Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.6% → 54.5%

Time: 33.6s

Alternatives: 9

Speedup: 151.0×

• could not determine a ground truth

  1. c0 = -1.772606063918739e+235
  2. w = -1.8041825833482409e-302
  3. h = -7.90792628849085e-251
  4. D = -1.9591251134644717e-236
  5. d = 7.500013407736391e+205
  6. M = -4.1380908369766765e-234

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 24.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 54.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 81.2%

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

   3. Simplified 78.4%

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

   4. Taylor expanded in c0 around inf 82.0%

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

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

   1. Initial program 0.0%

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

   3. Simplified 2.4%

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

   4. Taylor expanded in c0 around -inf 2.9%

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

      1. associate-*r* 2.9%

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

      2. neg-mul-1 2.9%

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

      3. distribute-lft1-in 2.9%

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

      4. metadata-eval 2.9%

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

      5. mul0-lft 37.7%

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

      6. distribute-lft-neg-in 37.7%

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

      7. distribute-rgt-neg-in 37.7%

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

      8. metadata-eval 37.7%

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

   6. Simplified 37.7%

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

   7. Taylor expanded in c0 around 0 44.9%

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

4. Final simplification 55.8%

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

Alternative 2: 54.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((D * D) * (w * h));
	double t_1 = (c0 / (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 = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((D * D) * (w * h))
	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 = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(D * D) * Float64(w * h)))
	t_1 = Float64(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 = 0.0;
	end
	return tmp
end

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(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, 0.0]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 81.2%

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

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

   1. Initial program 0.0%

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

   3. Simplified 2.4%

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

   4. Taylor expanded in c0 around -inf 2.9%

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

      1. associate-*r* 2.9%

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

      2. neg-mul-1 2.9%

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

      3. distribute-lft1-in 2.9%

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

      4. metadata-eval 2.9%

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

      5. mul0-lft 37.7%

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

      6. distribute-lft-neg-in 37.7%

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

      7. distribute-rgt-neg-in 37.7%

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

      8. metadata-eval 37.7%

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

   6. Simplified 37.7%

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

   7. Taylor expanded in c0 around 0 44.9%

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

4. Final simplification 55.6%

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

Alternative 3: 43.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{d}{D}\right)}^{2}\\ \mathbf{if}\;M \cdot M \leq 2 \cdot 10^{-316}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \cdot M \leq 2.6 \cdot 10^{-302}:\\ \;\;\;\;\frac{\frac{c0}{w} \cdot \left(2 \cdot \left(t_0 \cdot \frac{\frac{c0}{w}}{h}\right)\right)}{2}\\ \mathbf{elif}\;M \cdot M \leq 9.8 \cdot 10^{-243}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \cdot M \leq 1.9 \cdot 10^{-72}:\\ \;\;\;\;\frac{\frac{c0}{w}}{2} \cdot \left(2 \cdot \frac{1}{\frac{w}{c0} \cdot \frac{h}{t_0}}\right)\\ \mathbf{elif}\;M \cdot M \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c0}{w} \cdot t_0\right) \cdot \frac{c0}{w \cdot h}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (pow (/ d D) 2.0)))
   (if (<= (* M M) 2e-316)
     0.0
     (if (<= (* M M) 2.6e-302)
       (/ (* (/ c0 w) (* 2.0 (* t_0 (/ (/ c0 w) h)))) 2.0)
       (if (<= (* M M) 9.8e-243)
         0.0
         (if (<= (* M M) 1.9e-72)
           (* (/ (/ c0 w) 2.0) (* 2.0 (/ 1.0 (* (/ w c0) (/ h t_0)))))
           (if (<= (* M M) 1.15e+77)
             0.0
             (* (* (/ c0 w) t_0) (/ c0 (* w h))))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = pow((d / D), 2.0);
	double tmp;
	if ((M * M) <= 2e-316) {
		tmp = 0.0;
	} else if ((M * M) <= 2.6e-302) {
		tmp = ((c0 / w) * (2.0 * (t_0 * ((c0 / w) / h)))) / 2.0;
	} else if ((M * M) <= 9.8e-243) {
		tmp = 0.0;
	} else if ((M * M) <= 1.9e-72) {
		tmp = ((c0 / w) / 2.0) * (2.0 * (1.0 / ((w / c0) * (h / t_0))));
	} else if ((M * M) <= 1.15e+77) {
		tmp = 0.0;
	} else {
		tmp = ((c0 / w) * t_0) * (c0 / (w * h));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (d_1 / d) ** 2.0d0
    if ((m * m) <= 2d-316) then
        tmp = 0.0d0
    else if ((m * m) <= 2.6d-302) then
        tmp = ((c0 / w) * (2.0d0 * (t_0 * ((c0 / w) / h)))) / 2.0d0
    else if ((m * m) <= 9.8d-243) then
        tmp = 0.0d0
    else if ((m * m) <= 1.9d-72) then
        tmp = ((c0 / w) / 2.0d0) * (2.0d0 * (1.0d0 / ((w / c0) * (h / t_0))))
    else if ((m * m) <= 1.15d+77) then
        tmp = 0.0d0
    else
        tmp = ((c0 / w) * t_0) * (c0 / (w * h))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = Math.pow((d / D), 2.0);
	double tmp;
	if ((M * M) <= 2e-316) {
		tmp = 0.0;
	} else if ((M * M) <= 2.6e-302) {
		tmp = ((c0 / w) * (2.0 * (t_0 * ((c0 / w) / h)))) / 2.0;
	} else if ((M * M) <= 9.8e-243) {
		tmp = 0.0;
	} else if ((M * M) <= 1.9e-72) {
		tmp = ((c0 / w) / 2.0) * (2.0 * (1.0 / ((w / c0) * (h / t_0))));
	} else if ((M * M) <= 1.15e+77) {
		tmp = 0.0;
	} else {
		tmp = ((c0 / w) * t_0) * (c0 / (w * h));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = math.pow((d / D), 2.0)
	tmp = 0
	if (M * M) <= 2e-316:
		tmp = 0.0
	elif (M * M) <= 2.6e-302:
		tmp = ((c0 / w) * (2.0 * (t_0 * ((c0 / w) / h)))) / 2.0
	elif (M * M) <= 9.8e-243:
		tmp = 0.0
	elif (M * M) <= 1.9e-72:
		tmp = ((c0 / w) / 2.0) * (2.0 * (1.0 / ((w / c0) * (h / t_0))))
	elif (M * M) <= 1.15e+77:
		tmp = 0.0
	else:
		tmp = ((c0 / w) * t_0) * (c0 / (w * h))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(d / D) ^ 2.0
	tmp = 0.0
	if (Float64(M * M) <= 2e-316)
		tmp = 0.0;
	elseif (Float64(M * M) <= 2.6e-302)
		tmp = Float64(Float64(Float64(c0 / w) * Float64(2.0 * Float64(t_0 * Float64(Float64(c0 / w) / h)))) / 2.0);
	elseif (Float64(M * M) <= 9.8e-243)
		tmp = 0.0;
	elseif (Float64(M * M) <= 1.9e-72)
		tmp = Float64(Float64(Float64(c0 / w) / 2.0) * Float64(2.0 * Float64(1.0 / Float64(Float64(w / c0) * Float64(h / t_0)))));
	elseif (Float64(M * M) <= 1.15e+77)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(c0 / w) * t_0) * Float64(c0 / Float64(w * h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (d / D) ^ 2.0;
	tmp = 0.0;
	if ((M * M) <= 2e-316)
		tmp = 0.0;
	elseif ((M * M) <= 2.6e-302)
		tmp = ((c0 / w) * (2.0 * (t_0 * ((c0 / w) / h)))) / 2.0;
	elseif ((M * M) <= 9.8e-243)
		tmp = 0.0;
	elseif ((M * M) <= 1.9e-72)
		tmp = ((c0 / w) / 2.0) * (2.0 * (1.0 / ((w / c0) * (h / t_0))));
	elseif ((M * M) <= 1.15e+77)
		tmp = 0.0;
	else
		tmp = ((c0 / w) * t_0) * (c0 / (w * h));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(M * M), $MachinePrecision], 2e-316], 0.0, If[LessEqual[N[(M * M), $MachinePrecision], 2.6e-302], N[(N[(N[(c0 / w), $MachinePrecision] * N[(2.0 * N[(t$95$0 * N[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[N[(M * M), $MachinePrecision], 9.8e-243], 0.0, If[LessEqual[N[(M * M), $MachinePrecision], 1.9e-72], N[(N[(N[(c0 / w), $MachinePrecision] / 2.0), $MachinePrecision] * N[(2.0 * N[(1.0 / N[(N[(w / c0), $MachinePrecision] * N[(h / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(M * M), $MachinePrecision], 1.15e+77], 0.0, N[(N[(N[(c0 / w), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 M M) < 2.000000017e-316 or 2.60000000000000011e-302 < (*.f64 M M) < 9.8e-243 or 1.90000000000000001e-72 < (*.f64 M M) < 1.14999999999999997e77

   1. Initial program 23.7%

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

   3. Simplified 25.4%

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

   4. Taylor expanded in c0 around -inf 10.4%

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

      1. associate-*r* 10.4%

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

      2. neg-mul-1 10.4%

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

      3. distribute-lft1-in 10.4%

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

      4. metadata-eval 10.4%

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

      5. mul0-lft 45.4%

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

      6. distribute-lft-neg-in 45.4%

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

      7. distribute-rgt-neg-in 45.4%

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

      8. metadata-eval 45.4%

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

   6. Simplified 45.4%

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

   7. Taylor expanded in c0 around 0 54.7%

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

   if 2.000000017e-316 < (*.f64 M M) < 2.60000000000000011e-302

   1. Initial program 60.6%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in h around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. *-commutative 0.0%

