Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.7% → 66.6%

Time: 33.2s

Alternatives: 9

Speedup: 16.8×

• could not determine a ground truth

  1. c0 = 4.694285449063578e+239
  2. w = -3.2827369181517987e-292
  3. h = 6.991761049295222e-252
  4. D = 1.7968160843530603e-252
  5. d = -3.025340737774981e+279
  6. M = -2.06075861798445e+21

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 24.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 66.6% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_1 (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))))
   (if (<= t_1 INFINITY) t_1 (* (/ 0.25 d) (/ D (/ d (* D (* h (* 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));
	double t_1 = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = (0.25 / d) * (D / (d / (D * (h * (M * M)))));
	}
	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 = (c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
	double tmp;
	if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = (0.25 / d) * (D / (d / (D * (h * (M * M)))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	t_1 = (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))
	tmp = 0
	if t_1 <= math.inf:
		tmp = t_1
	else:
		tmp = (0.25 / d) * (D / (d / (D * (h * (M * M)))))
	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(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(Float64(0.25 / d) * Float64(D / Float64(d / Float64(D * Float64(h * Float64(M * M))))));
	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 = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	tmp = 0.0;
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = (0.25 / d) * (D / (d / (D * (h * (M * M)))));
	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[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[(N[(0.25 / d), $MachinePrecision] * N[(D / N[(d / N[(D * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 84.1%

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

   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 0.8%

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

      1. fma-def 0.8%

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

      2. times-frac 1.4%

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

      3. unpow2 1.4%

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

      4. unpow2 1.4%

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

      5. *-commutative 1.4%

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

      6. unpow2 1.4%

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

      7. associate-*r* 1.4%

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

   4. Simplified 42.2%

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

   5. Taylor expanded in c0 around 0 46.8%

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

      1. associate-*r/ 46.8%

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

      2. *-commutative 46.8%

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

      3. *-commutative 46.8%

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

      4. unpow2 46.8%

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

      5. unpow2 46.8%

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

      6. unpow2 46.8%

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

   7. Simplified 46.8%

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

      1. *-un-lft-identity 46.8%

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

      2. times-frac 56.6%

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

      3. associate-*l* 66.4%

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

   9. Applied egg-rr 66.4%

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

       1. *-lft-identity 66.4%

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

       2. associate-/l* 67.7%

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

   11. Simplified 67.7%

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

4. Final simplification 74.0%

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

Alternative 2: 43.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := h \cdot \left(M \cdot M\right)\\ t_1 := \left(w \cdot h\right) \cdot \left(D \cdot D\right)\\ t_2 := \frac{c0}{2 \cdot w}\\ t_3 := t_2 \cdot \left(\frac{\frac{c0}{h}}{w} \cdot {\left(\frac{d}{D}\right)}^{2} + \left(d \cdot \frac{d}{D \cdot D}\right) \cdot \frac{c0}{w \cdot h}\right)\\ t_4 := t_2 \cdot \left(2 \cdot \left(d \cdot \frac{c0 \cdot d}{t_1}\right)\right)\\ t_5 := \frac{0.25}{d} \cdot \frac{D}{\frac{d}{D \cdot t_0}}\\ \mathbf{if}\;c0 \leq -6.2 \cdot 10^{+265}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;c0 \leq -1.4 \cdot 10^{+231}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;c0 \leq -0.0018:\\ \;\;\;\;t_3\\ \mathbf{elif}\;c0 \leq -5.5 \cdot 10^{-116}:\\ \;\;\;\;\frac{c0}{w} \cdot \frac{0.5 \cdot \frac{{\left(\frac{D}{d}\right)}^{2}}{\frac{c0}{w \cdot t_0}}}{2}\\ \mathbf{elif}\;c0 \leq -5.8 \cdot 10^{-199}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;c0 \leq 9.8 \cdot 10^{-119}:\\ \;\;\;\;t_2 \cdot \left(c0 \cdot 0\right)\\ \mathbf{elif}\;c0 \leq 7.5 \cdot 10^{-97}:\\ \;\;\;\;t_2 \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot D}{d \cdot d} \cdot \left(h \cdot \left(\left(M \cdot M\right) \cdot \frac{w}{c0}\right)\right), \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{t_1}\right)\\ \mathbf{elif}\;c0 \leq 1.6 \cdot 10^{+216}:\\ \;\;\;\;t_5\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* h (* M M)))
        (t_1 (* (* w h) (* D D)))
        (t_2 (/ c0 (* 2.0 w)))
        (t_3
         (*
          t_2
          (+
           (* (/ (/ c0 h) w) (pow (/ d D) 2.0))
           (* (* d (/ d (* D D))) (/ c0 (* w h))))))
        (t_4 (* t_2 (* 2.0 (* d (/ (* c0 d) t_1)))))
        (t_5 (* (/ 0.25 d) (/ D (/ d (* D t_0))))))
   (if (<= c0 -6.2e+265)
     t_4
     (if (<= c0 -1.4e+231)
       t_5
       (if (<= c0 -0.0018)
         t_3
         (if (<= c0 -5.5e-116)
           (* (/ c0 w) (/ (* 0.5 (/ (pow (/ D d) 2.0) (/ c0 (* w t_0)))) 2.0))
           (if (<= c0 -5.8e-199)
             t_4
             (if (<= c0 9.8e-119)
               (* t_2 (* c0 0.0))
               (if (<= c0 7.5e-97)
                 (*
                  t_2
                  (fma
                   -0.5
                   (* (/ (* D D) (* d d)) (* h (* (* M M) (/ w c0))))
                   (/ (* 2.0 (* c0 (* d d))) t_1)))
                 (if (<= c0 1.6e+216) t_5 t_3))))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = h * (M * M);
	double t_1 = (w * h) * (D * D);
	double t_2 = c0 / (2.0 * w);
	double t_3 = t_2 * ((((c0 / h) / w) * pow((d / D), 2.0)) + ((d * (d / (D * D))) * (c0 / (w * h))));
	double t_4 = t_2 * (2.0 * (d * ((c0 * d) / t_1)));
	double t_5 = (0.25 / d) * (D / (d / (D * t_0)));
	double tmp;
	if (c0 <= -6.2e+265) {
		tmp = t_4;
	} else if (c0 <= -1.4e+231) {
		tmp = t_5;
	} else if (c0 <= -0.0018) {
		tmp = t_3;
	} else if (c0 <= -5.5e-116) {
		tmp = (c0 / w) * ((0.5 * (pow((D / d), 2.0) / (c0 / (w * t_0)))) / 2.0);
	} else if (c0 <= -5.8e-199) {
		tmp = t_4;
	} else if (c0 <= 9.8e-119) {
		tmp = t_2 * (c0 * 0.0);
	} else if (c0 <= 7.5e-97) {
		tmp = t_2 * fma(-0.5, (((D * D) / (d * d)) * (h * ((M * M) * (w / c0)))), ((2.0 * (c0 * (d * d))) / t_1));
	} else if (c0 <= 1.6e+216) {
		tmp = t_5;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(h * Float64(M * M))
	t_1 = Float64(Float64(w * h) * Float64(D * D))
	t_2 = Float64(c0 / Float64(2.0 * w))
	t_3 = Float64(t_2 * Float64(Float64(Float64(Float64(c0 / h) / w) * (Float64(d / D) ^ 2.0)) + Float64(Float64(d * Float64(d / Float64(D * D))) * Float64(c0 / Float64(w * h)))))
	t_4 = Float64(t_2 * Float64(2.0 * Float64(d * Float64(Float64(c0 * d) / t_1))))
	t_5 = Float64(Float64(0.25 / d) * Float64(D / Float64(d / Float64(D * t_0))))
	tmp = 0.0
	if (c0 <= -6.2e+265)
		tmp = t_4;
	elseif (c0 <= -1.4e+231)
		tmp = t_5;
	elseif (c0 <= -0.0018)
		tmp = t_3;
	elseif (c0 <= -5.5e-116)
		tmp = Float64(Float64(c0 / w) * Float64(Float64(0.5 * Float64((Float64(D / d) ^ 2.0) / Float64(c0 / Float64(w * t_0)))) / 2.0));
	elseif (c0 <= -5.8e-199)
		tmp = t_4;
	elseif (c0 <= 9.8e-119)
		tmp = Float64(t_2 * Float64(c0 * 0.0));
	elseif (c0 <= 7.5e-97)
		tmp = Float64(t_2 * fma(-0.5, Float64(Float64(Float64(D * D) / Float64(d * d)) * Float64(h * Float64(Float64(M * M) * Float64(w / c0)))), Float64(Float64(2.0 * Float64(c0 * Float64(d * d))) / t_1)));
	elseif (c0 <= 1.6e+216)
		tmp = t_5;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[(N[(N[(c0 / h), $MachinePrecision] / w), $MachinePrecision] * N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(d * N[(d / N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 * N[(2.0 * N[(d * N[(N[(c0 * d), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(0.25 / d), $MachinePrecision] * N[(D / N[(d / N[(D * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c0, -6.2e+265], t$95$4, If[LessEqual[c0, -1.4e+231], t$95$5, If[LessEqual[c0, -0.0018], t$95$3, If[LessEqual[c0, -5.5e-116], N[(N[(c0 / w), $MachinePrecision] * N[(N[(0.5 * N[(N[Power[N[(D / d), $MachinePrecision], 2.0], $MachinePrecision] / N[(c0 / N[(w * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[c0, -5.8e-199], t$95$4, If[LessEqual[c0, 9.8e-119], N[(t$95$2 * N[(c0 * 0.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[c0, 7.5e-97], N[(t$95$2 * N[(-0.5 * N[(N[(N[(D * D), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * N[(h * N[(N[(M * M), $MachinePrecision] * N[(w / c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c0, 1.6e+216], t$95$5, t$95$3]]]]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if c0 < -6.20000000000000016e265 or -5.4999999999999998e-116 < c0 < -5.8e-199

   1. Initial program 44.1%

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

      1. associate-*l* 43.9%

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

      2. difference-of-squares 57.2%

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

      3. associate-*l* 57.3%

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

      4. associate-*l* 57.3%

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

   3. Simplified 57.3%

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

   4. Taylor expanded in c0 around inf 58.0%

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

      1. *-commutative 58.0%

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

      2. associate-*r* 57.9%

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

      3. unpow2 57.9%

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

      4. associate-*r/ 57.9%

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

      5. unpow2 57.9%

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

      6. associate-*r* 68.2%

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

      7. associate-*r/ 68.2%

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

      8. unpow2 68.2%

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

      9. associate-*r* 71.4%

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

      10. *-commutative 71.4%

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

      11. unpow2 71.4%

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

   6. Simplified 71.4%

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

   if -6.20000000000000016e265 < c0 < -1.4e231 or 7.5e-97 < c0 < 1.59999999999999985e216

