

       1. associate-/r/ 49.1%

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

       2. unpow2 49.1%

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

       3. associate-*l/ 50.3%

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

       4. *-commutative 50.3%

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

       5. associate-/l* 48.6%

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

       6. associate-*r/ 48.0%

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

       7. unpow2 48.0%

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

   15. Simplified 48.0%

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

   if 5.00000000000000008e140 < (*.f64 D D)

   1. Initial program 25.6%

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

      1. times-frac 25.5%

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

      2. fma-def 23.5%

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

      3. times-frac 23.6%

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

      4. difference-of-squares 25.7%

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

   3. Simplified 25.9%

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

      1. fma-udef 28.0%

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

      2. pow2 28.0%

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

      3. associate-*l* 25.9%

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

      4. div-inv 23.9%

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

      5. clear-num 23.9%

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

      6. associate-*r/ 25.6%

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

      7. *-commutative 25.6%

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

   5. Applied egg-rr 40.8%

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

      1. distribute-lft-in 40.7%

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

      2. *-commutative 40.7%

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

      3. associate-*l/ 38.7%

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

      4. *-commutative 38.7%

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

      5. associate-*r/ 34.5%

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

      6. associate-/l* 36.7%

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

      7. associate-*l/ 36.7%

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

   7. Applied egg-rr 36.7%

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

      1. distribute-lft-out 36.9%

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

      2. times-frac 36.7%

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

      3. associate-/r/ 34.7%

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

      4. associate-*l/ 32.5%

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

      5. associate-*r/ 34.7%

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

      6. times-frac 38.8%

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

   9. Simplified 38.8%

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

   10. Taylor expanded in c0 around inf 25.8%

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

       1. times-frac 26.0%

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

       2. unpow2 26.0%

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

       3. unpow2 26.0%

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

       4. times-frac 51.6%

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

       5. unpow2 51.6%

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

       6. unpow2 51.6%

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

       7. *-commutative 51.6%

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

       8. unpow2 51.6%

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

   12. Simplified 51.6%

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

       1. unpow2 51.6%

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

   14. Applied egg-rr 51.6%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \cdot D \leq 5 \cdot 10^{+140}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \frac{\frac{D \cdot \left(M \cdot M\right)}{\frac{d}{h}}}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot c0}{h \cdot \left(w \cdot w\right)}\\ \end{array} \]

Alternative 8: 42.6% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ 0.25 \cdot \left(\frac{D}{\frac{d}{D}} \cdot \frac{h}{\frac{d}{M \cdot M}}\right) \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (* 0.25 (* (/ D (/ d D)) (/ h (/ d (* M M))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return 0.25 * ((D / (d / D)) * (h / (d / (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
    code = 0.25d0 * ((d / (d_1 / d)) * (h / (d_1 / (m * m))))
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return 0.25 * ((D / (d / D)) * (h / (d / (M * M))));
}

Python

def code(c0, w, h, D, d, M):
	return 0.25 * ((D / (d / D)) * (h / (d / (M * M))))

Julia

function code(c0, w, h, D, d, M)
	return Float64(0.25 * Float64(Float64(D / Float64(d / D)) * Float64(h / Float64(d / Float64(M * M)))))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = 0.25 * ((D / (d / D)) * (h / (d / (M * M))));
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(0.25 * N[(N[(D / N[(d / D), $MachinePrecision]), $MachinePrecision] * N[(h / N[(d / N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 27.4%

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

2. Taylor expanded in c0 around -inf 4.5%

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

   1. fma-def 4.5%

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

   2. times-frac 4.9%

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

   3. *-commutative 4.9%

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

   4. unpow2 4.9%

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

   5. unpow2 4.9%

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

   6. unpow2 4.9%

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

   7. associate-*r* 4.9%

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

4. Simplified 24.2%

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

5. Taylor expanded in c0 around 0 31.7%

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

   1. unpow2 31.7%

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

   2. unpow2 31.7%

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

   3. unpow2 31.7%

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

7. Simplified 31.7%

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

   1. *-un-lft-identity 31.7%

      \[\leadsto 0.25 \cdot \color{blue}{\left(1 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}\right)} \]

   2. times-frac 39.2%

      \[\leadsto 0.25 \cdot \left(1 \cdot \color{blue}{\left(\frac{D \cdot D}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)}\right) \]