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

      4. times-frac 0.0%

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

      5. associate-/r* 0.0%

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

      6. times-frac 0.0%

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

      7. *-commutative 0.0%

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

      8. *-commutative 0.0%

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

      9. times-frac 0.0%

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

      10. associate-/r* 0.0%

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

   6. Simplified 60.6%

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

      1. expm1-log1p-u 60.6%

         \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{{d}^{2}}{{D}^{2}}, \frac{c0}{h \cdot w}, \frac{0.5 \cdot \left({D}^{2} \cdot \left(w \cdot 0\right)\right)}{c0 \cdot {d}^{2}}\right)\right)\right)}\right) \]

      2. expm1-udef 60.6%

         \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{{d}^{2}}{{D}^{2}}, \frac{c0}{h \cdot w}, \frac{0.5 \cdot \left({D}^{2} \cdot \left(w \cdot 0\right)\right)}{c0 \cdot {d}^{2}}\right)\right)} - 1}\right) \]

   8. Applied egg-rr 82.0%

      \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{\frac{c0}{h}}{w}, {\left(\frac{d}{D}\right)}^{2}, \frac{\frac{0}{c0}}{{d}^{2}}\right)\right)} - 1}\right) \]
   9. Step-by-step derivation

      1. expm1-def 97.9%

         \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{\frac{c0}{h}}{w}, {\left(\frac{d}{D}\right)}^{2}, \frac{\frac{0}{c0}}{{d}^{2}}\right)\right)\right)}\right) \]

      2. expm1-log1p 99.7%

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

      3. associate-/l/ 98.6%

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

      4. *-commutative 98.6%

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

      5. div0 98.6%

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

      6. div0 98.6%

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

   10. Simplified 98.6%

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

       1. associate-/r* 98.4%

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

       2. pow2 98.4%

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

       3. fma-def 98.4%

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

       4. associate-*l/ 98.3%

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

       5. frac-times 98.6%

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

       6. pow2 98.6%

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

       7. *-commutative 98.6%

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

       8. div0 98.6%

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

       9. fma-def 98.6%

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

       10. associate-*l/ 98.6%

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

       11. frac-times 99.7%

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

       12. fma-def 99.7%

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

       13. pow2 99.7%

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

       14. div0 99.7%

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

   12. Applied egg-rr 99.7%

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

       1. fma-def 99.7%

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

       2. +-rgt-identity 99.7%

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

       3. count-2 99.7%

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

       4. associate-*l/ 99.7%

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

       5. associate-*r/ 99.7%

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

       6. associate-/r* 98.3%

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

       7. *-commutative 98.3%

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

       8. associate-*r/ 98.4%

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

   14. Simplified 98.4%

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

       1. associate-*l/ 98.4%

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

       2. associate-/l/ 99.7%

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

   16. Applied egg-rr 99.7%

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

   if 9.8e-243 < (*.f64 M M) < 1.90000000000000001e-72

   1. Initial program 42.5%

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

   3. Simplified 54.6%

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

   4. Taylor expanded in h around 0 0.2%

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

      1. +-commutative 0.2%

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

      2. *-commutative 0.2%

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

      3. *-commutative 0.2%

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

      4. times-frac 3.8%

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

      5. associate-/r* 3.8%

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

      6. times-frac 0.2%

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

      7. *-commutative 0.2%

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

      8. *-commutative 0.2%

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

      9. times-frac 3.8%

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

      10. associate-/r* 3.8%

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

   6. Simplified 46.4%

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

      1. expm1-log1p-u 45.8%

         \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{{d}^{2}}{{D}^{2}}, \frac{c0}{h \cdot w}, \frac{0.5 \cdot \left({D}^{2} \cdot \left(w \cdot 0\right)\right)}{c0 \cdot {d}^{2}}\right)\right)\right)}\right) \]

      2. expm1-udef 45.8%

         \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{{d}^{2}}{{D}^{2}}, \frac{c0}{h \cdot w}, \frac{0.5 \cdot \left({D}^{2} \cdot \left(w \cdot 0\right)\right)}{c0 \cdot {d}^{2}}\right)\right)} - 1}\right) \]

   8. Applied egg-rr 49.7%

      \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{\frac{c0}{h}}{w}, {\left(\frac{d}{D}\right)}^{2}, \frac{\frac{0}{c0}}{{d}^{2}}\right)\right)} - 1}\right) \]
   9. Step-by-step derivation

      1. expm1-def 49.7%

         \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{\frac{c0}{h}}{w}, {\left(\frac{d}{D}\right)}^{2}, \frac{\frac{0}{c0}}{{d}^{2}}\right)\right)\right)}\right) \]

      2. expm1-log1p 50.4%

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

      3. associate-/l/ 46.6%

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

      4. *-commutative 46.6%

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

      5. div0 46.6%

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

      6. div0 54.6%

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

   10. Simplified 54.6%

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

       1. associate-/r* 54.7%

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

       2. pow2 54.7%

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

       3. fma-def 54.7%

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

       4. associate-*l/ 54.7%

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

       5. frac-times 54.6%

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

       6. pow2 54.6%

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

       7. *-commutative 54.6%

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

       8. div0 54.6%

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

       9. fma-def 54.6%

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

       10. associate-*l/ 54.6%

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

       11. frac-times 58.7%

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

       12. fma-def 58.7%

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

       13. pow2 58.7%

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

       14. div0 58.7%

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

   12. Applied egg-rr 58.7%

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

       1. fma-def 58.7%

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

       2. +-rgt-identity 58.7%

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

       3. count-2 58.7%

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

       4. associate-*l/ 58.5%

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

       5. associate-*r/ 58.6%

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

       6. associate-/r* 54.7%

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

       7. *-commutative 54.7%

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

       8. associate-*r/ 54.7%

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

   14. Simplified 54.7%

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

       1. pow2 54.7%

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

       2. associate-/r* 58.4%

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

       3. add-cbrt-cube 47.3%

          \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \left(2 \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \color{blue}{\sqrt[3]{\left(\frac{\frac{c0}{h}}{w} \cdot \frac{\frac{c0}{h}}{w}\right) \cdot \frac{\frac{c0}{h}}{w}}}\right)\right) \]

       4. unpow3 47.3%

          \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \left(2 \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \sqrt[3]{\color{blue}{{\left(\frac{\frac{c0}{h}}{w}\right)}^{3}}}\right)\right) \]

       5. *-commutative 47.3%

          \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \left(2 \cdot \color{blue}{\left(\sqrt[3]{{\left(\frac{\frac{c0}{h}}{w}\right)}^{3}} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}\right) \]

       6. unpow3 47.3%

          \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \left(2 \cdot \left(\sqrt[3]{\color{blue}{\left(\frac{\frac{c0}{h}}{w} \cdot \frac{\frac{c0}{h}}{w}\right) \cdot \frac{\frac{c0}{h}}{w}}} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\right) \]

       7. add-cbrt-cube 58.4%

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

       8. associate-/l/ 54.7%

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

       9. frac-times 46.6%

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

       10. times-frac 43.0%

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

       11. clear-num 43.0%

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

       12. associate-*l* 39.3%

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

       13. pow2 39.3%

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

       14. pow2 39.3%

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

   16. Applied egg-rr 39.3%

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

       1. times-frac 39.4%

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

       2. associate-/l* 50.7%

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

       3. unpow2 50.7%

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

       4. unpow2 50.7%

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

       5. times-frac 58.8%

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

       6. unpow2 58.8%

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

   18. Simplified 58.8%

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

   if 1.14999999999999997e77 < (*.f64 M M)

   1. Initial program 17.0%

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

   3. Simplified 41.0%

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

   4. Taylor expanded in h around 0 3.5%

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

      1. +-commutative 3.5%

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

      2. *-commutative 3.5%

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

      3. *-commutative 3.5%

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

      4. times-frac 3.5%

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

      5. associate-/r* 3.5%

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

      6. times-frac 3.2%

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

      7. *-commutative 3.2%

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

      8. *-commutative 3.2%

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

      9. times-frac 3.5%

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

      10. associate-/r* 3.5%

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

   6. Simplified 36.5%

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

      1. expm1-log1p-u 31.4%

         \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{{d}^{2}}{{D}^{2}}, \frac{c0}{h \cdot w}, \frac{0.5 \cdot \left({D}^{2} \cdot \left(w \cdot 0\right)\right)}{c0 \cdot {d}^{2}}\right)\right)\right)}\right) \]

      2. expm1-udef 31.4%

         \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{{d}^{2}}{{D}^{2}}, \frac{c0}{h \cdot w}, \frac{0.5 \cdot \left({D}^{2} \cdot \left(w \cdot 0\right)\right)}{c0 \cdot {d}^{2}}\right)\right)} - 1}\right) \]

   8. Applied egg-rr 35.5%

      \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{\frac{c0}{h}}{w}, {\left(\frac{d}{D}\right)}^{2}, \frac{\frac{0}{c0}}{{d}^{2}}\right)\right)} - 1}\right) \]
   9. Step-by-step derivation

      1. expm1-def 35.5%

         \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{\frac{c0}{h}}{w}, {\left(\frac{d}{D}\right)}^{2}, \frac{\frac{0}{c0}}{{d}^{2}}\right)\right)\right)}\right) \]

      2. expm1-log1p 43.0%

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

      3. associate-/l/ 43.0%

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

      4. *-commutative 43.0%

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

      5. div0 43.0%

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

      6. div0 49.2%

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

   10. Simplified 49.2%

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

       1. associate-/r* 49.2%

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

       2. pow2 49.2%

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

       3. fma-def 49.2%

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

       4. associate-*l/ 49.2%

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

       5. frac-times 49.0%

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

       6. pow2 49.0%

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

       7. *-commutative 49.0%

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

       8. div0 49.0%

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

       9. fma-def 49.0%

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

       10. associate-*l/ 49.0%

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

       11. frac-times 49.1%

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

       12. fma-def 49.1%

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

       13. pow2 49.1%

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

       14. div0 49.1%

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

   12. Applied egg-rr 49.1%

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

       1. fma-def 49.1%

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

       2. +-rgt-identity 49.1%

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

       3. count-2 49.1%

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

       4. associate-*l/ 49.1%

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

       5. associate-*r/ 49.1%

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

       6. associate-/r* 49.3%

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

       7. *-commutative 49.3%

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

       8. associate-*r/ 49.2%

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

   14. Simplified 49.2%

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

   16. Applied egg-rr 21.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\frac{c0}{w} \cdot 0.5\right) \cdot \left(2 \cdot \left({\left(\frac{d}{D}\right)}^{2} \cdot \frac{\frac{c0}{w}}{h}\right)\right)\right)} - 1} \]
   17. Step-by-step derivation