   1. Initial program 20.4%

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

   2. Taylor expanded in c0 around -inf 6.5%

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

      1. fma-def 6.5%

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

      2. times-frac 7.9%

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

      3. unpow2 7.9%

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

      4. unpow2 7.9%

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

      5. *-commutative 7.9%

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

      6. unpow2 7.9%

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

      7. associate-*r* 7.9%

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

   4. Simplified 44.2%

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

   5. Taylor expanded in c0 around 0 47.6%

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

      1. associate-*r/ 47.6%

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

      2. *-commutative 47.6%

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

      3. *-commutative 47.6%

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

      4. unpow2 47.6%

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

      5. unpow2 47.6%

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

      6. unpow2 47.6%

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

   7. Simplified 47.6%

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

      1. *-un-lft-identity 47.6%

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

      2. times-frac 51.0%

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

      3. associate-*l* 63.5%

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

   9. Applied egg-rr 63.5%

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

       1. *-lft-identity 63.5%

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

       2. associate-/l* 66.5%

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

   11. Simplified 66.5%

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

   if -1.4e231 < c0 < -0.0018 or 1.59999999999999985e216 < c0

   1. Initial program 42.9%

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

      1. times-frac 41.7%

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

      2. fma-def 41.7%

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

      3. times-frac 43.0%

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

      4. difference-of-squares 52.7%

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

   3. Simplified 51.4%

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

      1. fma-udef 52.6%

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

      2. associate-/l/ 52.6%

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

      3. pow2 52.6%

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

      4. associate-*l* 53.8%

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

      5. div-inv 53.8%

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

      6. clear-num 53.8%

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

      7. associate-*r/ 53.8%

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

      8. *-commutative 53.8%

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

   5. Applied egg-rr 63.9%

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

   6. Taylor expanded in c0 around inf 52.6%

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

      1. times-frac 52.8%

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

      2. unpow2 52.8%

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

      3. associate-*r/ 61.8%

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

      4. unpow2 61.8%

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

   8. Simplified 61.8%

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

   if -0.0018 < c0 < -5.4999999999999998e-116

   1. Initial program 28.3%

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

   2. Taylor expanded in c0 around -inf 8.0%

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

      1. fma-def 8.0%

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

      2. times-frac 8.0%

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

      3. unpow2 8.0%

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

      4. unpow2 8.0%

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

      5. *-commutative 8.0%

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

      6. unpow2 8.0%

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

      7. associate-*r* 8.0%

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

   4. Simplified 56.8%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{D \cdot D}{d \cdot d} \cdot \frac{w \cdot \left(h \cdot \left(M \cdot M\right)\right)}{c0}, c0 \cdot 0\right)} \]
   5. Step-by-step derivation

      1. associate-*r/ 56.8%

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

      2. times-frac 60.9%

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

   6. Applied egg-rr 60.9%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(0.5, \color{blue}{\frac{\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(w \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{c0}}, c0 \cdot 0\right) \]
   7. Step-by-step derivation

      1. associate-*l/ 60.9%

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

      2. pow2 60.9%

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

      3. mul0-rgt 60.9%

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

      4. *-commutative 60.9%

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

   8. Applied egg-rr 60.9%

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

      1. times-frac 60.9%

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

      2. fma-udef 60.9%

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

      3. +-rgt-identity 60.9%

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

      4. associate-/l* 60.9%

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

   10. Simplified 60.9%

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

   if -5.8e-199 < c0 < 9.8e-119

   1. Initial program 19.3%

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

      1. associate-*l* 19.3%

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

      2. difference-of-squares 19.3%

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

      3. associate-*l* 19.3%

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

      4. associate-*l* 19.3%

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

   3. Simplified 19.3%

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

   4. Taylor expanded in c0 around -inf 2.3%

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

      1. associate-*r* 2.3%

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

      2. distribute-rgt1-in 2.3%

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

      3. metadata-eval 2.3%

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

      4. mul0-lft 57.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot \color{blue}{0}\right) \cdot c0\right) \]

      5. metadata-eval 57.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{0} \cdot c0\right) \]

      6. mul0-lft 4.4%

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

      7. metadata-eval 4.4%

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

      8. distribute-lft1-in 4.4%

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

      9. *-commutative 4.4%

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

      10. distribute-lft1-in 4.4%

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

      11. metadata-eval 4.4%

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

      12. mul0-lft 57.4%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{0}\right) \]

   6. Simplified 57.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]

   if 9.8e-119 < c0 < 7.5e-97

   1. Initial program 81.3%

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

      1. associate-*l* 65.0%

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

      2. difference-of-squares 65.0%

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

      3. associate-*l* 63.9%

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

      4. associate-*l* 61.3%

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

   3. Simplified 61.3%

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

   5. Applied egg-rr 5.1%

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

      1. unpow2 5.1%

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

      2. associate--r- 5.6%

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

      3. +-inverses 6.0%

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

      4. unpow2 6.0%

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

      5. associate-/l/ 6.0%

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

      6. associate-/l/ 6.0%

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

   7. Simplified 6.0%

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

   8. Taylor expanded in M around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. *-commutative 99.7%

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

      3. fma-def 99.7%

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

   10. Simplified 99.7%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(-0.5, \frac{D \cdot D}{d \cdot d} \cdot \left(h \cdot \left(\left(M \cdot M\right) \cdot \frac{w}{c0}\right)\right), \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -6.2 \cdot 10^{+265}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(d \cdot \frac{c0 \cdot d}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right)\\ \mathbf{elif}\;c0 \leq -1.4 \cdot 10^{+231}:\\ \;\;\;\;\frac{0.25}{d} \cdot \frac{D}{\frac{d}{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}}\\ \mathbf{elif}\;c0 \leq -0.0018:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{h}}{w} \cdot {\left(\frac{d}{D}\right)}^{2} + \left(d \cdot \frac{d}{D \cdot D}\right) \cdot \frac{c0}{w \cdot h}\right)\\ \mathbf{elif}\;c0 \leq -5.5 \cdot 10^{-116}:\\ \;\;\;\;\frac{c0}{w} \cdot \frac{0.5 \cdot \frac{{\left(\frac{D}{d}\right)}^{2}}{\frac{c0}{w \cdot \left(h \cdot \left(M \cdot M\right)\right)}}}{2}\\ \mathbf{elif}\;c0 \leq -5.8 \cdot 10^{-199}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(d \cdot \frac{c0 \cdot d}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right)\\ \mathbf{elif}\;c0 \leq 9.8 \cdot 10^{-119}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(c0 \cdot 0\right)\\ \mathbf{elif}\;c0 \leq 7.5 \cdot 10^{-97}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot D}{d \cdot d} \cdot \left(h \cdot \left(\left(M \cdot M\right) \cdot \frac{w}{c0}\right)\right), \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\\ \mathbf{elif}\;c0 \leq 1.6 \cdot 10^{+216}:\\ \;\;\;\;\frac{0.25}{d} \cdot \frac{D}{\frac{d}{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{h}}{w} \cdot {\left(\frac{d}{D}\right)}^{2} + \left(d \cdot \frac{d}{D \cdot D}\right) \cdot \frac{c0}{w \cdot h}\right)\\ \end{array} \]