9. Applied egg-rr 39.2%

   \[\leadsto 0.25 \cdot \color{blue}{\left(1 \cdot \left(\frac{D \cdot D}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\right)} \]
10. Step-by-step derivation

    1. *-lft-identity 39.2%

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

    2. associate-/l* 41.4%

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

    3. unpow2 41.4%

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

    4. associate-/l* 41.8%

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

    5. unpow2 41.8%

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

11. Simplified 41.8%

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

12. Final simplification 41.8%

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

Alternative 9: 45.3% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ 0.25 \cdot \left(D \cdot \frac{\frac{D \cdot \left(M \cdot M\right)}{\frac{d}{h}}}{d}\right) \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (* 0.25 (* D (/ (/ (* D (* M M)) (/ d h)) d))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return 0.25 * (D * (((D * (M * M)) / (d / h)) / d));
}

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 = 0.25d0 * (d * (((d * (m * m)) / (d_1 / h)) / d_1))
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return 0.25 * (D * (((D * (M * M)) / (d / h)) / d));
}

Python

def code(c0, w, h, D, d, M):
	return 0.25 * (D * (((D * (M * M)) / (d / h)) / d))

Julia

function code(c0, w, h, D, d, M)
	return Float64(0.25 * Float64(D * Float64(Float64(Float64(D * Float64(M * M)) / Float64(d / h)) / d)))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = 0.25 * (D * (((D * (M * M)) / (d / h)) / d));
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(0.25 * N[(D * N[(N[(N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision] / N[(d / h), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 27.4%

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

2. Taylor expanded in c0 around -inf 4.5%

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

   1. fma-def 4.5%

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

   2. times-frac 4.9%

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

   3. *-commutative 4.9%

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

   4. unpow2 4.9%

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

   5. unpow2 4.9%

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

   6. unpow2 4.9%

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

   7. associate-*r* 4.9%

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

4. Simplified 24.2%

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

5. Taylor expanded in c0 around 0 31.7%

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

   1. unpow2 31.7%

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

   2. unpow2 31.7%

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

   3. unpow2 31.7%

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

7. Simplified 31.7%

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

   1. *-un-lft-identity 31.7%

      \[\leadsto 0.25 \cdot \color{blue}{\left(1 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}\right)} \]

   2. times-frac 39.2%

      \[\leadsto 0.25 \cdot \left(1 \cdot \color{blue}{\left(\frac{D \cdot D}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)}\right) \]

9. Applied egg-rr 39.2%

   \[\leadsto 0.25 \cdot \color{blue}{\left(1 \cdot \left(\frac{D \cdot D}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\right)} \]
10. Step-by-step derivation

    1. *-lft-identity 39.2%

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

    2. associate-/l* 41.4%

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

    3. unpow2 41.4%

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

    4. associate-/l* 41.8%

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

    5. unpow2 41.8%

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

11. Simplified 41.8%

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

    1. associate-*l/ 43.1%

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

    2. associate-/r/ 42.6%

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

13. Applied egg-rr 42.6%

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

    1. associate-/r/ 45.4%

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

    2. unpow2 45.4%

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

    3. associate-*l/ 47.1%

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

    4. *-commutative 47.1%

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

    5. associate-/l* 44.9%

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

    6. associate-*r/ 44.8%

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

    7. unpow2 44.8%

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

15. Simplified 44.8%

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

16. Final simplification 44.8%

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

Alternative 10: 33.3% 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 27.4%

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

   1. times-frac 26.9%

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

   2. fma-def 26.1%

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

   3. associate-/r* 26.2%

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

   4. difference-of-squares 31.3%

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

3. Simplified 37.5%

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

4. Taylor expanded in c0 around -inf 3.1%

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

   1. associate-*r* 3.1%

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

   2. distribute-rgt1-in 3.1%

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

   3. metadata-eval 3.1%

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

   4. mul0-lft 27.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot \color{blue}{0}\right) \cdot c0\right) \]

   5. metadata-eval 27.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{0} \cdot c0\right) \]

   6. mul0-lft 3.1%

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

   7. metadata-eval 3.1%

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

   8. distribute-lft1-in 3.1%

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

   9. *-commutative 3.1%

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

   10. distribute-lft1-in 3.1%

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

   11. metadata-eval 3.1%

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

   12. mul0-lft 27.0%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{0}\right) \]

6. Simplified 27.0%

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]

7. Taylor expanded in c0 around 0 32.5%

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

8. Final simplification 32.5%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023176 
(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))))))