       1. expm1-def 21.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{c0}{w} \cdot 0.5\right) \cdot \left(2 \cdot \left({\left(\frac{d}{D}\right)}^{2} \cdot \frac{\frac{c0}{w}}{h}\right)\right)\right)\right)} \]

       2. expm1-log1p 49.2%

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

       3. associate-*r* 49.2%

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

       4. associate-*r* 50.2%

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

       5. associate-*l* 50.2%

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

       6. metadata-eval 50.2%

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

       7. *-rgt-identity 50.2%

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

       8. unpow2 50.2%

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

       9. times-frac 36.5%

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

       10. unpow2 36.5%

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

       11. unpow2 36.5%

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

       12. times-frac 37.4%

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

       13. *-commutative 37.4%

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

       14. *-commutative 37.4%

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

       15. times-frac 36.5%

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

       16. unpow2 36.5%

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

       17. unpow2 36.5%

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

       18. times-frac 50.2%

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

       19. unpow2 50.2%

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

       20. associate-/l/ 50.1%

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

   18. Simplified 50.1%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot M \leq 2 \cdot 10^{-316}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \cdot M \leq 2.6 \cdot 10^{-302}:\\ \;\;\;\;\frac{\frac{c0}{w} \cdot \left(2 \cdot \left({\left(\frac{d}{D}\right)}^{2} \cdot \frac{\frac{c0}{w}}{h}\right)\right)}{2}\\ \mathbf{elif}\;M \cdot M \leq 9.8 \cdot 10^{-243}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \cdot M \leq 1.9 \cdot 10^{-72}:\\ \;\;\;\;\frac{\frac{c0}{w}}{2} \cdot \left(2 \cdot \frac{1}{\frac{w}{c0} \cdot \frac{h}{{\left(\frac{d}{D}\right)}^{2}}}\right)\\ \mathbf{elif}\;M \cdot M \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c0}{w} \cdot {\left(\frac{d}{D}\right)}^{2}\right) \cdot \frac{c0}{w \cdot h}\\ \end{array} \]

Alternative 4: 38.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{d}{D}\right)}^{2}\\ t_1 := \frac{\frac{c0}{w} \cdot \left(2 \cdot \left(t_0 \cdot \frac{\frac{c0}{w}}{h}\right)\right)}{2}\\ \mathbf{if}\;M \leq 1.9 \cdot 10^{-158}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 7 \cdot 10^{-149}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;M \leq 1.3 \cdot 10^{-121}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 3.2 \cdot 10^{-37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;M \leq 1.5 \cdot 10^{+38}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c0}{w} \cdot t_0\right) \cdot \frac{c0}{w \cdot h}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (pow (/ d D) 2.0))
        (t_1 (/ (* (/ c0 w) (* 2.0 (* t_0 (/ (/ c0 w) h)))) 2.0)))
   (if (<= M 1.9e-158)
     0.0
     (if (<= M 7e-149)
       t_1
       (if (<= M 1.3e-121)
         0.0
         (if (<= M 3.2e-37)
           t_1
           (if (<= M 1.5e+38) 0.0 (* (* (/ c0 w) t_0) (/ c0 (* w h))))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = pow((d / D), 2.0);
	double t_1 = ((c0 / w) * (2.0 * (t_0 * ((c0 / w) / h)))) / 2.0;
	double tmp;
	if (M <= 1.9e-158) {
		tmp = 0.0;
	} else if (M <= 7e-149) {
		tmp = t_1;
	} else if (M <= 1.3e-121) {
		tmp = 0.0;
	} else if (M <= 3.2e-37) {
		tmp = t_1;
	} else if (M <= 1.5e+38) {
		tmp = 0.0;
	} else {
		tmp = ((c0 / w) * t_0) * (c0 / (w * h));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (d_1 / d) ** 2.0d0
    t_1 = ((c0 / w) * (2.0d0 * (t_0 * ((c0 / w) / h)))) / 2.0d0
    if (m <= 1.9d-158) then
        tmp = 0.0d0
    else if (m <= 7d-149) then
        tmp = t_1
    else if (m <= 1.3d-121) then
        tmp = 0.0d0
    else if (m <= 3.2d-37) then
        tmp = t_1
    else if (m <= 1.5d+38) then
        tmp = 0.0d0
    else
        tmp = ((c0 / w) * t_0) * (c0 / (w * h))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = Math.pow((d / D), 2.0);
	double t_1 = ((c0 / w) * (2.0 * (t_0 * ((c0 / w) / h)))) / 2.0;
	double tmp;
	if (M <= 1.9e-158) {
		tmp = 0.0;
	} else if (M <= 7e-149) {
		tmp = t_1;
	} else if (M <= 1.3e-121) {
		tmp = 0.0;
	} else if (M <= 3.2e-37) {
		tmp = t_1;
	} else if (M <= 1.5e+38) {
		tmp = 0.0;
	} else {
		tmp = ((c0 / w) * t_0) * (c0 / (w * h));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = math.pow((d / D), 2.0)
	t_1 = ((c0 / w) * (2.0 * (t_0 * ((c0 / w) / h)))) / 2.0
	tmp = 0
	if M <= 1.9e-158:
		tmp = 0.0
	elif M <= 7e-149:
		tmp = t_1
	elif M <= 1.3e-121:
		tmp = 0.0
	elif M <= 3.2e-37:
		tmp = t_1
	elif M <= 1.5e+38:
		tmp = 0.0
	else:
		tmp = ((c0 / w) * t_0) * (c0 / (w * h))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(d / D) ^ 2.0
	t_1 = Float64(Float64(Float64(c0 / w) * Float64(2.0 * Float64(t_0 * Float64(Float64(c0 / w) / h)))) / 2.0)
	tmp = 0.0
	if (M <= 1.9e-158)
		tmp = 0.0;
	elseif (M <= 7e-149)
		tmp = t_1;
	elseif (M <= 1.3e-121)
		tmp = 0.0;
	elseif (M <= 3.2e-37)
		tmp = t_1;
	elseif (M <= 1.5e+38)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(c0 / w) * t_0) * Float64(c0 / Float64(w * h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (d / D) ^ 2.0;
	t_1 = ((c0 / w) * (2.0 * (t_0 * ((c0 / w) / h)))) / 2.0;
	tmp = 0.0;
	if (M <= 1.9e-158)
		tmp = 0.0;
	elseif (M <= 7e-149)
		tmp = t_1;
	elseif (M <= 1.3e-121)
		tmp = 0.0;
	elseif (M <= 3.2e-37)
		tmp = t_1;
	elseif (M <= 1.5e+38)
		tmp = 0.0;
	else
		tmp = ((c0 / w) * t_0) * (c0 / (w * h));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(c0 / w), $MachinePrecision] * N[(2.0 * N[(t$95$0 * N[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[M, 1.9e-158], 0.0, If[LessEqual[M, 7e-149], t$95$1, If[LessEqual[M, 1.3e-121], 0.0, If[LessEqual[M, 3.2e-37], t$95$1, If[LessEqual[M, 1.5e+38], 0.0, N[(N[(N[(c0 / w), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if M < 1.8999999999999999e-158 or 7e-149 < M < 1.29999999999999993e-121 or 3.1999999999999999e-37 < M < 1.5000000000000001e38

   1. Initial program 24.7%

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

   3. Simplified 25.8%

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

   4. Taylor expanded in c0 around -inf 6.8%

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

      1. associate-*r* 6.8%

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

      2. neg-mul-1 6.8%

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

      3. distribute-lft1-in 6.8%

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

      4. metadata-eval 6.8%

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

      5. mul0-lft 34.3%

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

      6. distribute-lft-neg-in 34.3%

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

      7. distribute-rgt-neg-in 34.3%

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

      8. metadata-eval 34.3%

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

   6. Simplified 34.3%

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

   7. Taylor expanded in c0 around 0 41.0%

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

   if 1.8999999999999999e-158 < M < 7e-149 or 1.29999999999999993e-121 < M < 3.1999999999999999e-37

   1. Initial program 33.5%

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

   3. Simplified 54.1%

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

   4. Taylor expanded in h around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. *-commutative 0.0%

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

      4. times-frac 0.0%

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

      5. associate-/r* 0.0%

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

      6. times-frac 0.0%

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

      7. *-commutative 0.0%

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

      8. *-commutative 0.0%

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

      9. times-frac 0.0%

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

      10. associate-/r* 0.0%

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

   6. Simplified 33.7%

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

      1. expm1-log1p-u 33.5%

         \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{{d}^{2}}{{D}^{2}}, \frac{c0}{h \cdot w}, \frac{0.5 \cdot \left({D}^{2} \cdot \left(w \cdot 0\right)\right)}{c0 \cdot {d}^{2}}\right)\right)\right)}\right) \]

      2. expm1-udef 33.5%

         \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{{d}^{2}}{{D}^{2}}, \frac{c0}{h \cdot w}, \frac{0.5 \cdot \left({D}^{2} \cdot \left(w \cdot 0\right)\right)}{c0 \cdot {d}^{2}}\right)\right)} - 1}\right) \]

   8. Applied egg-rr 39.4%

      \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{\frac{c0}{h}}{w}, {\left(\frac{d}{D}\right)}^{2}, \frac{\frac{0}{c0}}{{d}^{2}}\right)\right)} - 1}\right) \]
   9. Step-by-step derivation

      1. expm1-def 39.4%

         \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{\frac{c0}{h}}{w}, {\left(\frac{d}{D}\right)}^{2}, \frac{\frac{0}{c0}}{{d}^{2}}\right)\right)\right)}\right) \]