Alternative 3: 43.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := h \cdot \left(M \cdot M\right)\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := t_1 \cdot \left(\frac{\frac{c0}{h}}{w} \cdot {\left(\frac{d}{D}\right)}^{2} + \left(d \cdot \frac{d}{D \cdot D}\right) \cdot \frac{c0}{w \cdot h}\right)\\ t_3 := t_1 \cdot \left(2 \cdot \left(d \cdot \frac{c0 \cdot d}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right)\\ t_4 := \frac{0.25}{d} \cdot \frac{D}{\frac{d}{D \cdot t_0}}\\ \mathbf{if}\;c0 \leq -6.5 \cdot 10^{+271}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;c0 \leq -2.05 \cdot 10^{+228}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;c0 \leq -0.000135:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c0 \leq -3.1 \cdot 10^{-116}:\\ \;\;\;\;\frac{c0}{w} \cdot \frac{0.5 \cdot \frac{{\left(\frac{D}{d}\right)}^{2}}{\frac{c0}{w \cdot t_0}}}{2}\\ \mathbf{elif}\;c0 \leq -4.6 \cdot 10^{-194}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;c0 \leq 1.3 \cdot 10^{-123}:\\ \;\;\;\;t_1 \cdot \left(c0 \cdot 0\right)\\ \mathbf{elif}\;c0 \leq 8.5 \cdot 10^{-97} \lor \neg \left(c0 \leq 4.8 \cdot 10^{+216}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* h (* M M)))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2
         (*
          t_1
          (+
           (* (/ (/ c0 h) w) (pow (/ d D) 2.0))
           (* (* d (/ d (* D D))) (/ c0 (* w h))))))
        (t_3 (* t_1 (* 2.0 (* d (/ (* c0 d) (* (* w h) (* D D)))))))
        (t_4 (* (/ 0.25 d) (/ D (/ d (* D t_0))))))
   (if (<= c0 -6.5e+271)
     t_3
     (if (<= c0 -2.05e+228)
       t_4
       (if (<= c0 -0.000135)
         t_2
         (if (<= c0 -3.1e-116)
           (* (/ c0 w) (/ (* 0.5 (/ (pow (/ D d) 2.0) (/ c0 (* w t_0)))) 2.0))
           (if (<= c0 -4.6e-194)
             t_3
             (if (<= c0 1.3e-123)
               (* t_1 (* c0 0.0))
               (if (or (<= c0 8.5e-97) (not (<= c0 4.8e+216))) t_2 t_4)))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = h * (M * M);
	double t_1 = c0 / (2.0 * w);
	double t_2 = t_1 * ((((c0 / h) / w) * pow((d / D), 2.0)) + ((d * (d / (D * D))) * (c0 / (w * h))));
	double t_3 = t_1 * (2.0 * (d * ((c0 * d) / ((w * h) * (D * D)))));
	double t_4 = (0.25 / d) * (D / (d / (D * t_0)));
	double tmp;
	if (c0 <= -6.5e+271) {
		tmp = t_3;
	} else if (c0 <= -2.05e+228) {
		tmp = t_4;
	} else if (c0 <= -0.000135) {
		tmp = t_2;
	} else if (c0 <= -3.1e-116) {
		tmp = (c0 / w) * ((0.5 * (pow((D / d), 2.0) / (c0 / (w * t_0)))) / 2.0);
	} else if (c0 <= -4.6e-194) {
		tmp = t_3;
	} else if (c0 <= 1.3e-123) {
		tmp = t_1 * (c0 * 0.0);
	} else if ((c0 <= 8.5e-97) || !(c0 <= 4.8e+216)) {
		tmp = t_2;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = h * (m * m)
    t_1 = c0 / (2.0d0 * w)
    t_2 = t_1 * ((((c0 / h) / w) * ((d_1 / d) ** 2.0d0)) + ((d_1 * (d_1 / (d * d))) * (c0 / (w * h))))
    t_3 = t_1 * (2.0d0 * (d_1 * ((c0 * d_1) / ((w * h) * (d * d)))))
    t_4 = (0.25d0 / d_1) * (d / (d_1 / (d * t_0)))
    if (c0 <= (-6.5d+271)) then
        tmp = t_3
    else if (c0 <= (-2.05d+228)) then
        tmp = t_4
    else if (c0 <= (-0.000135d0)) then
        tmp = t_2
    else if (c0 <= (-3.1d-116)) then
        tmp = (c0 / w) * ((0.5d0 * (((d / d_1) ** 2.0d0) / (c0 / (w * t_0)))) / 2.0d0)
    else if (c0 <= (-4.6d-194)) then
        tmp = t_3
    else if (c0 <= 1.3d-123) then
        tmp = t_1 * (c0 * 0.0d0)
    else if ((c0 <= 8.5d-97) .or. (.not. (c0 <= 4.8d+216))) then
        tmp = t_2
    else
        tmp = t_4
    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 = h * (M * M);
	double t_1 = c0 / (2.0 * w);
	double t_2 = t_1 * ((((c0 / h) / w) * Math.pow((d / D), 2.0)) + ((d * (d / (D * D))) * (c0 / (w * h))));
	double t_3 = t_1 * (2.0 * (d * ((c0 * d) / ((w * h) * (D * D)))));
	double t_4 = (0.25 / d) * (D / (d / (D * t_0)));
	double tmp;
	if (c0 <= -6.5e+271) {
		tmp = t_3;
	} else if (c0 <= -2.05e+228) {
		tmp = t_4;
	} else if (c0 <= -0.000135) {
		tmp = t_2;
	} else if (c0 <= -3.1e-116) {
		tmp = (c0 / w) * ((0.5 * (Math.pow((D / d), 2.0) / (c0 / (w * t_0)))) / 2.0);
	} else if (c0 <= -4.6e-194) {
		tmp = t_3;
	} else if (c0 <= 1.3e-123) {
		tmp = t_1 * (c0 * 0.0);
	} else if ((c0 <= 8.5e-97) || !(c0 <= 4.8e+216)) {
		tmp = t_2;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = h * (M * M)
	t_1 = c0 / (2.0 * w)
	t_2 = t_1 * ((((c0 / h) / w) * math.pow((d / D), 2.0)) + ((d * (d / (D * D))) * (c0 / (w * h))))
	t_3 = t_1 * (2.0 * (d * ((c0 * d) / ((w * h) * (D * D)))))
	t_4 = (0.25 / d) * (D / (d / (D * t_0)))
	tmp = 0
	if c0 <= -6.5e+271:
		tmp = t_3
	elif c0 <= -2.05e+228:
		tmp = t_4
	elif c0 <= -0.000135:
		tmp = t_2
	elif c0 <= -3.1e-116:
		tmp = (c0 / w) * ((0.5 * (math.pow((D / d), 2.0) / (c0 / (w * t_0)))) / 2.0)
	elif c0 <= -4.6e-194:
		tmp = t_3
	elif c0 <= 1.3e-123:
		tmp = t_1 * (c0 * 0.0)
	elif (c0 <= 8.5e-97) or not (c0 <= 4.8e+216):
		tmp = t_2
	else:
		tmp = t_4
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(h * Float64(M * M))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(t_1 * Float64(Float64(Float64(Float64(c0 / h) / w) * (Float64(d / D) ^ 2.0)) + Float64(Float64(d * Float64(d / Float64(D * D))) * Float64(c0 / Float64(w * h)))))
	t_3 = Float64(t_1 * Float64(2.0 * Float64(d * Float64(Float64(c0 * d) / Float64(Float64(w * h) * Float64(D * D))))))
	t_4 = Float64(Float64(0.25 / d) * Float64(D / Float64(d / Float64(D * t_0))))
	tmp = 0.0
	if (c0 <= -6.5e+271)
		tmp = t_3;
	elseif (c0 <= -2.05e+228)
		tmp = t_4;
	elseif (c0 <= -0.000135)
		tmp = t_2;
	elseif (c0 <= -3.1e-116)
		tmp = Float64(Float64(c0 / w) * Float64(Float64(0.5 * Float64((Float64(D / d) ^ 2.0) / Float64(c0 / Float64(w * t_0)))) / 2.0));
	elseif (c0 <= -4.6e-194)
		tmp = t_3;
	elseif (c0 <= 1.3e-123)
		tmp = Float64(t_1 * Float64(c0 * 0.0));
	elseif ((c0 <= 8.5e-97) || !(c0 <= 4.8e+216))
		tmp = t_2;
	else
		tmp = t_4;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = h * (M * M);
	t_1 = c0 / (2.0 * w);
	t_2 = t_1 * ((((c0 / h) / w) * ((d / D) ^ 2.0)) + ((d * (d / (D * D))) * (c0 / (w * h))));
	t_3 = t_1 * (2.0 * (d * ((c0 * d) / ((w * h) * (D * D)))));
	t_4 = (0.25 / d) * (D / (d / (D * t_0)));
	tmp = 0.0;
	if (c0 <= -6.5e+271)
		tmp = t_3;
	elseif (c0 <= -2.05e+228)
		tmp = t_4;
	elseif (c0 <= -0.000135)
		tmp = t_2;
	elseif (c0 <= -3.1e-116)
		tmp = (c0 / w) * ((0.5 * (((D / d) ^ 2.0) / (c0 / (w * t_0)))) / 2.0);
	elseif (c0 <= -4.6e-194)
		tmp = t_3;
	elseif (c0 <= 1.3e-123)
		tmp = t_1 * (c0 * 0.0);
	elseif ((c0 <= 8.5e-97) || ~((c0 <= 4.8e+216)))
		tmp = t_2;
	else
		tmp = t_4;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(N[(N[(c0 / h), $MachinePrecision] / w), $MachinePrecision] * N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(d * N[(d / N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[(2.0 * N[(d * N[(N[(c0 * d), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(0.25 / d), $MachinePrecision] * N[(D / N[(d / N[(D * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c0, -6.5e+271], t$95$3, If[LessEqual[c0, -2.05e+228], t$95$4, If[LessEqual[c0, -0.000135], t$95$2, If[LessEqual[c0, -3.1e-116], N[(N[(c0 / w), $MachinePrecision] * N[(N[(0.5 * N[(N[Power[N[(D / d), $MachinePrecision], 2.0], $MachinePrecision] / N[(c0 / N[(w * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[c0, -4.6e-194], t$95$3, If[LessEqual[c0, 1.3e-123], N[(t$95$1 * N[(c0 * 0.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[c0, 8.5e-97], N[Not[LessEqual[c0, 4.8e+216]], $MachinePrecision]], t$95$2, t$95$4]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if c0 < -6.49999999999999998e271 or -3.10000000000000018e-116 < c0 < -4.60000000000000005e-194

   1. Initial program 44.1%

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

      1. associate-*l* 43.9%

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

      2. difference-of-squares 57.2%

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

      3. associate-*l* 57.3%

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

      4. associate-*l* 57.3%

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

   3. Simplified 57.3%

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

   4. Taylor expanded in c0 around inf 58.0%

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

      1. *-commutative 58.0%

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

      2. associate-*r* 57.9%

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

      3. unpow2 57.9%

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

      4. associate-*r/ 57.9%

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

      5. unpow2 57.9%

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

      6. associate-*r* 68.2%

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

      7. associate-*r/ 68.2%

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

      8. unpow2 68.2%

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

      9. associate-*r* 71.4%

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

      10. *-commutative 71.4%

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

      11. unpow2 71.4%

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

   6. Simplified 71.4%

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

   if -6.49999999999999998e271 < c0 < -2.05e228 or 8.5000000000000002e-97 < c0 < 4.7999999999999999e216

   1. Initial program 20.4%

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

   2. Taylor expanded in c0 around -inf 6.5%

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

      1. fma-def 6.5%

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

      2. times-frac 7.9%

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

      3. unpow2 7.9%

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

      4. unpow2 7.9%

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

      5. *-commutative 7.9%

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

      6. unpow2 7.9%

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

      7. associate-*r* 7.9%

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

   4. Simplified 44.2%

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

   5. Taylor expanded in c0 around 0 47.6%

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

      1. associate-*r/ 47.6%

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

      2. *-commutative 47.6%

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

      3. *-commutative 47.6%

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

      4. unpow2 47.6%

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

      5. unpow2 47.6%

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

      6. unpow2 47.6%

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

   7. Simplified 47.6%

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

      1. *-un-lft-identity 47.6%

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

      2. times-frac 51.0%

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

      3. associate-*l* 63.5%

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

   9. Applied egg-rr 63.5%

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

       1. *-lft-identity 63.5%

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

       2. associate-/l* 66.5%

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

   11. Simplified 66.5%

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

   if -2.05e228 < c0 < -1.35000000000000002e-4 or 1.29999999999999998e-123 < c0 < 8.5000000000000002e-97 or 4.7999999999999999e216 < c0

   1. Initial program 45.1%

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

      1. times-frac 44.0%

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

      2. fma-def 44.0%

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

      3. times-frac 45.2%

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

      4. difference-of-squares 54.3%

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

   3. Simplified 53.1%

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

      1. fma-udef 54.2%

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

      2. associate-/l/ 54.2%

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

      3. pow2 54.2%

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

      4. associate-*l* 55.3%

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

      5. div-inv 55.3%

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

      6. clear-num 55.3%

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

      7. associate-*r/ 55.4%

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

      8. *-commutative 55.4%

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

   5. Applied egg-rr 64.9%

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

   6. Taylor expanded in c0 around inf 55.2%

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

      1. times-frac 55.5%

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

      2. unpow2 55.5%

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

      3. associate-*r/ 63.9%

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

      4. unpow2 63.9%

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

   8. Simplified 63.9%

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

   if -1.35000000000000002e-4 < c0 < -3.10000000000000018e-116

   1. Initial program 28.3%

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

   2. Taylor expanded in c0 around -inf 8.0%

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

      1. fma-def 8.0%

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

      2. times-frac 8.0%

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

      3. unpow2 8.0%

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

      4. unpow2 8.0%

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

      5. *-commutative 8.0%

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

      6. unpow2 8.0%

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

      7. associate-*r* 8.0%

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

   4. Simplified 56.8%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{D \cdot D}{d \cdot d} \cdot \frac{w \cdot \left(h \cdot \left(M \cdot M\right)\right)}{c0}, c0 \cdot 0\right)} \]
   5. Step-by-step derivation