      2. expm1-log1p 40.4%

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

      3. associate-/l/ 40.0%

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

      4. *-commutative 40.0%

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

      5. div0 40.0%

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

      6. div0 54.0%

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

   10. Simplified 54.0%

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

       1. associate-/r* 54.0%

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

       2. pow2 54.0%

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

       3. fma-def 54.0%

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

       4. associate-*l/ 54.0%

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

       5. frac-times 54.0%

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

       6. pow2 54.0%

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

       7. *-commutative 54.0%

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

       8. div0 54.0%

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

       9. fma-def 54.0%

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

       10. associate-*l/ 54.0%

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

       11. frac-times 54.6%

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

       12. fma-def 54.6%

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

       13. pow2 54.6%

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

       14. div0 54.6%

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

   12. Applied egg-rr 54.6%

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

       1. fma-def 54.6%

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

       2. +-rgt-identity 54.6%

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

       3. count-2 54.6%

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

       4. associate-*l/ 54.2%

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

       5. associate-*r/ 54.2%

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

       6. associate-/r* 54.0%

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

       7. *-commutative 54.0%

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

       8. associate-*r/ 54.0%

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

   14. Simplified 54.0%

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

       1. associate-*l/ 54.0%

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

       2. associate-/l/ 54.3%

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

   16. Applied egg-rr 54.3%

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

   if 1.5000000000000001e38 < M

   1. Initial program 16.6%

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

   3. Simplified 39.3%

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

   4. Taylor expanded in h around 0 7.4%

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

      1. +-commutative 7.4%

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

      2. *-commutative 7.4%

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

      3. *-commutative 7.4%

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

      4. times-frac 7.4%

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

      5. associate-/r* 7.4%

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

      6. times-frac 6.8%

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

      7. *-commutative 6.8%

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

      8. *-commutative 6.8%

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

      9. times-frac 7.4%

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

      10. associate-/r* 7.4%

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

   6. Simplified 37.4%

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

      1. expm1-log1p-u 29.7%

         \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{{d}^{2}}{{D}^{2}}, \frac{c0}{h \cdot w}, \frac{0.5 \cdot \left({D}^{2} \cdot \left(w \cdot 0\right)\right)}{c0 \cdot {d}^{2}}\right)\right)\right)}\right) \]

      2. expm1-udef 29.7%

         \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{{d}^{2}}{{D}^{2}}, \frac{c0}{h \cdot w}, \frac{0.5 \cdot \left({D}^{2} \cdot \left(w \cdot 0\right)\right)}{c0 \cdot {d}^{2}}\right)\right)} - 1}\right) \]

   8. Applied egg-rr 34.1%

      \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{\frac{c0}{h}}{w}, {\left(\frac{d}{D}\right)}^{2}, \frac{\frac{0}{c0}}{{d}^{2}}\right)\right)} - 1}\right) \]
   9. Step-by-step derivation

      1. expm1-def 34.1%

         \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{\frac{c0}{h}}{w}, {\left(\frac{d}{D}\right)}^{2}, \frac{\frac{0}{c0}}{{d}^{2}}\right)\right)\right)}\right) \]

      2. expm1-log1p 44.5%

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

      3. associate-/l/ 44.5%

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

      4. *-commutative 44.5%

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

      5. div0 44.5%

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

      6. div0 51.3%

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

   10. Simplified 51.3%

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

       1. associate-/r* 51.3%

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

       2. pow2 51.3%

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

       3. fma-def 51.3%

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

       4. associate-*l/ 51.3%

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

       5. frac-times 50.8%

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

       6. pow2 50.8%

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

       7. *-commutative 50.8%

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

       8. div0 50.8%

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

       9. fma-def 50.8%

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

       10. associate-*l/ 50.9%

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

       11. frac-times 50.9%

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

       12. fma-def 50.9%

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

       13. pow2 50.9%

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

       14. div0 50.9%

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

   12. Applied egg-rr 50.9%

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

       1. fma-def 50.9%

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

       2. +-rgt-identity 50.9%

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

       3. count-2 50.9%

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

       4. associate-*l/ 50.9%

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

       5. associate-*r/ 50.9%

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

       6. associate-/r* 51.4%

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

       7. *-commutative 51.4%

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

       8. associate-*r/ 51.3%

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

   14. Simplified 51.3%

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

   16. Applied egg-rr 14.2%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\frac{c0}{w} \cdot 0.5\right) \cdot \left(2 \cdot \left({\left(\frac{d}{D}\right)}^{2} \cdot \frac{\frac{c0}{w}}{h}\right)\right)\right)} - 1} \]
   17. Step-by-step derivation

       1. expm1-def 14.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{c0}{w} \cdot 0.5\right) \cdot \left(2 \cdot \left({\left(\frac{d}{D}\right)}^{2} \cdot \frac{\frac{c0}{w}}{h}\right)\right)\right)\right)} \]

       2. expm1-log1p 51.3%

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

       3. associate-*r* 51.3%

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

       4. associate-*r* 51.3%

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

       5. associate-*l* 51.3%

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

       6. metadata-eval 51.3%

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

       7. *-rgt-identity 51.3%

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

       8. unpow2 51.3%

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

       9. times-frac 37.4%

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

       10. unpow2 37.4%

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

       11. unpow2 37.4%

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

       12. times-frac 37.3%

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

       13. *-commutative 37.3%

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

       14. *-commutative 37.3%

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

       15. times-frac 37.4%

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

       16. unpow2 37.4%

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

       17. unpow2 37.4%

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

       18. times-frac 51.3%

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

       19. unpow2 51.3%

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

       20. associate-/l/ 51.3%

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

   18. Simplified 51.3%

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

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.9 \cdot 10^{-158}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 7 \cdot 10^{-149}:\\ \;\;\;\;\frac{\frac{c0}{w} \cdot \left(2 \cdot \left({\left(\frac{d}{D}\right)}^{2} \cdot \frac{\frac{c0}{w}}{h}\right)\right)}{2}\\ \mathbf{elif}\;M \leq 1.3 \cdot 10^{-121}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 3.2 \cdot 10^{-37}:\\ \;\;\;\;\frac{\frac{c0}{w} \cdot \left(2 \cdot \left({\left(\frac{d}{D}\right)}^{2} \cdot \frac{\frac{c0}{w}}{h}\right)\right)}{2}\\ \mathbf{elif}\;M \leq 1.5 \cdot 10^{+38}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c0}{w} \cdot {\left(\frac{d}{D}\right)}^{2}\right) \cdot \frac{c0}{w \cdot h}\\ \end{array} \]

Alternative 5: 38.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{w \cdot h}\\ t_1 := \left(\frac{c0}{w} \cdot {\left(\frac{d}{D}\right)}^{2}\right) \cdot t_0\\ \mathbf{if}\;M \leq 1.72 \cdot 10^{-158}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 6.8 \cdot 10^{-155}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;M \leq 1.3 \cdot 10^{-121}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 5.5 \cdot 10^{-38}:\\ \;\;\;\;\frac{\frac{c0}{w}}{2} \cdot \left(2 \cdot \left(t_0 \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\right)\\ \mathbf{elif}\;M \leq 2.2 \cdot 10^{+38}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* w h))) (t_1 (* (* (/ c0 w) (pow (/ d D) 2.0)) t_0)))
   (if (<= M 1.72e-158)
     0.0
     (if (<= M 6.8e-155)
       t_1
       (if (<= M 1.3e-121)
         0.0
         (if (<= M 5.5e-38)
           (* (/ (/ c0 w) 2.0) (* 2.0 (* t_0 (* (/ d D) (/ d D)))))
           (if (<= M 2.2e+38) 0.0 t_1)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (w * h);
	double t_1 = ((c0 / w) * pow((d / D), 2.0)) * t_0;
	double tmp;
	if (M <= 1.72e-158) {
		tmp = 0.0;
	} else if (M <= 6.8e-155) {
		tmp = t_1;
	} else if (M <= 1.3e-121) {
		tmp = 0.0;
	} else if (M <= 5.5e-38) {
		tmp = ((c0 / w) / 2.0) * (2.0 * (t_0 * ((d / D) * (d / D))));
	} else if (M <= 2.2e+38) {
		tmp = 0.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c0 / (w * h)
    t_1 = ((c0 / w) * ((d_1 / d) ** 2.0d0)) * t_0
    if (m <= 1.72d-158) then
        tmp = 0.0d0
    else if (m <= 6.8d-155) then
        tmp = t_1
    else if (m <= 1.3d-121) then
        tmp = 0.0d0
    else if (m <= 5.5d-38) then
        tmp = ((c0 / w) / 2.0d0) * (2.0d0 * (t_0 * ((d_1 / d) * (d_1 / d))))
    else if (m <= 2.2d+38) then
        tmp = 0.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (w * h);
	double t_1 = ((c0 / w) * Math.pow((d / D), 2.0)) * t_0;
	double tmp;
	if (M <= 1.72e-158) {
		tmp = 0.0;
	} else if (M <= 6.8e-155) {
		tmp = t_1;
	} else if (M <= 1.3e-121) {
		tmp = 0.0;
	} else if (M <= 5.5e-38) {
		tmp = ((c0 / w) / 2.0) * (2.0 * (t_0 * ((d / D) * (d / D))));
	} else if (M <= 2.2e+38) {
		tmp = 0.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (w * h)
	t_1 = ((c0 / w) * math.pow((d / D), 2.0)) * t_0
	tmp = 0
	if M <= 1.72e-158:
		tmp = 0.0
	elif M <= 6.8e-155:
		tmp = t_1
	elif M <= 1.3e-121:
		tmp = 0.0
	elif M <= 5.5e-38:
		tmp = ((c0 / w) / 2.0) * (2.0 * (t_0 * ((d / D) * (d / D))))
	elif M <= 2.2e+38:
		tmp = 0.0
	else:
		tmp = t_1
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(w * h))
	t_1 = Float64(Float64(Float64(c0 / w) * (Float64(d / D) ^ 2.0)) * t_0)
	tmp = 0.0
	if (M <= 1.72e-158)
		tmp = 0.0;
	elseif (M <= 6.8e-155)
		tmp = t_1;
	elseif (M <= 1.3e-121)
		tmp = 0.0;
	elseif (M <= 5.5e-38)
		tmp = Float64(Float64(Float64(c0 / w) / 2.0) * Float64(2.0 * Float64(t_0 * Float64(Float64(d / D) * Float64(d / D)))));
	elseif (M <= 2.2e+38)
		tmp = 0.0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (w * h);
	t_1 = ((c0 / w) * ((d / D) ^ 2.0)) * t_0;
	tmp = 0.0;
	if (M <= 1.72e-158)
		tmp = 0.0;
	elseif (M <= 6.8e-155)
		tmp = t_1;
	elseif (M <= 1.3e-121)
		tmp = 0.0;
	elseif (M <= 5.5e-38)
		tmp = ((c0 / w) / 2.0) * (2.0 * (t_0 * ((d / D) * (d / D))));
	elseif (M <= 2.2e+38)
		tmp = 0.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(c0 / w), $MachinePrecision] * N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[M, 1.72e-158], 0.0, If[LessEqual[M, 6.8e-155], t$95$1, If[LessEqual[M, 1.3e-121], 0.0, If[LessEqual[M, 5.5e-38], N[(N[(N[(c0 / w), $MachinePrecision] / 2.0), $MachinePrecision] * N[(2.0 * N[(t$95$0 * N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 2.2e+38], 0.0, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if M < 1.72e-158 or 6.8e-155 < M < 1.29999999999999993e-121 or 5.50000000000000005e-38 < M < 2.20000000000000006e38

   1. Initial program 24.7%

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

   3. Simplified 25.8%

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

   4. Taylor expanded in c0 around -inf 6.8%

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

      1. associate-*r* 6.8%

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

      2. neg-mul-1 6.8%

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

      3. distribute-lft1-in 6.8%

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

      4. metadata-eval 6.8%

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

      5. mul0-lft 34.3%

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

      6. distribute-lft-neg-in 34.3%

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

      7. distribute-rgt-neg-in 34.3%

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

      8. metadata-eval 34.3%

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

   6. Simplified 34.3%

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

   7. Taylor expanded in c0 around 0 41.0%

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

   if 1.72e-158 < M < 6.8e-155 or 2.20000000000000006e38 < M

   1. Initial program 18.1%

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

   3. Simplified 41.9%

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

   4. Taylor expanded in h around 0 7.1%

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

      1. +-commutative 7.1%

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

      2. *-commutative 7.1%

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

      3. *-commutative 7.1%

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

      4. times-frac 7.1%

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

      5. associate-/r* 7.1%

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

      6. times-frac 6.5%

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

      7. *-commutative 6.5%

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

      8. *-commutative 6.5%

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

      9. times-frac 7.1%

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

      10. associate-/r* 7.1%

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

   6. Simplified 38.0%

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

      1. expm1-log1p-u 30.6%

         \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{{d}^{2}}{{D}^{2}}, \frac{c0}{h \cdot w}, \frac{0.5 \cdot \left({D}^{2} \cdot \left(w \cdot 0\right)\right)}{c0 \cdot {d}^{2}}\right)\right)\right)}\right) \]

      2. expm1-udef 30.6%

         \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{{d}^{2}}{{D}^{2}}, \frac{c0}{h \cdot w}, \frac{0.5 \cdot \left({D}^{2} \cdot \left(w \cdot 0\right)\right)}{c0 \cdot {d}^{2}}\right)\right)} - 1}\right) \]

   8. Applied egg-rr 36.8%

      \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{\frac{c0}{h}}{w}, {\left(\frac{d}{D}\right)}^{2}, \frac{\frac{0}{c0}}{{d}^{2}}\right)\right)} - 1}\right) \]
   9. Step-by-step derivation

      1. expm1-def 36.8%

         \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{\frac{c0}{h}}{w}, {\left(\frac{d}{D}\right)}^{2}, \frac{\frac{0}{c0}}{{d}^{2}}\right)\right)\right)}\right) \]

      2. expm1-log1p 46.8%

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

      3. associate-/l/ 46.8%

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

      4. *-commutative 46.8%

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

      5. div0 46.8%

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

      6. div0 53.3%

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

   10. Simplified 53.3%

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

       1. associate-/r* 53.3%

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

       2. pow2 53.3%

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

       3. fma-def 53.3%

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

       4. associate-*l/ 53.3%

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

       5. frac-times 52.8%

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

       6. pow2 52.8%

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

       7. *-commutative 52.8%

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

       8. div0 52.8%

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

       9. fma-def 52.8%

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

       10. associate-*l/ 52.9%

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

       11. frac-times 53.0%

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

       12. fma-def 53.0%

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

       13. pow2 53.0%

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

       14. div0 53.0%

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

   12. Applied egg-rr 53.0%

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

       1. fma-def 53.0%

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

       2. +-rgt-identity 53.0%

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

       3. count-2 53.0%

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

       4. associate-*l/ 53.0%

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

       5. associate-*r/ 52.9%

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

       6. associate-/r* 53.3%

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

       7. *-commutative 53.3%

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

       8. associate-*r/ 53.3%

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

   14. Simplified 53.3%

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

   16. Applied egg-rr 13.6%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\frac{c0}{w} \cdot 0.5\right) \cdot \left(2 \cdot \left({\left(\frac{d}{D}\right)}^{2} \cdot \frac{\frac{c0}{w}}{h}\right)\right)\right)} - 1} \]
   17. Step-by-step derivation

       1. expm1-def 16.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{c0}{w} \cdot 0.5\right) \cdot \left(2 \cdot \left({\left(\frac{d}{D}\right)}^{2} \cdot \frac{\frac{c0}{w}}{h}\right)\right)\right)\right)} \]

       2. expm1-log1p 53.4%

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

       3. associate-*r* 53.4%

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

       4. associate-*r* 53.4%

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

       5. associate-*l* 53.4%

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

       6. metadata-eval 53.4%

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

       7. *-rgt-identity 53.4%

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

       8. unpow2 53.4%

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

       9. times-frac 38.0%

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

       10. unpow2 38.0%

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

       11. unpow2 38.0%

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

       12. times-frac 37.9%

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

       13. *-commutative 37.9%

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

       14. *-commutative 37.9%

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

       15. times-frac 38.0%

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

       16. unpow2 38.0%

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

       17. unpow2 38.0%

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

       18. times-frac 53.4%

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

       19. unpow2 53.4%

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

       20. associate-/l/ 53.3%

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

   18. Simplified 53.3%

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

   if 1.29999999999999993e-121 < M < 5.50000000000000005e-38

   1. Initial program 30.8%

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

   3. Simplified 47.1%

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

   4. Taylor expanded in h around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. *-commutative 0.0%

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

      4. times-frac 0.0%

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

      5. associate-/r* 0.0%

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

      6. times-frac 0.0%

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

      7. *-commutative 0.0%

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

      8. *-commutative 0.0%

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

      9. times-frac 0.0%

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

      10. associate-/r* 0.0%

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

   6. Simplified 31.0%

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

      1. expm1-log1p-u 30.8%

         \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{{d}^{2}}{{D}^{2}}, \frac{c0}{h \cdot w}, \frac{0.5 \cdot \left({D}^{2} \cdot \left(w \cdot 0\right)\right)}{c0 \cdot {d}^{2}}\right)\right)\right)}\right) \]

      2. expm1-udef 30.8%

         \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{{d}^{2}}{{D}^{2}}, \frac{c0}{h \cdot w}, \frac{0.5 \cdot \left({D}^{2} \cdot \left(w \cdot 0\right)\right)}{c0 \cdot {d}^{2}}\right)\right)} - 1}\right) \]

   8. Applied egg-rr 30.8%

      \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{\frac{c0}{h}}{w}, {\left(\frac{d}{D}\right)}^{2}, \frac{\frac{0}{c0}}{{d}^{2}}\right)\right)} - 1}\right) \]
   9. Step-by-step derivation

      1. expm1-def 30.8%

         \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{\frac{c0}{h}}{w}, {\left(\frac{d}{D}\right)}^{2}, \frac{\frac{0}{c0}}{{d}^{2}}\right)\right)\right)}\right) \]

      2. expm1-log1p 31.2%

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

      3. associate-/l/ 31.2%

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

      4. *-commutative 31.2%

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

      5. div0 31.2%

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

      6. div0 47.3%

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

   10. Simplified 47.3%

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

       1. associate-/r* 47.4%

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

       2. pow2 47.4%

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

       3. fma-def 47.4%

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

       4. associate-*l/ 47.4%

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

       5. frac-times 47.3%

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

       6. pow2 47.3%

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

       7. *-commutative 47.3%

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

       8. div0 47.3%

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

       9. fma-def 47.3%

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

       10. associate-*l/ 47.3%

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

       11. frac-times 47.6%

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

       12. fma-def 47.6%

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

       13. pow2 47.6%

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

       14. div0 47.6%

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

   12. Applied egg-rr 47.6%

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

       1. fma-def 47.6%

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

       2. +-rgt-identity 47.6%

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

       3. count-2 47.6%

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

       4. associate-*l/ 47.2%

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

       5. associate-*r/ 47.3%

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

       6. associate-/r* 47.4%

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

       7. *-commutative 47.4%

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

       8. associate-*r/ 47.4%

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

   14. Simplified 47.4%

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

       1. pow2 47.4%

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

   16. Applied egg-rr 47.4%

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

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.72 \cdot 10^{-158}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 6.8 \cdot 10^{-155}:\\ \;\;\;\;\left(\frac{c0}{w} \cdot {\left(\frac{d}{D}\right)}^{2}\right) \cdot \frac{c0}{w \cdot h}\\ \mathbf{elif}\;M \leq 1.3 \cdot 10^{-121}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 5.5 \cdot 10^{-38}:\\ \;\;\;\;\frac{\frac{c0}{w}}{2} \cdot \left(2 \cdot \left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\right)\\ \mathbf{elif}\;M \leq 2.2 \cdot 10^{+38}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c0}{w} \cdot {\left(\frac{d}{D}\right)}^{2}\right) \cdot \frac{c0}{w \cdot h}\\ \end{array} \]