      1. associate-*r/ 56.8%

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

      2. times-frac 60.9%

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

   6. Applied egg-rr 60.9%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(0.5, \color{blue}{\frac{\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(w \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{c0}}, c0 \cdot 0\right) \]
   7. Step-by-step derivation

      1. associate-*l/ 60.9%

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

      2. pow2 60.9%

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

      3. mul0-rgt 60.9%

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

      4. *-commutative 60.9%

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

   8. Applied egg-rr 60.9%

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

      1. times-frac 60.9%

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

      2. fma-udef 60.9%

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

      3. +-rgt-identity 60.9%

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

      4. associate-/l* 60.9%

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

   10. Simplified 60.9%

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

   if -4.60000000000000005e-194 < c0 < 1.29999999999999998e-123

   1. Initial program 19.3%

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

      1. associate-*l* 19.3%

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

      2. difference-of-squares 19.3%

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

      3. associate-*l* 19.3%

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

      4. associate-*l* 19.3%

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

   3. Simplified 19.3%

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

   4. Taylor expanded in c0 around -inf 2.3%

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

      1. associate-*r* 2.3%

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

      2. distribute-rgt1-in 2.3%

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

      3. metadata-eval 2.3%

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

      4. mul0-lft 57.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot \color{blue}{0}\right) \cdot c0\right) \]

      5. metadata-eval 57.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{0} \cdot c0\right) \]

      6. mul0-lft 4.4%

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

      7. metadata-eval 4.4%

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

      8. distribute-lft1-in 4.4%

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

      9. *-commutative 4.4%

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

      10. distribute-lft1-in 4.4%

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

      11. metadata-eval 4.4%

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

      12. mul0-lft 57.4%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{0}\right) \]

   6. Simplified 57.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -6.5 \cdot 10^{+271}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(d \cdot \frac{c0 \cdot d}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right)\\ \mathbf{elif}\;c0 \leq -2.05 \cdot 10^{+228}:\\ \;\;\;\;\frac{0.25}{d} \cdot \frac{D}{\frac{d}{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}}\\ \mathbf{elif}\;c0 \leq -0.000135:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{h}}{w} \cdot {\left(\frac{d}{D}\right)}^{2} + \left(d \cdot \frac{d}{D \cdot D}\right) \cdot \frac{c0}{w \cdot h}\right)\\ \mathbf{elif}\;c0 \leq -3.1 \cdot 10^{-116}:\\ \;\;\;\;\frac{c0}{w} \cdot \frac{0.5 \cdot \frac{{\left(\frac{D}{d}\right)}^{2}}{\frac{c0}{w \cdot \left(h \cdot \left(M \cdot M\right)\right)}}}{2}\\ \mathbf{elif}\;c0 \leq -4.6 \cdot 10^{-194}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(d \cdot \frac{c0 \cdot d}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right)\\ \mathbf{elif}\;c0 \leq 1.3 \cdot 10^{-123}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(c0 \cdot 0\right)\\ \mathbf{elif}\;c0 \leq 8.5 \cdot 10^{-97} \lor \neg \left(c0 \leq 4.8 \cdot 10^{+216}\right):\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{h}}{w} \cdot {\left(\frac{d}{D}\right)}^{2} + \left(d \cdot \frac{d}{D \cdot D}\right) \cdot \frac{c0}{w \cdot h}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{d} \cdot \frac{D}{\frac{d}{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}}\\ \end{array} \]

Alternative 4: 43.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := h \cdot \left(M \cdot M\right)\\ t_1 := \left(w \cdot h\right) \cdot \left(D \cdot D\right)\\ t_2 := \frac{c0}{2 \cdot w}\\ t_3 := t_2 \cdot \left(2 \cdot \left(d \cdot \frac{c0 \cdot d}{t_1}\right)\right)\\ t_4 := \frac{0.25}{d} \cdot \frac{D}{\frac{d}{D \cdot t_0}}\\ \mathbf{if}\;c0 \leq -4.4 \cdot 10^{+267}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;c0 \leq -1.28 \cdot 10^{+227}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;c0 \leq -0.0025:\\ \;\;\;\;t_3\\ \mathbf{elif}\;c0 \leq -8.5 \cdot 10^{-115}:\\ \;\;\;\;\frac{c0}{w} \cdot \frac{0.5 \cdot \frac{{\left(\frac{D}{d}\right)}^{2}}{\frac{c0}{w \cdot t_0}}}{2}\\ \mathbf{elif}\;c0 \leq -8 \cdot 10^{-204}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;c0 \leq 1.15 \cdot 10^{-120}:\\ \;\;\;\;t_2 \cdot \left(c0 \cdot 0\right)\\ \mathbf{elif}\;c0 \leq 3.1 \cdot 10^{-95}:\\ \;\;\;\;t_2 \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{t_1}\\ \mathbf{elif}\;c0 \leq 7.4 \cdot 10^{+221}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* h (* M M)))
        (t_1 (* (* w h) (* D D)))
        (t_2 (/ c0 (* 2.0 w)))
        (t_3 (* t_2 (* 2.0 (* d (/ (* c0 d) t_1)))))
        (t_4 (* (/ 0.25 d) (/ D (/ d (* D t_0))))))
   (if (<= c0 -4.4e+267)
     t_3
     (if (<= c0 -1.28e+227)
       t_4
       (if (<= c0 -0.0025)
         t_3
         (if (<= c0 -8.5e-115)
           (* (/ c0 w) (/ (* 0.5 (/ (pow (/ D d) 2.0) (/ c0 (* w t_0)))) 2.0))
           (if (<= c0 -8e-204)
             t_3
             (if (<= c0 1.15e-120)
               (* t_2 (* c0 0.0))
               (if (<= c0 3.1e-95)
                 (* t_2 (/ (* 2.0 (* c0 (* d d))) t_1))
                 (if (<= c0 7.4e+221) t_4 t_3))))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = h * (M * M);
	double t_1 = (w * h) * (D * D);
	double t_2 = c0 / (2.0 * w);
	double t_3 = t_2 * (2.0 * (d * ((c0 * d) / t_1)));
	double t_4 = (0.25 / d) * (D / (d / (D * t_0)));
	double tmp;
	if (c0 <= -4.4e+267) {
		tmp = t_3;
	} else if (c0 <= -1.28e+227) {
		tmp = t_4;
	} else if (c0 <= -0.0025) {
		tmp = t_3;
	} else if (c0 <= -8.5e-115) {
		tmp = (c0 / w) * ((0.5 * (pow((D / d), 2.0) / (c0 / (w * t_0)))) / 2.0);
	} else if (c0 <= -8e-204) {
		tmp = t_3;
	} else if (c0 <= 1.15e-120) {
		tmp = t_2 * (c0 * 0.0);
	} else if (c0 <= 3.1e-95) {
		tmp = t_2 * ((2.0 * (c0 * (d * d))) / t_1);
	} else if (c0 <= 7.4e+221) {
		tmp = t_4;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = h * (m * m)
    t_1 = (w * h) * (d * d)
    t_2 = c0 / (2.0d0 * w)
    t_3 = t_2 * (2.0d0 * (d_1 * ((c0 * d_1) / t_1)))
    t_4 = (0.25d0 / d_1) * (d / (d_1 / (d * t_0)))
    if (c0 <= (-4.4d+267)) then
        tmp = t_3
    else if (c0 <= (-1.28d+227)) then
        tmp = t_4
    else if (c0 <= (-0.0025d0)) then
        tmp = t_3
    else if (c0 <= (-8.5d-115)) then
        tmp = (c0 / w) * ((0.5d0 * (((d / d_1) ** 2.0d0) / (c0 / (w * t_0)))) / 2.0d0)
    else if (c0 <= (-8d-204)) then
        tmp = t_3
    else if (c0 <= 1.15d-120) then
        tmp = t_2 * (c0 * 0.0d0)
    else if (c0 <= 3.1d-95) then
        tmp = t_2 * ((2.0d0 * (c0 * (d_1 * d_1))) / t_1)
    else if (c0 <= 7.4d+221) then
        tmp = t_4
    else
        tmp = t_3
    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 = h * (M * M);
	double t_1 = (w * h) * (D * D);
	double t_2 = c0 / (2.0 * w);
	double t_3 = t_2 * (2.0 * (d * ((c0 * d) / t_1)));
	double t_4 = (0.25 / d) * (D / (d / (D * t_0)));
	double tmp;
	if (c0 <= -4.4e+267) {
		tmp = t_3;
	} else if (c0 <= -1.28e+227) {
		tmp = t_4;
	} else if (c0 <= -0.0025) {
		tmp = t_3;
	} else if (c0 <= -8.5e-115) {
		tmp = (c0 / w) * ((0.5 * (Math.pow((D / d), 2.0) / (c0 / (w * t_0)))) / 2.0);
	} else if (c0 <= -8e-204) {
		tmp = t_3;
	} else if (c0 <= 1.15e-120) {
		tmp = t_2 * (c0 * 0.0);
	} else if (c0 <= 3.1e-95) {
		tmp = t_2 * ((2.0 * (c0 * (d * d))) / t_1);
	} else if (c0 <= 7.4e+221) {
		tmp = t_4;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = h * (M * M)
	t_1 = (w * h) * (D * D)
	t_2 = c0 / (2.0 * w)
	t_3 = t_2 * (2.0 * (d * ((c0 * d) / t_1)))
	t_4 = (0.25 / d) * (D / (d / (D * t_0)))
	tmp = 0
	if c0 <= -4.4e+267:
		tmp = t_3
	elif c0 <= -1.28e+227:
		tmp = t_4
	elif c0 <= -0.0025:
		tmp = t_3
	elif c0 <= -8.5e-115:
		tmp = (c0 / w) * ((0.5 * (math.pow((D / d), 2.0) / (c0 / (w * t_0)))) / 2.0)
	elif c0 <= -8e-204:
		tmp = t_3
	elif c0 <= 1.15e-120:
		tmp = t_2 * (c0 * 0.0)
	elif c0 <= 3.1e-95:
		tmp = t_2 * ((2.0 * (c0 * (d * d))) / t_1)
	elif c0 <= 7.4e+221:
		tmp = t_4
	else:
		tmp = t_3
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(h * Float64(M * M))
	t_1 = Float64(Float64(w * h) * Float64(D * D))
	t_2 = Float64(c0 / Float64(2.0 * w))
	t_3 = Float64(t_2 * Float64(2.0 * Float64(d * Float64(Float64(c0 * d) / t_1))))
	t_4 = Float64(Float64(0.25 / d) * Float64(D / Float64(d / Float64(D * t_0))))
	tmp = 0.0
	if (c0 <= -4.4e+267)
		tmp = t_3;
	elseif (c0 <= -1.28e+227)
		tmp = t_4;
	elseif (c0 <= -0.0025)
		tmp = t_3;
	elseif (c0 <= -8.5e-115)
		tmp = Float64(Float64(c0 / w) * Float64(Float64(0.5 * Float64((Float64(D / d) ^ 2.0) / Float64(c0 / Float64(w * t_0)))) / 2.0));
	elseif (c0 <= -8e-204)
		tmp = t_3;
	elseif (c0 <= 1.15e-120)
		tmp = Float64(t_2 * Float64(c0 * 0.0));
	elseif (c0 <= 3.1e-95)
		tmp = Float64(t_2 * Float64(Float64(2.0 * Float64(c0 * Float64(d * d))) / t_1));
	elseif (c0 <= 7.4e+221)
		tmp = t_4;
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = h * (M * M);
	t_1 = (w * h) * (D * D);
	t_2 = c0 / (2.0 * w);
	t_3 = t_2 * (2.0 * (d * ((c0 * d) / t_1)));
	t_4 = (0.25 / d) * (D / (d / (D * t_0)));
	tmp = 0.0;
	if (c0 <= -4.4e+267)
		tmp = t_3;
	elseif (c0 <= -1.28e+227)
		tmp = t_4;
	elseif (c0 <= -0.0025)
		tmp = t_3;
	elseif (c0 <= -8.5e-115)
		tmp = (c0 / w) * ((0.5 * (((D / d) ^ 2.0) / (c0 / (w * t_0)))) / 2.0);
	elseif (c0 <= -8e-204)
		tmp = t_3;
	elseif (c0 <= 1.15e-120)
		tmp = t_2 * (c0 * 0.0);
	elseif (c0 <= 3.1e-95)
		tmp = t_2 * ((2.0 * (c0 * (d * d))) / t_1);
	elseif (c0 <= 7.4e+221)
		tmp = t_4;
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(2.0 * N[(d * N[(N[(c0 * d), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(0.25 / d), $MachinePrecision] * N[(D / N[(d / N[(D * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c0, -4.4e+267], t$95$3, If[LessEqual[c0, -1.28e+227], t$95$4, If[LessEqual[c0, -0.0025], t$95$3, If[LessEqual[c0, -8.5e-115], N[(N[(c0 / w), $MachinePrecision] * N[(N[(0.5 * N[(N[Power[N[(D / d), $MachinePrecision], 2.0], $MachinePrecision] / N[(c0 / N[(w * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[c0, -8e-204], t$95$3, If[LessEqual[c0, 1.15e-120], N[(t$95$2 * N[(c0 * 0.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[c0, 3.1e-95], N[(t$95$2 * N[(N[(2.0 * N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[c0, 7.4e+221], t$95$4, t$95$3]]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if c0 < -4.4000000000000002e267 or -1.27999999999999991e227 < c0 < -0.00250000000000000005 or -8.49999999999999953e-115 < c0 < -8.00000000000000001e-204 or 7.40000000000000002e221 < c0