Alternative 6: 40.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 1.55 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{c0}{w}}{2} \cdot \left(2 \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot \frac{d}{D}}{D}\right)\right)\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{+23}:\\ \;\;\;\;0\\ \mathbf{elif}\;d \leq 9.5 \cdot 10^{+164}:\\ \;\;\;\;\frac{\frac{c0}{w}}{\frac{1}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= d 1.55e-6)
   (* (/ (/ c0 w) 2.0) (* 2.0 (* (/ c0 (* w h)) (/ (* d (/ d D)) D))))
   (if (<= d 2.7e+23)
     0.0
     (if (<= d 9.5e+164)
       (/ (/ c0 w) (/ 1.0 (* c0 (/ (pow (/ d D) 2.0) (* w h)))))
       0.0))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (d <= 1.55e-6) {
		tmp = ((c0 / w) / 2.0) * (2.0 * ((c0 / (w * h)) * ((d * (d / D)) / D)));
	} else if (d <= 2.7e+23) {
		tmp = 0.0;
	} else if (d <= 9.5e+164) {
		tmp = (c0 / w) / (1.0 / (c0 * (pow((d / D), 2.0) / (w * h))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (d_1 <= 1.55d-6) then
        tmp = ((c0 / w) / 2.0d0) * (2.0d0 * ((c0 / (w * h)) * ((d_1 * (d_1 / d)) / d)))
    else if (d_1 <= 2.7d+23) then
        tmp = 0.0d0
    else if (d_1 <= 9.5d+164) then
        tmp = (c0 / w) / (1.0d0 / (c0 * (((d_1 / d) ** 2.0d0) / (w * h))))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (d <= 1.55e-6) {
		tmp = ((c0 / w) / 2.0) * (2.0 * ((c0 / (w * h)) * ((d * (d / D)) / D)));
	} else if (d <= 2.7e+23) {
		tmp = 0.0;
	} else if (d <= 9.5e+164) {
		tmp = (c0 / w) / (1.0 / (c0 * (Math.pow((d / D), 2.0) / (w * h))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if d <= 1.55e-6:
		tmp = ((c0 / w) / 2.0) * (2.0 * ((c0 / (w * h)) * ((d * (d / D)) / D)))
	elif d <= 2.7e+23:
		tmp = 0.0
	elif d <= 9.5e+164:
		tmp = (c0 / w) / (1.0 / (c0 * (math.pow((d / D), 2.0) / (w * h))))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (d <= 1.55e-6)
		tmp = Float64(Float64(Float64(c0 / w) / 2.0) * Float64(2.0 * Float64(Float64(c0 / Float64(w * h)) * Float64(Float64(d * Float64(d / D)) / D))));
	elseif (d <= 2.7e+23)
		tmp = 0.0;
	elseif (d <= 9.5e+164)
		tmp = Float64(Float64(c0 / w) / Float64(1.0 / Float64(c0 * Float64((Float64(d / D) ^ 2.0) / Float64(w * h)))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (d <= 1.55e-6)
		tmp = ((c0 / w) / 2.0) * (2.0 * ((c0 / (w * h)) * ((d * (d / D)) / D)));
	elseif (d <= 2.7e+23)
		tmp = 0.0;
	elseif (d <= 9.5e+164)
		tmp = (c0 / w) / (1.0 / (c0 * (((d / D) ^ 2.0) / (w * h))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[d, 1.55e-6], N[(N[(N[(c0 / w), $MachinePrecision] / 2.0), $MachinePrecision] * N[(2.0 * N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[(N[(d * N[(d / D), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.7e+23], 0.0, If[LessEqual[d, 9.5e+164], N[(N[(c0 / w), $MachinePrecision] / N[(1.0 / N[(c0 * N[(N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision] / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if d < 1.55e-6

   1. Initial program 25.4%

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

   3. Simplified 40.4%

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

   4. Taylor expanded in h around 0 6.4%

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

      1. +-commutative 6.4%

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

      2. *-commutative 6.4%

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

      3. *-commutative 6.4%

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

      4. times-frac 7.0%

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

      5. associate-/r* 7.0%

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

      6. times-frac 5.3%

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

      7. *-commutative 5.3%

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

      8. *-commutative 5.3%

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

      9. times-frac 7.0%

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

      10. associate-/r* 7.0%

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

   6. Simplified 32.5%

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

      1. expm1-log1p-u 30.0%

         \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{{d}^{2}}{{D}^{2}}, \frac{c0}{h \cdot w}, \frac{0.5 \cdot \left({D}^{2} \cdot \left(w \cdot 0\right)\right)}{c0 \cdot {d}^{2}}\right)\right)\right)}\right) \]

      2. expm1-udef 29.5%

         \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{{d}^{2}}{{D}^{2}}, \frac{c0}{h \cdot w}, \frac{0.5 \cdot \left({D}^{2} \cdot \left(w \cdot 0\right)\right)}{c0 \cdot {d}^{2}}\right)\right)} - 1}\right) \]

   8. Applied egg-rr 32.8%

      \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{\frac{c0}{h}}{w}, {\left(\frac{d}{D}\right)}^{2}, \frac{\frac{0}{c0}}{{d}^{2}}\right)\right)} - 1}\right) \]
   9. Step-by-step derivation

      1. expm1-def 34.3%

         \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{\frac{c0}{h}}{w}, {\left(\frac{d}{D}\right)}^{2}, \frac{\frac{0}{c0}}{{d}^{2}}\right)\right)\right)}\right) \]

      2. expm1-log1p 38.0%

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

      3. associate-/l/ 38.5%

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

      4. *-commutative 38.5%

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

      5. div0 38.5%

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

      6. div0 46.3%

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

   10. Simplified 46.3%

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

       1. associate-/r* 46.9%

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

       2. pow2 46.9%

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

       3. fma-def 46.9%

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

       4. associate-*l/ 46.3%

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

       5. frac-times 44.4%

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

       6. pow2 44.4%

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

       7. *-commutative 44.4%

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

       8. div0 44.4%

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

       9. fma-def 44.4%

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

       10. associate-*l/ 44.4%

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

       11. frac-times 45.1%

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

       12. fma-def 45.1%

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

       13. pow2 45.1%

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

       14. div0 45.1%

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

   12. Applied egg-rr 45.1%

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

       1. fma-def 45.1%

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

       2. +-rgt-identity 45.1%

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

       3. count-2 45.1%

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

       4. associate-*l/ 45.1%

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

       5. associate-*r/ 46.3%

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

       6. associate-/r* 46.3%

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

       7. *-commutative 46.3%

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

       8. associate-*r/ 46.9%

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

   14. Simplified 46.9%

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

       1. pow2 46.9%

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

       2. associate-*r/ 47.4%

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

   16. Applied egg-rr 47.4%

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

   if 1.55e-6 < d < 2.6999999999999999e23 or 9.49999999999999976e164 < d

   1. Initial program 15.6%

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

   3. Simplified 15.6%

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

   4. Taylor expanded in c0 around -inf 1.7%

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

      1. associate-*r* 1.7%

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

      2. neg-mul-1 1.7%

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

      3. distribute-lft1-in 1.7%

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

      4. metadata-eval 1.7%

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

      5. mul0-lft 46.0%

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

      6. distribute-lft-neg-in 46.0%

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

      7. distribute-rgt-neg-in 46.0%

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

      8. metadata-eval 46.0%

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

   6. Simplified 46.0%

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

   7. Taylor expanded in c0 around 0 53.7%

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

   if 2.6999999999999999e23 < d < 9.49999999999999976e164

   1. Initial program 29.4%

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

   3. Simplified 40.5%

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

   4. Taylor expanded in h around 0 11.0%

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

      1. +-commutative 11.0%

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

      2. *-commutative 11.0%

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

      3. *-commutative 11.0%

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

      4. times-frac 11.0%

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

      5. associate-/r* 11.0%

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

      6. times-frac 8.4%

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

      7. *-commutative 8.4%

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

      8. *-commutative 8.4%

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

      9. times-frac 11.0%

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

      10. associate-/r* 11.0%

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

   6. Simplified 41.2%

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

      1. expm1-log1p-u 35.2%

         \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{{d}^{2}}{{D}^{2}}, \frac{c0}{h \cdot w}, \frac{0.5 \cdot \left({D}^{2} \cdot \left(w \cdot 0\right)\right)}{c0 \cdot {d}^{2}}\right)\right)\right)}\right) \]

      2. expm1-udef 35.2%

         \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{{d}^{2}}{{D}^{2}}, \frac{c0}{h \cdot w}, \frac{0.5 \cdot \left({D}^{2} \cdot \left(w \cdot 0\right)\right)}{c0 \cdot {d}^{2}}\right)\right)} - 1}\right) \]

   8. Applied egg-rr 38.3%

      \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{\frac{c0}{h}}{w}, {\left(\frac{d}{D}\right)}^{2}, \frac{\frac{0}{c0}}{{d}^{2}}\right)\right)} - 1}\right) \]
   9. Step-by-step derivation

      1. expm1-def 40.3%

         \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{\frac{c0}{h}}{w}, {\left(\frac{d}{D}\right)}^{2}, \frac{\frac{0}{c0}}{{d}^{2}}\right)\right)\right)}\right) \]