   1. Initial program 43.6%

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

      1. associate-*l* 43.6%

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

      2. difference-of-squares 54.3%

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

      3. associate-*l* 54.3%

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

      4. associate-*l* 55.2%

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

   3. Simplified 55.2%

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

   4. Taylor expanded in c0 around inf 54.4%

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

      1. *-commutative 54.4%

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

      2. associate-*r* 54.5%

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

      3. unpow2 54.5%

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

      4. associate-*r/ 54.5%

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

      5. unpow2 54.5%

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

      6. associate-*r* 61.1%

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

      7. associate-*r/ 61.0%

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

      8. unpow2 61.0%

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

      9. associate-*r* 61.7%

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

      10. *-commutative 61.7%

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

      11. unpow2 61.7%

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

   6. Simplified 61.7%

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

   if -4.4000000000000002e267 < c0 < -1.27999999999999991e227 or 3.09999999999999992e-95 < c0 < 7.40000000000000002e221

   1. Initial program 20.1%

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

   2. Taylor expanded in c0 around -inf 6.4%

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

      1. fma-def 6.4%

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

      2. times-frac 7.8%

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

      3. unpow2 7.8%

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

      4. unpow2 7.8%

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

      5. *-commutative 7.8%

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

      6. unpow2 7.8%

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

      7. associate-*r* 7.8%

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

   4. Simplified 43.5%

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

   5. Taylor expanded in c0 around 0 46.9%

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

      1. associate-*r/ 46.9%

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

      2. *-commutative 46.9%

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

      3. *-commutative 46.9%

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

      4. unpow2 46.9%

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

      5. unpow2 46.9%

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

      6. unpow2 46.9%

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

   7. Simplified 46.9%

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

      1. *-un-lft-identity 46.9%

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

      2. times-frac 50.2%

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

      3. associate-*l* 62.5%

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

   9. Applied egg-rr 62.5%

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

       1. *-lft-identity 62.5%

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

       2. associate-/l* 65.5%

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

   11. Simplified 65.5%

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

   if -0.00250000000000000005 < c0 < -8.49999999999999953e-115

   1. Initial program 28.3%

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

   2. Taylor expanded in c0 around -inf 8.0%

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

      1. fma-def 8.0%

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

      2. times-frac 8.0%

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

      3. unpow2 8.0%

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

      4. unpow2 8.0%

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

      5. *-commutative 8.0%

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

      6. unpow2 8.0%

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

      7. associate-*r* 8.0%

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

   4. Simplified 56.8%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{D \cdot D}{d \cdot d} \cdot \frac{w \cdot \left(h \cdot \left(M \cdot M\right)\right)}{c0}, c0 \cdot 0\right)} \]
   5. Step-by-step derivation

      1. associate-*r/ 56.8%

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

      2. times-frac 60.9%

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

   6. Applied egg-rr 60.9%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(0.5, \color{blue}{\frac{\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(w \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{c0}}, c0 \cdot 0\right) \]
   7. Step-by-step derivation

      1. associate-*l/ 60.9%

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

      2. pow2 60.9%

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

      3. mul0-rgt 60.9%

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

      4. *-commutative 60.9%

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

   8. Applied egg-rr 60.9%

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

      1. times-frac 60.9%

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

      2. fma-udef 60.9%

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

      3. +-rgt-identity 60.9%

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

      4. associate-/l* 60.9%

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

   10. Simplified 60.9%

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

   if -8.00000000000000001e-204 < c0 < 1.14999999999999993e-120

   1. Initial program 19.3%

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

      1. associate-*l* 19.3%

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

      2. difference-of-squares 19.3%

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

      3. associate-*l* 19.3%

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

      4. associate-*l* 19.3%

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

   3. Simplified 19.3%

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

   4. Taylor expanded in c0 around -inf 2.3%

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

      1. associate-*r* 2.3%

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

      2. distribute-rgt1-in 2.3%

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

      3. metadata-eval 2.3%

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

      4. mul0-lft 57.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot \color{blue}{0}\right) \cdot c0\right) \]

      5. metadata-eval 57.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{0} \cdot c0\right) \]

      6. mul0-lft 4.4%

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

      7. metadata-eval 4.4%

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

      8. distribute-lft1-in 4.4%

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

      9. *-commutative 4.4%

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

      10. distribute-lft1-in 4.4%

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

      11. metadata-eval 4.4%

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

      12. mul0-lft 57.4%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{0}\right) \]

   6. Simplified 57.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]

   if 1.14999999999999993e-120 < c0 < 3.09999999999999992e-95

   1. Initial program 81.3%

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

      1. times-frac 81.3%

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

      2. fma-def 81.3%

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

      3. times-frac 81.3%

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

      4. difference-of-squares 81.3%

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

   3. Simplified 81.3%

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

      1. fma-udef 81.3%

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

      2. associate-/l/ 81.3%

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

      3. pow2 81.3%

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

      4. associate-*l* 81.3%

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

      5. div-inv 81.3%

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

      6. clear-num 81.3%

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

      7. associate-*r/ 81.3%

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

      8. *-commutative 81.3%

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

   5. Applied egg-rr 81.3%

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

   6. Taylor expanded in c0 around inf 99.1%

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

      1. associate-*r/ 99.1%

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

      2. unpow2 99.1%

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

      3. unpow2 99.1%

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

   8. Simplified 99.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -4.4 \cdot 10^{+267}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(d \cdot \frac{c0 \cdot d}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right)\\ \mathbf{elif}\;c0 \leq -1.28 \cdot 10^{+227}:\\ \;\;\;\;\frac{0.25}{d} \cdot \frac{D}{\frac{d}{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}}\\ \mathbf{elif}\;c0 \leq -0.0025:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(d \cdot \frac{c0 \cdot d}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right)\\ \mathbf{elif}\;c0 \leq -8.5 \cdot 10^{-115}:\\ \;\;\;\;\frac{c0}{w} \cdot \frac{0.5 \cdot \frac{{\left(\frac{D}{d}\right)}^{2}}{\frac{c0}{w \cdot \left(h \cdot \left(M \cdot M\right)\right)}}}{2}\\ \mathbf{elif}\;c0 \leq -8 \cdot 10^{-204}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(d \cdot \frac{c0 \cdot d}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right)\\ \mathbf{elif}\;c0 \leq 1.15 \cdot 10^{-120}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(c0 \cdot 0\right)\\ \mathbf{elif}\;c0 \leq 3.1 \cdot 10^{-95}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{elif}\;c0 \leq 7.4 \cdot 10^{+221}:\\ \;\;\;\;\frac{0.25}{d} \cdot \frac{D}{\frac{d}{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(d \cdot \frac{c0 \cdot d}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right)\\ \end{array} \]