      2. expm1-log1p 46.5%

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

      3. associate-/l/ 46.5%

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

      4. *-commutative 46.5%

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

      5. div0 46.5%

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

      6. div0 46.5%

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

   10. Simplified 46.5%

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

       1. associate-/r* 46.5%

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

       2. pow2 46.5%

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

       3. fma-def 46.5%

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

       4. associate-*l/ 46.5%

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

       5. frac-times 43.8%

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

       6. pow2 43.8%

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

       7. *-commutative 43.8%

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

       8. div0 43.8%

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

       9. fma-def 43.8%

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

       10. associate-*l/ 46.3%

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

       11. frac-times 46.3%

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

       12. fma-def 46.3%

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

       13. pow2 46.3%

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

       14. div0 46.3%

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

   12. Applied egg-rr 46.3%

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

       1. fma-def 46.3%

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

       2. +-rgt-identity 46.3%

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

       3. count-2 46.3%

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

       4. associate-*l/ 46.3%

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

       5. associate-*r/ 46.3%

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

       6. associate-/r* 48.9%

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

       7. *-commutative 48.9%

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

       8. associate-*r/ 46.5%

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

   14. Simplified 46.5%

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

       1. associate-*l/ 46.5%

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

       2. associate-/l/ 46.5%

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

   16. Applied egg-rr 46.5%

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

       1. associate-/l* 46.5%

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

       2. associate-/r* 46.5%

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

       3. metadata-eval 46.5%

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

       4. associate-/l/ 46.5%

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

       5. associate-*r/ 49.0%

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

       6. *-commutative 49.0%

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

       7. associate-*r/ 49.0%

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

   18. Simplified 49.0%

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

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.55 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{c0}{w}}{2} \cdot \left(2 \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot \frac{d}{D}}{D}\right)\right)\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{+23}:\\ \;\;\;\;0\\ \mathbf{elif}\;d \leq 9.5 \cdot 10^{+164}:\\ \;\;\;\;\frac{\frac{c0}{w}}{\frac{1}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7: 41.8% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 1.05 \cdot 10^{-6} \lor \neg \left(d \leq 1.12 \cdot 10^{+27}\right) \land d \leq 1.55 \cdot 10^{+165}:\\ \;\;\;\;\frac{\frac{c0}{w}}{2} \cdot \left(2 \cdot \left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (or (<= d 1.05e-6) (and (not (<= d 1.12e+27)) (<= d 1.55e+165)))
   (* (/ (/ c0 w) 2.0) (* 2.0 (* (/ c0 (* w h)) (* (/ d D) (/ d D)))))
   0.0))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((d <= 1.05e-6) || (!(d <= 1.12e+27) && (d <= 1.55e+165))) {
		tmp = ((c0 / w) / 2.0) * (2.0 * ((c0 / (w * h)) * ((d / D) * (d / D))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((d_1 <= 1.05d-6) .or. (.not. (d_1 <= 1.12d+27)) .and. (d_1 <= 1.55d+165)) then
        tmp = ((c0 / w) / 2.0d0) * (2.0d0 * ((c0 / (w * h)) * ((d_1 / d) * (d_1 / d))))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((d <= 1.05e-6) || (!(d <= 1.12e+27) && (d <= 1.55e+165))) {
		tmp = ((c0 / w) / 2.0) * (2.0 * ((c0 / (w * h)) * ((d / D) * (d / D))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (d <= 1.05e-6) or (not (d <= 1.12e+27) and (d <= 1.55e+165)):
		tmp = ((c0 / w) / 2.0) * (2.0 * ((c0 / (w * h)) * ((d / D) * (d / D))))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if ((d <= 1.05e-6) || (!(d <= 1.12e+27) && (d <= 1.55e+165)))
		tmp = Float64(Float64(Float64(c0 / w) / 2.0) * Float64(2.0 * Float64(Float64(c0 / Float64(w * h)) * Float64(Float64(d / D) * Float64(d / D)))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((d <= 1.05e-6) || (~((d <= 1.12e+27)) && (d <= 1.55e+165)))
		tmp = ((c0 / w) / 2.0) * (2.0 * ((c0 / (w * h)) * ((d / D) * (d / D))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[Or[LessEqual[d, 1.05e-6], And[N[Not[LessEqual[d, 1.12e+27]], $MachinePrecision], LessEqual[d, 1.55e+165]]], N[(N[(N[(c0 / w), $MachinePrecision] / 2.0), $MachinePrecision] * N[(2.0 * N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if d < 1.0499999999999999e-6 or 1.12e27 < d < 1.5500000000000001e165

   1. Initial program 26.2%

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

   3. Simplified 40.4%

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

   4. Taylor expanded in h around 0 7.3%

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

      1. +-commutative 7.3%

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

      2. *-commutative 7.3%

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

      3. *-commutative 7.3%

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

      4. times-frac 7.8%

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

      5. associate-/r* 7.8%

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

      6. times-frac 5.9%

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

      7. *-commutative 5.9%

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

      8. *-commutative 5.9%

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

      9. times-frac 7.8%

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

      10. associate-/r* 7.8%

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

   6. Simplified 34.2%

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

      1. expm1-log1p-u 31.0%

         \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{{d}^{2}}{{D}^{2}}, \frac{c0}{h \cdot w}, \frac{0.5 \cdot \left({D}^{2} \cdot \left(w \cdot 0\right)\right)}{c0 \cdot {d}^{2}}\right)\right)\right)}\right) \]

      2. expm1-udef 30.6%

         \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{{d}^{2}}{{D}^{2}}, \frac{c0}{h \cdot w}, \frac{0.5 \cdot \left({D}^{2} \cdot \left(w \cdot 0\right)\right)}{c0 \cdot {d}^{2}}\right)\right)} - 1}\right) \]

   8. Applied egg-rr 33.9%

      \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{\frac{c0}{h}}{w}, {\left(\frac{d}{D}\right)}^{2}, \frac{\frac{0}{c0}}{{d}^{2}}\right)\right)} - 1}\right) \]
   9. Step-by-step derivation

      1. expm1-def 35.5%

         \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{\frac{c0}{h}}{w}, {\left(\frac{d}{D}\right)}^{2}, \frac{\frac{0}{c0}}{{d}^{2}}\right)\right)\right)}\right) \]

      2. expm1-log1p 39.6%

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

      3. associate-/l/ 40.1%

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

      4. *-commutative 40.1%

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

      5. div0 40.1%

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

      6. div0 46.3%

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

   10. Simplified 46.3%

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

       1. associate-/r* 46.8%

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

       2. pow2 46.8%

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

       3. fma-def 46.8%

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

       4. associate-*l/ 46.3%

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

       5. frac-times 44.3%

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

       6. pow2 44.3%

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

       7. *-commutative 44.3%

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

       8. div0 44.3%

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

       9. fma-def 44.3%

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

       10. associate-*l/ 44.8%

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

       11. frac-times 45.3%

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

       12. fma-def 45.3%

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

       13. pow2 45.3%

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

       14. div0 45.3%

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

   12. Applied egg-rr 45.3%

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

       1. fma-def 45.3%

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

       2. +-rgt-identity 45.3%

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

       3. count-2 45.3%

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

       4. associate-*l/ 45.3%

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

       5. associate-*r/ 46.3%

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

       6. associate-/r* 46.8%

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

       7. *-commutative 46.8%

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

       8. associate-*r/ 46.8%

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

   14. Simplified 46.8%

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

       1. pow2 46.8%

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

   16. Applied egg-rr 46.8%

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

   if 1.0499999999999999e-6 < d < 1.12e27 or 1.5500000000000001e165 < d

   1. Initial program 15.6%

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

   3. Simplified 15.6%

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

   4. Taylor expanded in c0 around -inf 1.7%

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

      1. associate-*r* 1.7%

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

      2. neg-mul-1 1.7%

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

      3. distribute-lft1-in 1.7%

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

      4. metadata-eval 1.7%

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

      5. mul0-lft 46.0%

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

      6. distribute-lft-neg-in 46.0%

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

      7. distribute-rgt-neg-in 46.0%

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

      8. metadata-eval 46.0%

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

   6. Simplified 46.0%

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

   7. Taylor expanded in c0 around 0 53.7%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.05 \cdot 10^{-6} \lor \neg \left(d \leq 1.12 \cdot 10^{+27}\right) \land d \leq 1.55 \cdot 10^{+165}:\\ \;\;\;\;\frac{\frac{c0}{w}}{2} \cdot \left(2 \cdot \left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 8: 40.8% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{w \cdot h}\\ t_1 := \frac{\frac{c0}{w}}{2}\\ \mathbf{if}\;d \leq 8.2 \cdot 10^{-7}:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \left(t_0 \cdot \frac{d \cdot \frac{d}{D}}{D}\right)\right)\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{+28}:\\ \;\;\;\;0\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{+165}:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \left(t_0 \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* w h))) (t_1 (/ (/ c0 w) 2.0)))
   (if (<= d 8.2e-7)
     (* t_1 (* 2.0 (* t_0 (/ (* d (/ d D)) D))))
     (if (<= d 1.4e+28)
       0.0
       (if (<= d 1.35e+165)
         (* t_1 (* 2.0 (* t_0 (* (/ d D) (/ d D)))))
         0.0)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (w * h);
	double t_1 = (c0 / w) / 2.0;
	double tmp;
	if (d <= 8.2e-7) {
		tmp = t_1 * (2.0 * (t_0 * ((d * (d / D)) / D)));
	} else if (d <= 1.4e+28) {
		tmp = 0.0;
	} else if (d <= 1.35e+165) {
		tmp = t_1 * (2.0 * (t_0 * ((d / D) * (d / D))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c0 / (w * h)
    t_1 = (c0 / w) / 2.0d0
    if (d_1 <= 8.2d-7) then
        tmp = t_1 * (2.0d0 * (t_0 * ((d_1 * (d_1 / d)) / d)))
    else if (d_1 <= 1.4d+28) then
        tmp = 0.0d0
    else if (d_1 <= 1.35d+165) then
        tmp = t_1 * (2.0d0 * (t_0 * ((d_1 / d) * (d_1 / d))))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (w * h);
	double t_1 = (c0 / w) / 2.0;
	double tmp;
	if (d <= 8.2e-7) {
		tmp = t_1 * (2.0 * (t_0 * ((d * (d / D)) / D)));
	} else if (d <= 1.4e+28) {
		tmp = 0.0;
	} else if (d <= 1.35e+165) {
		tmp = t_1 * (2.0 * (t_0 * ((d / D) * (d / D))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (w * h)
	t_1 = (c0 / w) / 2.0
	tmp = 0
	if d <= 8.2e-7:
		tmp = t_1 * (2.0 * (t_0 * ((d * (d / D)) / D)))
	elif d <= 1.4e+28:
		tmp = 0.0
	elif d <= 1.35e+165:
		tmp = t_1 * (2.0 * (t_0 * ((d / D) * (d / D))))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(w * h))
	t_1 = Float64(Float64(c0 / w) / 2.0)
	tmp = 0.0
	if (d <= 8.2e-7)
		tmp = Float64(t_1 * Float64(2.0 * Float64(t_0 * Float64(Float64(d * Float64(d / D)) / D))));
	elseif (d <= 1.4e+28)
		tmp = 0.0;
	elseif (d <= 1.35e+165)
		tmp = Float64(t_1 * Float64(2.0 * Float64(t_0 * Float64(Float64(d / D) * Float64(d / D)))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (w * h);
	t_1 = (c0 / w) / 2.0;
	tmp = 0.0;
	if (d <= 8.2e-7)
		tmp = t_1 * (2.0 * (t_0 * ((d * (d / D)) / D)));
	elseif (d <= 1.4e+28)
		tmp = 0.0;
	elseif (d <= 1.35e+165)
		tmp = t_1 * (2.0 * (t_0 * ((d / D) * (d / D))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 / w), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[d, 8.2e-7], N[(t$95$1 * N[(2.0 * N[(t$95$0 * N[(N[(d * N[(d / D), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.4e+28], 0.0, If[LessEqual[d, 1.35e+165], N[(t$95$1 * N[(2.0 * N[(t$95$0 * N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]]]