Alternative 5: 43.5% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.25}{d} \cdot \frac{D}{\frac{d}{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := t_1 \cdot \left(2 \cdot \left(d \cdot \frac{c0 \cdot d}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right)\\ t_3 := t_1 \cdot \left(c0 \cdot 0\right)\\ \mathbf{if}\;c0 \leq -5.5 \cdot 10^{+265}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c0 \leq -5.5 \cdot 10^{+226}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c0 \leq -8 \cdot 10^{-5}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c0 \leq -8.5 \cdot 10^{-115}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;c0 \leq -2.8 \cdot 10^{-185}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c0 \leq 1.15 \cdot 10^{-120}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;c0 \leq 10^{-96} \lor \neg \left(c0 \leq 6.9 \cdot 10^{+217}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ 0.25 d) (/ D (/ d (* D (* h (* M M)))))))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (* t_1 (* 2.0 (* d (/ (* c0 d) (* (* w h) (* D D)))))))
        (t_3 (* t_1 (* c0 0.0))))
   (if (<= c0 -5.5e+265)
     t_2
     (if (<= c0 -5.5e+226)
       t_0
       (if (<= c0 -8e-5)
         t_2
         (if (<= c0 -8.5e-115)
           t_3
           (if (<= c0 -2.8e-185)
             t_2
             (if (<= c0 1.15e-120)
               t_3
               (if (or (<= c0 1e-96) (not (<= c0 6.9e+217))) t_2 t_0)))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (0.25 / d) * (D / (d / (D * (h * (M * M)))));
	double t_1 = c0 / (2.0 * w);
	double t_2 = t_1 * (2.0 * (d * ((c0 * d) / ((w * h) * (D * D)))));
	double t_3 = t_1 * (c0 * 0.0);
	double tmp;
	if (c0 <= -5.5e+265) {
		tmp = t_2;
	} else if (c0 <= -5.5e+226) {
		tmp = t_0;
	} else if (c0 <= -8e-5) {
		tmp = t_2;
	} else if (c0 <= -8.5e-115) {
		tmp = t_3;
	} else if (c0 <= -2.8e-185) {
		tmp = t_2;
	} else if (c0 <= 1.15e-120) {
		tmp = t_3;
	} else if ((c0 <= 1e-96) || !(c0 <= 6.9e+217)) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (0.25d0 / d_1) * (d / (d_1 / (d * (h * (m * m)))))
    t_1 = c0 / (2.0d0 * w)
    t_2 = t_1 * (2.0d0 * (d_1 * ((c0 * d_1) / ((w * h) * (d * d)))))
    t_3 = t_1 * (c0 * 0.0d0)
    if (c0 <= (-5.5d+265)) then
        tmp = t_2
    else if (c0 <= (-5.5d+226)) then
        tmp = t_0
    else if (c0 <= (-8d-5)) then
        tmp = t_2
    else if (c0 <= (-8.5d-115)) then
        tmp = t_3
    else if (c0 <= (-2.8d-185)) then
        tmp = t_2
    else if (c0 <= 1.15d-120) then
        tmp = t_3
    else if ((c0 <= 1d-96) .or. (.not. (c0 <= 6.9d+217))) then
        tmp = t_2
    else
        tmp = t_0
    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 / (d / (D * (h * (M * M)))));
	double t_1 = c0 / (2.0 * w);
	double t_2 = t_1 * (2.0 * (d * ((c0 * d) / ((w * h) * (D * D)))));
	double t_3 = t_1 * (c0 * 0.0);
	double tmp;
	if (c0 <= -5.5e+265) {
		tmp = t_2;
	} else if (c0 <= -5.5e+226) {
		tmp = t_0;
	} else if (c0 <= -8e-5) {
		tmp = t_2;
	} else if (c0 <= -8.5e-115) {
		tmp = t_3;
	} else if (c0 <= -2.8e-185) {
		tmp = t_2;
	} else if (c0 <= 1.15e-120) {
		tmp = t_3;
	} else if ((c0 <= 1e-96) || !(c0 <= 6.9e+217)) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (0.25 / d) * (D / (d / (D * (h * (M * M)))))
	t_1 = c0 / (2.0 * w)
	t_2 = t_1 * (2.0 * (d * ((c0 * d) / ((w * h) * (D * D)))))
	t_3 = t_1 * (c0 * 0.0)
	tmp = 0
	if c0 <= -5.5e+265:
		tmp = t_2
	elif c0 <= -5.5e+226:
		tmp = t_0
	elif c0 <= -8e-5:
		tmp = t_2
	elif c0 <= -8.5e-115:
		tmp = t_3
	elif c0 <= -2.8e-185:
		tmp = t_2
	elif c0 <= 1.15e-120:
		tmp = t_3
	elif (c0 <= 1e-96) or not (c0 <= 6.9e+217):
		tmp = t_2
	else:
		tmp = t_0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(0.25 / d) * Float64(D / Float64(d / Float64(D * Float64(h * Float64(M * M))))))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(t_1 * Float64(2.0 * Float64(d * Float64(Float64(c0 * d) / Float64(Float64(w * h) * Float64(D * D))))))
	t_3 = Float64(t_1 * Float64(c0 * 0.0))
	tmp = 0.0
	if (c0 <= -5.5e+265)
		tmp = t_2;
	elseif (c0 <= -5.5e+226)
		tmp = t_0;
	elseif (c0 <= -8e-5)
		tmp = t_2;
	elseif (c0 <= -8.5e-115)
		tmp = t_3;
	elseif (c0 <= -2.8e-185)
		tmp = t_2;
	elseif (c0 <= 1.15e-120)
		tmp = t_3;
	elseif ((c0 <= 1e-96) || !(c0 <= 6.9e+217))
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (0.25 / d) * (D / (d / (D * (h * (M * M)))));
	t_1 = c0 / (2.0 * w);
	t_2 = t_1 * (2.0 * (d * ((c0 * d) / ((w * h) * (D * D)))));
	t_3 = t_1 * (c0 * 0.0);
	tmp = 0.0;
	if (c0 <= -5.5e+265)
		tmp = t_2;
	elseif (c0 <= -5.5e+226)
		tmp = t_0;
	elseif (c0 <= -8e-5)
		tmp = t_2;
	elseif (c0 <= -8.5e-115)
		tmp = t_3;
	elseif (c0 <= -2.8e-185)
		tmp = t_2;
	elseif (c0 <= 1.15e-120)
		tmp = t_3;
	elseif ((c0 <= 1e-96) || ~((c0 <= 6.9e+217)))
		tmp = t_2;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(0.25 / d), $MachinePrecision] * N[(D / N[(d / N[(D * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(2.0 * N[(d * N[(N[(c0 * d), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[(c0 * 0.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c0, -5.5e+265], t$95$2, If[LessEqual[c0, -5.5e+226], t$95$0, If[LessEqual[c0, -8e-5], t$95$2, If[LessEqual[c0, -8.5e-115], t$95$3, If[LessEqual[c0, -2.8e-185], t$95$2, If[LessEqual[c0, 1.15e-120], t$95$3, If[Or[LessEqual[c0, 1e-96], N[Not[LessEqual[c0, 6.9e+217]], $MachinePrecision]], t$95$2, t$95$0]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if c0 < -5.4999999999999997e265 or -5.5000000000000005e226 < c0 < -8.00000000000000065e-5 or -8.49999999999999953e-115 < c0 < -2.79999999999999991e-185 or 1.14999999999999993e-120 < c0 < 9.9999999999999991e-97 or 6.89999999999999969e217 < c0

   1. Initial program 45.2%

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

      1. associate-*l* 44.5%

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

      2. difference-of-squares 54.8%

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

      3. associate-*l* 54.7%

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

      4. associate-*l* 55.5%

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

   3. Simplified 55.5%

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

   4. Taylor expanded in c0 around inf 56.3%

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

      1. *-commutative 56.3%

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

      2. associate-*r* 55.6%

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

      3. unpow2 55.6%

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

      4. associate-*r/ 55.6%

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

      5. unpow2 55.6%

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

      6. associate-*r* 61.9%

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

      7. associate-*r/ 61.8%

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

      8. unpow2 61.8%

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

      9. associate-*r* 63.3%

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

      10. *-commutative 63.3%

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

      11. unpow2 63.3%

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

   6. Simplified 63.3%

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

   if -5.4999999999999997e265 < c0 < -5.5000000000000005e226 or 9.9999999999999991e-97 < c0 < 6.89999999999999969e217

   1. Initial program 20.1%

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

   2. Taylor expanded in c0 around -inf 6.4%

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

      1. fma-def 6.4%

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

      2. times-frac 7.8%

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

      3. unpow2 7.8%

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

      4. unpow2 7.8%

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

      5. *-commutative 7.8%

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

      6. unpow2 7.8%

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

      7. associate-*r* 7.8%

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

   4. Simplified 43.5%

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

   5. Taylor expanded in c0 around 0 46.9%

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

      1. associate-*r/ 46.9%

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

      2. *-commutative 46.9%

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

      3. *-commutative 46.9%

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

      4. unpow2 46.9%

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

      5. unpow2 46.9%

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

      6. unpow2 46.9%

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

   7. Simplified 46.9%

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

      1. *-un-lft-identity 46.9%

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

      2. times-frac 50.2%

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

      3. associate-*l* 62.5%

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

   9. Applied egg-rr 62.5%

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

       1. *-lft-identity 62.5%

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

       2. associate-/l* 65.5%

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

   11. Simplified 65.5%

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

   if -8.00000000000000065e-5 < c0 < -8.49999999999999953e-115 or -2.79999999999999991e-185 < c0 < 1.14999999999999993e-120

   1. Initial program 22.3%

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

      1. associate-*l* 21.0%

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

      2. difference-of-squares 21.0%

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

      3. associate-*l* 21.0%

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

      4. associate-*l* 22.3%

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

   3. Simplified 22.3%

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

   4. Taylor expanded in c0 around -inf 4.2%

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

      1. associate-*r* 4.2%

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

      2. distribute-rgt1-in 4.2%

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

      3. metadata-eval 4.2%

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

      4. mul0-lft 57.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot \color{blue}{0}\right) \cdot c0\right) \]

      5. metadata-eval 57.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{0} \cdot c0\right) \]

      6. mul0-lft 7.0%

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

      7. metadata-eval 7.0%

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

      8. distribute-lft1-in 7.0%

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

      9. *-commutative 7.0%

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

      10. distribute-lft1-in 7.0%

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

      11. metadata-eval 7.0%

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

      12. mul0-lft 57.5%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{0}\right) \]

   6. Simplified 57.5%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -5.5 \cdot 10^{+265}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(d \cdot \frac{c0 \cdot d}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right)\\ \mathbf{elif}\;c0 \leq -5.5 \cdot 10^{+226}:\\ \;\;\;\;\frac{0.25}{d} \cdot \frac{D}{\frac{d}{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}}\\ \mathbf{elif}\;c0 \leq -8 \cdot 10^{-5}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(d \cdot \frac{c0 \cdot d}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right)\\ \mathbf{elif}\;c0 \leq -8.5 \cdot 10^{-115}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(c0 \cdot 0\right)\\ \mathbf{elif}\;c0 \leq -2.8 \cdot 10^{-185}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(d \cdot \frac{c0 \cdot d}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right)\\ \mathbf{elif}\;c0 \leq 1.15 \cdot 10^{-120}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(c0 \cdot 0\right)\\ \mathbf{elif}\;c0 \leq 10^{-96} \lor \neg \left(c0 \leq 6.9 \cdot 10^{+217}\right):\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(d \cdot \frac{c0 \cdot d}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{d} \cdot \frac{D}{\frac{d}{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}}\\ \end{array} \]

Alternative 6: 43.2% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(w \cdot h\right) \cdot \left(D \cdot D\right)\\ t_1 := \frac{0.25}{d} \cdot \frac{D}{\frac{d}{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}}\\ t_2 := \frac{c0}{2 \cdot w}\\ t_3 := t_2 \cdot \left(2 \cdot \left(d \cdot \frac{c0 \cdot d}{t_0}\right)\right)\\ t_4 := t_2 \cdot \left(c0 \cdot 0\right)\\ \mathbf{if}\;c0 \leq -5.8 \cdot 10^{+266}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;c0 \leq -3.2 \cdot 10^{+230}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c0 \leq -0.00185:\\ \;\;\;\;t_3\\ \mathbf{elif}\;c0 \leq -3.15 \cdot 10^{-117}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;c0 \leq -4.6 \cdot 10^{-185}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;c0 \leq 1.2 \cdot 10^{-122}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;c0 \leq 2.1 \cdot 10^{-95}:\\ \;\;\;\;t_2 \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{t_0}\\ \mathbf{elif}\;c0 \leq 9.2 \cdot 10^{+219}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (* w h) (* D D)))
        (t_1 (* (/ 0.25 d) (/ D (/ d (* D (* h (* M M)))))))
        (t_2 (/ c0 (* 2.0 w)))
        (t_3 (* t_2 (* 2.0 (* d (/ (* c0 d) t_0)))))
        (t_4 (* t_2 (* c0 0.0))))
   (if (<= c0 -5.8e+266)
     t_3
     (if (<= c0 -3.2e+230)
       t_1
       (if (<= c0 -0.00185)
         t_3
         (if (<= c0 -3.15e-117)
           t_4
           (if (<= c0 -4.6e-185)
             t_3
             (if (<= c0 1.2e-122)
               t_4
               (if (<= c0 2.1e-95)
                 (* t_2 (/ (* 2.0 (* c0 (* d d))) t_0))
                 (if (<= c0 9.2e+219) t_1 t_3))))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (w * h) * (D * D);
	double t_1 = (0.25 / d) * (D / (d / (D * (h * (M * M)))));
	double t_2 = c0 / (2.0 * w);
	double t_3 = t_2 * (2.0 * (d * ((c0 * d) / t_0)));
	double t_4 = t_2 * (c0 * 0.0);
	double tmp;
	if (c0 <= -5.8e+266) {
		tmp = t_3;
	} else if (c0 <= -3.2e+230) {
		tmp = t_1;
	} else if (c0 <= -0.00185) {
		tmp = t_3;
	} else if (c0 <= -3.15e-117) {
		tmp = t_4;
	} else if (c0 <= -4.6e-185) {
		tmp = t_3;
	} else if (c0 <= 1.2e-122) {
		tmp = t_4;
	} else if (c0 <= 2.1e-95) {
		tmp = t_2 * ((2.0 * (c0 * (d * d))) / t_0);
	} else if (c0 <= 9.2e+219) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = (w * h) * (d * d)
    t_1 = (0.25d0 / d_1) * (d / (d_1 / (d * (h * (m * m)))))
    t_2 = c0 / (2.0d0 * w)
    t_3 = t_2 * (2.0d0 * (d_1 * ((c0 * d_1) / t_0)))
    t_4 = t_2 * (c0 * 0.0d0)
    if (c0 <= (-5.8d+266)) then
        tmp = t_3
    else if (c0 <= (-3.2d+230)) then
        tmp = t_1
    else if (c0 <= (-0.00185d0)) then
        tmp = t_3
    else if (c0 <= (-3.15d-117)) then
        tmp = t_4
    else if (c0 <= (-4.6d-185)) then
        tmp = t_3
    else if (c0 <= 1.2d-122) then
        tmp = t_4
    else if (c0 <= 2.1d-95) then
        tmp = t_2 * ((2.0d0 * (c0 * (d_1 * d_1))) / t_0)
    else if (c0 <= 9.2d+219) then
        tmp = t_1
    else
        tmp = t_3
    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 = (w * h) * (D * D);
	double t_1 = (0.25 / d) * (D / (d / (D * (h * (M * M)))));
	double t_2 = c0 / (2.0 * w);
	double t_3 = t_2 * (2.0 * (d * ((c0 * d) / t_0)));
	double t_4 = t_2 * (c0 * 0.0);
	double tmp;
	if (c0 <= -5.8e+266) {
		tmp = t_3;
	} else if (c0 <= -3.2e+230) {
		tmp = t_1;
	} else if (c0 <= -0.00185) {
		tmp = t_3;
	} else if (c0 <= -3.15e-117) {
		tmp = t_4;
	} else if (c0 <= -4.6e-185) {
		tmp = t_3;
	} else if (c0 <= 1.2e-122) {
		tmp = t_4;
	} else if (c0 <= 2.1e-95) {
		tmp = t_2 * ((2.0 * (c0 * (d * d))) / t_0);
	} else if (c0 <= 9.2e+219) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (w * h) * (D * D)
	t_1 = (0.25 / d) * (D / (d / (D * (h * (M * M)))))
	t_2 = c0 / (2.0 * w)
	t_3 = t_2 * (2.0 * (d * ((c0 * d) / t_0)))
	t_4 = t_2 * (c0 * 0.0)
	tmp = 0
	if c0 <= -5.8e+266:
		tmp = t_3
	elif c0 <= -3.2e+230:
		tmp = t_1
	elif c0 <= -0.00185:
		tmp = t_3
	elif c0 <= -3.15e-117:
		tmp = t_4
	elif c0 <= -4.6e-185:
		tmp = t_3
	elif c0 <= 1.2e-122:
		tmp = t_4
	elif c0 <= 2.1e-95:
		tmp = t_2 * ((2.0 * (c0 * (d * d))) / t_0)
	elif c0 <= 9.2e+219:
		tmp = t_1
	else:
		tmp = t_3
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(w * h) * Float64(D * D))
	t_1 = Float64(Float64(0.25 / d) * Float64(D / Float64(d / Float64(D * Float64(h * Float64(M * M))))))
	t_2 = Float64(c0 / Float64(2.0 * w))
	t_3 = Float64(t_2 * Float64(2.0 * Float64(d * Float64(Float64(c0 * d) / t_0))))
	t_4 = Float64(t_2 * Float64(c0 * 0.0))
	tmp = 0.0
	if (c0 <= -5.8e+266)
		tmp = t_3;
	elseif (c0 <= -3.2e+230)
		tmp = t_1;
	elseif (c0 <= -0.00185)
		tmp = t_3;
	elseif (c0 <= -3.15e-117)
		tmp = t_4;
	elseif (c0 <= -4.6e-185)
		tmp = t_3;
	elseif (c0 <= 1.2e-122)
		tmp = t_4;
	elseif (c0 <= 2.1e-95)
		tmp = Float64(t_2 * Float64(Float64(2.0 * Float64(c0 * Float64(d * d))) / t_0));
	elseif (c0 <= 9.2e+219)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (w * h) * (D * D);
	t_1 = (0.25 / d) * (D / (d / (D * (h * (M * M)))));
	t_2 = c0 / (2.0 * w);
	t_3 = t_2 * (2.0 * (d * ((c0 * d) / t_0)));
	t_4 = t_2 * (c0 * 0.0);
	tmp = 0.0;
	if (c0 <= -5.8e+266)
		tmp = t_3;
	elseif (c0 <= -3.2e+230)
		tmp = t_1;
	elseif (c0 <= -0.00185)
		tmp = t_3;
	elseif (c0 <= -3.15e-117)
		tmp = t_4;
	elseif (c0 <= -4.6e-185)
		tmp = t_3;
	elseif (c0 <= 1.2e-122)
		tmp = t_4;
	elseif (c0 <= 2.1e-95)
		tmp = t_2 * ((2.0 * (c0 * (d * d))) / t_0);
	elseif (c0 <= 9.2e+219)
		tmp = t_1;
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.25 / d), $MachinePrecision] * N[(D / N[(d / N[(D * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(2.0 * N[(d * N[(N[(c0 * d), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 * N[(c0 * 0.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c0, -5.8e+266], t$95$3, If[LessEqual[c0, -3.2e+230], t$95$1, If[LessEqual[c0, -0.00185], t$95$3, If[LessEqual[c0, -3.15e-117], t$95$4, If[LessEqual[c0, -4.6e-185], t$95$3, If[LessEqual[c0, 1.2e-122], t$95$4, If[LessEqual[c0, 2.1e-95], N[(t$95$2 * N[(N[(2.0 * N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[c0, 9.2e+219], t$95$1, t$95$3]]]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if c0 < -5.80000000000000035e266 or -3.2e230 < c0 < -0.0018500000000000001 or -3.1499999999999999e-117 < c0 < -4.6000000000000002e-185 or 9.2000000000000004e219 < c0

   1. Initial program 43.6%

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

      1. associate-*l* 43.6%

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

      2. difference-of-squares 54.3%

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

      3. associate-*l* 54.3%

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

      4. associate-*l* 55.2%

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

   3. Simplified 55.2%

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

   4. Taylor expanded in c0 around inf 54.4%

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

      1. *-commutative 54.4%

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

      2. associate-*r* 54.5%

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

      3. unpow2 54.5%

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

      4. associate-*r/ 54.5%

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

      5. unpow2 54.5%

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

      6. associate-*r* 61.1%

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

      7. associate-*r/ 61.0%

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

      8. unpow2 61.0%

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

      9. associate-*r* 61.7%

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

      10. *-commutative 61.7%

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

      11. unpow2 61.7%

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

   6. Simplified 61.7%

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

   if -5.80000000000000035e266 < c0 < -3.2e230 or 2.1e-95 < c0 < 9.2000000000000004e219

   1. Initial program 20.1%

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

   2. Taylor expanded in c0 around -inf 6.4%

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

      1. fma-def 6.4%

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

      2. times-frac 7.8%

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

      3. unpow2 7.8%

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

      4. unpow2 7.8%

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

      5. *-commutative 7.8%

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

      6. unpow2 7.8%

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

      7. associate-*r* 7.8%

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

   4. Simplified 43.5%

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

   5. Taylor expanded in c0 around 0 46.9%

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

      1. associate-*r/ 46.9%

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

      2. *-commutative 46.9%

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

      3. *-commutative 46.9%

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

      4. unpow2 46.9%

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

      5. unpow2 46.9%

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

      6. unpow2 46.9%

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

   7. Simplified 46.9%

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

      1. *-un-lft-identity 46.9%

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

      2. times-frac 50.2%

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

      3. associate-*l* 62.5%

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

   9. Applied egg-rr 62.5%

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

       1. *-lft-identity 62.5%

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

       2. associate-/l* 65.5%

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

   11. Simplified 65.5%

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

   if -0.0018500000000000001 < c0 < -3.1499999999999999e-117 or -4.6000000000000002e-185 < c0 < 1.19999999999999994e-122