Derivation

1. Split input into 3 regimes

2. if d < 8.1999999999999998e-7

   1. Initial program 25.4%

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

   3. Simplified 40.4%

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

   4. Taylor expanded in h around 0 6.4%

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

      1. +-commutative 6.4%

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

      2. *-commutative 6.4%

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

      3. *-commutative 6.4%

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

      4. times-frac 7.0%

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

      5. associate-/r* 7.0%

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

      6. times-frac 5.3%

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

      7. *-commutative 5.3%

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

      8. *-commutative 5.3%

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

      9. times-frac 7.0%

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

      10. associate-/r* 7.0%

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

   6. Simplified 32.5%

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

      1. expm1-log1p-u 30.0%

         \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{{d}^{2}}{{D}^{2}}, \frac{c0}{h \cdot w}, \frac{0.5 \cdot \left({D}^{2} \cdot \left(w \cdot 0\right)\right)}{c0 \cdot {d}^{2}}\right)\right)\right)}\right) \]

      2. expm1-udef 29.5%

         \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{{d}^{2}}{{D}^{2}}, \frac{c0}{h \cdot w}, \frac{0.5 \cdot \left({D}^{2} \cdot \left(w \cdot 0\right)\right)}{c0 \cdot {d}^{2}}\right)\right)} - 1}\right) \]

   8. Applied egg-rr 32.8%

      \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{\frac{c0}{h}}{w}, {\left(\frac{d}{D}\right)}^{2}, \frac{\frac{0}{c0}}{{d}^{2}}\right)\right)} - 1}\right) \]
   9. Step-by-step derivation

      1. expm1-def 34.3%

         \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{\frac{c0}{h}}{w}, {\left(\frac{d}{D}\right)}^{2}, \frac{\frac{0}{c0}}{{d}^{2}}\right)\right)\right)}\right) \]

      2. expm1-log1p 38.0%

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

      3. associate-/l/ 38.5%

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

      4. *-commutative 38.5%

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

      5. div0 38.5%

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

      6. div0 46.3%

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

   10. Simplified 46.3%

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

       1. associate-/r* 46.9%

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

       2. pow2 46.9%

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

       3. fma-def 46.9%

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

       4. associate-*l/ 46.3%

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

       5. frac-times 44.4%

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

       6. pow2 44.4%

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

       7. *-commutative 44.4%

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

       8. div0 44.4%

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

       9. fma-def 44.4%

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

       10. associate-*l/ 44.4%

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

       11. frac-times 45.1%

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

       12. fma-def 45.1%

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

       13. pow2 45.1%

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

       14. div0 45.1%

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

   12. Applied egg-rr 45.1%

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

       1. fma-def 45.1%

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

       2. +-rgt-identity 45.1%

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

       3. count-2 45.1%

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

       4. associate-*l/ 45.1%

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

       5. associate-*r/ 46.3%

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

       6. associate-/r* 46.3%

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

       7. *-commutative 46.3%

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

       8. associate-*r/ 46.9%

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

   14. Simplified 46.9%

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

       1. pow2 46.9%

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

       2. associate-*r/ 47.4%

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

   16. Applied egg-rr 47.4%

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

   if 8.1999999999999998e-7 < d < 1.4000000000000001e28 or 1.35e165 < d

   1. Initial program 15.6%

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

   3. Simplified 15.6%

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

   4. Taylor expanded in c0 around -inf 1.7%

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

      1. associate-*r* 1.7%

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

      2. neg-mul-1 1.7%

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

      3. distribute-lft1-in 1.7%

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

      4. metadata-eval 1.7%

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

      5. mul0-lft 46.0%

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

      6. distribute-lft-neg-in 46.0%

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

      7. distribute-rgt-neg-in 46.0%

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

      8. metadata-eval 46.0%

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

   6. Simplified 46.0%

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

   7. Taylor expanded in c0 around 0 53.7%

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

   if 1.4000000000000001e28 < d < 1.35e165

   1. Initial program 29.4%

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

   3. Simplified 40.5%

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

   4. Taylor expanded in h around 0 11.0%

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

      1. +-commutative 11.0%

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

      2. *-commutative 11.0%

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

      3. *-commutative 11.0%

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

      4. times-frac 11.0%

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

      5. associate-/r* 11.0%

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

      6. times-frac 8.4%

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

      7. *-commutative 8.4%

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

      8. *-commutative 8.4%

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

      9. times-frac 11.0%

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

      10. associate-/r* 11.0%

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

   6. Simplified 41.2%

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

      1. expm1-log1p-u 35.2%

         \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{{d}^{2}}{{D}^{2}}, \frac{c0}{h \cdot w}, \frac{0.5 \cdot \left({D}^{2} \cdot \left(w \cdot 0\right)\right)}{c0 \cdot {d}^{2}}\right)\right)\right)}\right) \]

      2. expm1-udef 35.2%

         \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{{d}^{2}}{{D}^{2}}, \frac{c0}{h \cdot w}, \frac{0.5 \cdot \left({D}^{2} \cdot \left(w \cdot 0\right)\right)}{c0 \cdot {d}^{2}}\right)\right)} - 1}\right) \]

   8. Applied egg-rr 38.3%

      \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{\frac{c0}{h}}{w}, {\left(\frac{d}{D}\right)}^{2}, \frac{\frac{0}{c0}}{{d}^{2}}\right)\right)} - 1}\right) \]
   9. Step-by-step derivation

      1. expm1-def 40.3%

         \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{\frac{c0}{h}}{w}, {\left(\frac{d}{D}\right)}^{2}, \frac{\frac{0}{c0}}{{d}^{2}}\right)\right)\right)}\right) \]

      2. expm1-log1p 46.5%

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

      3. associate-/l/ 46.5%

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

      4. *-commutative 46.5%

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

      5. div0 46.5%

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

      6. div0 46.5%

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

   10. Simplified 46.5%

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

       1. associate-/r* 46.5%

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

       2. pow2 46.5%

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

       3. fma-def 46.5%

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

       4. associate-*l/ 46.5%

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

       5. frac-times 43.8%

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

       6. pow2 43.8%

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

       7. *-commutative 43.8%

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

       8. div0 43.8%

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

       9. fma-def 43.8%

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

       10. associate-*l/ 46.3%

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

       11. frac-times 46.3%

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

       12. fma-def 46.3%

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

       13. pow2 46.3%

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

       14. div0 46.3%

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

   12. Applied egg-rr 46.3%

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

       1. fma-def 46.3%

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

       2. +-rgt-identity 46.3%

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

       3. count-2 46.3%

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

       4. associate-*l/ 46.3%

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

       5. associate-*r/ 46.3%

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

       6. associate-/r* 48.9%

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

       7. *-commutative 48.9%

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

       8. associate-*r/ 46.5%

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

   14. Simplified 46.5%

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

       1. pow2 46.5%

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

   16. Applied egg-rr 46.5%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 8.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{c0}{w}}{2} \cdot \left(2 \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot \frac{d}{D}}{D}\right)\right)\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{+28}:\\ \;\;\;\;0\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{+165}:\\ \;\;\;\;\frac{\frac{c0}{w}}{2} \cdot \left(2 \cdot \left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9: 33.7% accurate, 151.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

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

C

double code(double c0, double w, double h, double D, double d, double M) {
	return 0.0;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = 0.0d0
end function

Java

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

Python

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

Julia

function code(c0, w, h, D, d, M)
	return 0.0
end

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := 0.0

Derivation

1. Initial program 23.8%

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

3. Simplified 24.7%

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

4. Taylor expanded in c0 around -inf 5.3%

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

   1. associate-*r* 5.3%

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

   2. neg-mul-1 5.3%

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

   3. distribute-lft1-in 5.3%

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

   4. metadata-eval 5.3%

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

   5. mul0-lft 30.2%

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

   6. distribute-lft-neg-in 30.2%

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

   7. distribute-rgt-neg-in 30.2%

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

   8. metadata-eval 30.2%

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

6. Simplified 30.2%

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

7. Taylor expanded in c0 around 0 35.4%

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

8. Final simplification 35.4%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023335 
(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))))))