   1. Initial program 22.3%

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

      1. associate-*l* 21.0%

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

      2. difference-of-squares 21.0%

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

      3. associate-*l* 21.0%

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

      4. associate-*l* 22.3%

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

   3. Simplified 22.3%

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

   4. Taylor expanded in c0 around -inf 4.2%

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

      1. associate-*r* 4.2%

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

      2. distribute-rgt1-in 4.2%

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

      3. metadata-eval 4.2%

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

      4. mul0-lft 57.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot \color{blue}{0}\right) \cdot c0\right) \]

      5. metadata-eval 57.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{0} \cdot c0\right) \]

      6. mul0-lft 7.0%

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

      7. metadata-eval 7.0%

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

      8. distribute-lft1-in 7.0%

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

      9. *-commutative 7.0%

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

      10. distribute-lft1-in 7.0%

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

      11. metadata-eval 7.0%

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

      12. mul0-lft 57.5%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{0}\right) \]

   6. Simplified 57.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]

   if 1.19999999999999994e-122 < c0 < 2.1e-95

   1. Initial program 81.3%

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

      1. times-frac 81.3%

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

      2. fma-def 81.3%

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

      3. times-frac 81.3%

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

      4. difference-of-squares 81.3%

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

   3. Simplified 81.3%

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

      1. fma-udef 81.3%

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

      2. associate-/l/ 81.3%

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

      3. pow2 81.3%

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

      4. associate-*l* 81.3%

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

      5. div-inv 81.3%

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

      6. clear-num 81.3%

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

      7. associate-*r/ 81.3%

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

      8. *-commutative 81.3%

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

   5. Applied egg-rr 81.3%

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

   6. Taylor expanded in c0 around inf 99.1%

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

      1. associate-*r/ 99.1%

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

      2. unpow2 99.1%

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

      3. unpow2 99.1%

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

   8. Simplified 99.1%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -5.8 \cdot 10^{+266}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(d \cdot \frac{c0 \cdot d}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right)\\ \mathbf{elif}\;c0 \leq -3.2 \cdot 10^{+230}:\\ \;\;\;\;\frac{0.25}{d} \cdot \frac{D}{\frac{d}{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}}\\ \mathbf{elif}\;c0 \leq -0.00185:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(d \cdot \frac{c0 \cdot d}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right)\\ \mathbf{elif}\;c0 \leq -3.15 \cdot 10^{-117}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(c0 \cdot 0\right)\\ \mathbf{elif}\;c0 \leq -4.6 \cdot 10^{-185}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(d \cdot \frac{c0 \cdot d}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right)\\ \mathbf{elif}\;c0 \leq 1.2 \cdot 10^{-122}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(c0 \cdot 0\right)\\ \mathbf{elif}\;c0 \leq 2.1 \cdot 10^{-95}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{elif}\;c0 \leq 9.2 \cdot 10^{+219}:\\ \;\;\;\;\frac{0.25}{d} \cdot \frac{D}{\frac{d}{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(d \cdot \frac{c0 \cdot d}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right)\\ \end{array} \]

Alternative 7: 33.7% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \cdot M \leq 2.9 \cdot 10^{-246}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(c0 \cdot 0\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= (* M M) 2.9e-246)
   (* (/ c0 (* 2.0 w)) (* c0 0.0))
   (* 0.25 (/ (* D D) (/ (* d d) (* h (* M M)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((M * M) <= 2.9e-246) {
		tmp = (c0 / (2.0 * w)) * (c0 * 0.0);
	} else {
		tmp = 0.25 * ((D * D) / ((d * d) / (h * (M * M))));
	}
	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 ((m * m) <= 2.9d-246) then
        tmp = (c0 / (2.0d0 * w)) * (c0 * 0.0d0)
    else
        tmp = 0.25d0 * ((d * d) / ((d_1 * d_1) / (h * (m * m))))
    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 ((M * M) <= 2.9e-246) {
		tmp = (c0 / (2.0 * w)) * (c0 * 0.0);
	} else {
		tmp = 0.25 * ((D * D) / ((d * d) / (h * (M * M))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (M * M) <= 2.9e-246:
		tmp = (c0 / (2.0 * w)) * (c0 * 0.0)
	else:
		tmp = 0.25 * ((D * D) / ((d * d) / (h * (M * M))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (Float64(M * M) <= 2.9e-246)
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(c0 * 0.0));
	else
		tmp = Float64(0.25 * Float64(Float64(D * D) / Float64(Float64(d * d) / Float64(h * Float64(M * M)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((M * M) <= 2.9e-246)
		tmp = (c0 / (2.0 * w)) * (c0 * 0.0);
	else
		tmp = 0.25 * ((D * D) / ((d * d) / (h * (M * M))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[N[(M * M), $MachinePrecision], 2.9e-246], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(c0 * 0.0), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(D * D), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] / N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M M) < 2.9e-246

   1. Initial program 39.7%

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

      1. associate-*l* 38.8%

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

      2. difference-of-squares 38.8%

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

      3. associate-*l* 38.8%

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

      4. associate-*l* 40.5%

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

   3. Simplified 40.5%

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

   4. Taylor expanded in c0 around -inf 7.9%

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

      1. associate-*r* 7.9%

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

      2. distribute-rgt1-in 7.9%

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

      3. metadata-eval 7.9%

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

      4. mul0-lft 43.2%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot \color{blue}{0}\right) \cdot c0\right) \]

      5. metadata-eval 43.2%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{0} \cdot c0\right) \]

      6. mul0-lft 12.0%

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

      7. metadata-eval 12.0%

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

      8. distribute-lft1-in 12.0%

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

      9. *-commutative 12.0%

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

      10. distribute-lft1-in 12.0%

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

      11. metadata-eval 12.0%

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

      12. mul0-lft 43.2%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{0}\right) \]

   6. Simplified 43.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]

   if 2.9e-246 < (*.f64 M M)

   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. Taylor expanded in c0 around -inf 3.6%

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

      1. fma-def 3.6%

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

      2. times-frac 3.6%

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

      3. unpow2 3.6%

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

      4. unpow2 3.6%

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

      5. *-commutative 3.6%

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

      6. unpow2 3.6%

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

      7. associate-*r* 3.6%

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

   4. Simplified 26.9%

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

   5. Taylor expanded in c0 around 0 31.9%

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

      1. associate-*r/ 31.9%

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

      2. *-commutative 31.9%

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

      3. *-commutative 31.9%

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

      4. unpow2 31.9%

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

      5. unpow2 31.9%

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

      6. unpow2 31.9%

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

   7. Simplified 31.9%

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

   8. Taylor expanded in D around 0 31.9%

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

      1. associate-/l* 33.1%

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

      2. unpow2 33.1%

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

      3. unpow2 33.1%

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

      4. *-commutative 33.1%

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

      5. unpow2 33.1%

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

   10. Simplified 33.1%

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

4. Final simplification 37.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot M \leq 2.9 \cdot 10^{-246}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(c0 \cdot 0\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\\ \end{array} \]

Alternative 8: 48.0% accurate, 10.1× speedup

Math

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

FPCore

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

C

double code(double c0, double w, double h, double D, double d, double M) {
	return (0.25 / d) * (D / (d / (D * (h * (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_1) * (d / (d_1 / (d * (h * (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 / (D * (h * (M * M)))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Taylor expanded in c0 around -inf 5.2%

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

   1. fma-def 5.2%

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

   2. times-frac 5.2%

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

   3. unpow2 5.2%

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

   4. unpow2 5.2%

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

   5. *-commutative 5.2%

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

   6. unpow2 5.2%

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

   7. associate-*r* 5.2%

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

4. Simplified 31.3%

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

5. Taylor expanded in c0 around 0 34.1%

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

   1. associate-*r/ 34.1%

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

   2. *-commutative 34.1%

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

   3. *-commutative 34.1%

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

   4. unpow2 34.1%

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

   5. unpow2 34.1%

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

   6. unpow2 34.1%

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

7. Simplified 34.1%

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

   1. *-un-lft-identity 34.1%

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

   2. times-frac 40.6%

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

   3. associate-*l* 47.0%

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

9. Applied egg-rr 47.0%

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

    1. *-lft-identity 47.0%

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

    2. associate-/l* 47.8%

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

11. Simplified 47.8%

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

12. Final simplification 47.8%

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

Alternative 9: 29.1% accurate, 16.8× speedup

Math

\[\begin{array}{l} \\ \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot 0\right) \end{array} \]

FPCore

(FPCore (c0 w h D d M) :precision binary64 (* (/ c0 (* 2.0 w)) (* c0 0.0)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return (c0 / (2.0 * w)) * (c0 * 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 = (c0 / (2.0d0 * w)) * (c0 * 0.0d0)
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return (c0 / (2.0 * w)) * (c0 * 0.0);
}

Python

def code(c0, w, h, D, d, M):
	return (c0 / (2.0 * w)) * (c0 * 0.0)

Julia

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

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = (c0 / (2.0 * w)) * (c0 * 0.0);
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(c0 * 0.0), $MachinePrecision]), $MachinePrecision]

Derivation

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. associate-*l* 31.5%

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

   2. difference-of-squares 38.2%

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

   3. associate-*l* 38.1%

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

   4. associate-*l* 40.0%

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

3. Simplified 40.0%

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

4. Taylor expanded in c0 around -inf 3.8%

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

   1. associate-*r* 3.8%

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

   2. distribute-rgt1-in 3.8%

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

   3. metadata-eval 3.8%

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

   4. mul0-lft 33.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot \color{blue}{0}\right) \cdot c0\right) \]

   5. metadata-eval 33.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{0} \cdot c0\right) \]

   6. mul0-lft 5.7%

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

   7. metadata-eval 5.7%

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

   8. distribute-lft1-in 5.7%

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

   9. *-commutative 5.7%

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

   10. distribute-lft1-in 5.7%

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

   11. metadata-eval 5.7%

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

   12. mul0-lft 33.4%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{0}\right) \]

6. Simplified 33.4%

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]

7. Final simplification 33.4%

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot 0\right) \]

Reproduce

herbie shell --seed 2023261 
(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))))))