Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.3% → 44.6%

Time: 41.3s

Alternatives: 10

Speedup: 151.0×

• 21.0% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. c0 = 2.6550144113355053e+245
  2. w = 1.5633902806303038e-208
  3. h = -1.8418212848580615e-267
  4. D = -1.883628975684403e-308
  5. d = 1.5168358290502748e+220
  6. M = 1.5245934983956088e+26

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: 24.3% 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: 44.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} t_0 := \frac{c0}{\frac{w}{{\left(\frac{d}{D}\right)}^{2} \cdot \left|\frac{c0}{w \cdot h}\right|}}\\ \mathbf{if}\;M_m \leq 1.05 \cdot 10^{-217}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;M_m \leq 4 \cdot 10^{-99}:\\ \;\;\;\;0\\ \mathbf{elif}\;M_m \leq 2.1 \cdot 10^{-40}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{w \cdot {\left(\frac{d}{D}\right)}^{-2}}{c0} \cdot \left(w \cdot h\right)}{c0}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (/ c0 (/ w (* (pow (/ d D) 2.0) (fabs (/ c0 (* w h))))))))
   (if (<= M_m 1.05e-217)
     t_0
     (if (<= M_m 4e-99)
       0.0
       (if (<= M_m 2.1e-40)
         t_0
         (/ 1.0 (/ (* (/ (* w (pow (/ d D) -2.0)) c0) (* w h)) c0)))))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = c0 / (w / (pow((d / D), 2.0) * fabs((c0 / (w * h)))));
	double tmp;
	if (M_m <= 1.05e-217) {
		tmp = t_0;
	} else if (M_m <= 4e-99) {
		tmp = 0.0;
	} else if (M_m <= 2.1e-40) {
		tmp = t_0;
	} else {
		tmp = 1.0 / ((((w * pow((d / D), -2.0)) / c0) * (w * h)) / c0);
	}
	return tmp;
}

Fortran

M_m = abs(M)
real(8) function code(c0, w, h, d, d_1, m_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_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 / (w / (((d_1 / d) ** 2.0d0) * abs((c0 / (w * h)))))
    if (m_m <= 1.05d-217) then
        tmp = t_0
    else if (m_m <= 4d-99) then
        tmp = 0.0d0
    else if (m_m <= 2.1d-40) then
        tmp = t_0
    else
        tmp = 1.0d0 / ((((w * ((d_1 / d) ** (-2.0d0))) / c0) * (w * h)) / c0)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = c0 / (w / (Math.pow((d / D), 2.0) * Math.abs((c0 / (w * h)))));
	double tmp;
	if (M_m <= 1.05e-217) {
		tmp = t_0;
	} else if (M_m <= 4e-99) {
		tmp = 0.0;
	} else if (M_m <= 2.1e-40) {
		tmp = t_0;
	} else {
		tmp = 1.0 / ((((w * Math.pow((d / D), -2.0)) / c0) * (w * h)) / c0);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	t_0 = c0 / (w / (math.pow((d / D), 2.0) * math.fabs((c0 / (w * h)))))
	tmp = 0
	if M_m <= 1.05e-217:
		tmp = t_0
	elif M_m <= 4e-99:
		tmp = 0.0
	elif M_m <= 2.1e-40:
		tmp = t_0
	else:
		tmp = 1.0 / ((((w * math.pow((d / D), -2.0)) / c0) * (w * h)) / c0)
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	t_0 = Float64(c0 / Float64(w / Float64((Float64(d / D) ^ 2.0) * abs(Float64(c0 / Float64(w * h))))))
	tmp = 0.0
	if (M_m <= 1.05e-217)
		tmp = t_0;
	elseif (M_m <= 4e-99)
		tmp = 0.0;
	elseif (M_m <= 2.1e-40)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(w * (Float64(d / D) ^ -2.0)) / c0) * Float64(w * h)) / c0));
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	t_0 = c0 / (w / (((d / D) ^ 2.0) * abs((c0 / (w * h)))));
	tmp = 0.0;
	if (M_m <= 1.05e-217)
		tmp = t_0;
	elseif (M_m <= 4e-99)
		tmp = 0.0;
	elseif (M_m <= 2.1e-40)
		tmp = t_0;
	else
		tmp = 1.0 / ((((w * ((d / D) ^ -2.0)) / c0) * (w * h)) / c0);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := Block[{t$95$0 = N[(c0 / N[(w / N[(N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision] * N[Abs[N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M$95$m, 1.05e-217], t$95$0, If[LessEqual[M$95$m, 4e-99], 0.0, If[LessEqual[M$95$m, 2.1e-40], t$95$0, N[(1.0 / N[(N[(N[(N[(w * N[Power[N[(d / D), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / c0), $MachinePrecision] * N[(w * h), $MachinePrecision]), $MachinePrecision] / c0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if M < 1.05e-217 or 4.0000000000000001e-99 < M < 2.10000000000000018e-40

   1. Initial program 27.2%

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

      1. +-commutative 27.2%

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

      2. +-commutative 27.2%

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

      3. times-frac 26.6%

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

      4. fma-neg 26.6%

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

   3. Simplified 27.3%

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

   5. Taylor expanded in c0 around inf 37.3%

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

      1. *-commutative 37.3%

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

      2. *-commutative 37.3%

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

      3. associate-*r* 37.2%

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

      4. associate-*r/ 37.3%

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

      5. associate-*r* 37.4%

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

      6. *-commutative 37.4%

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

      7. *-commutative 37.4%

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

      8. associate-/r* 37.5%

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

      9. unpow2 37.5%

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

      10. associate-*r/ 44.7%

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

      11. unpow2 44.7%

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

      12. associate-/l/ 47.8%

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

      13. associate-*r/ 47.2%

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

      14. associate-*l/ 48.4%

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

      15. unpow2 48.4%

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

   7. Simplified 48.4%

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

   9. Applied egg-rr 21.4%

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

       1. expm1-def 23.8%

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

       2. expm1-log1p 47.4%

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

       3. associate-*l* 47.4%

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

       4. associate-*l/ 46.3%

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

       5. associate-/l* 48.0%

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

       6. *-commutative 48.0%

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

       7. associate-/l* 48.1%

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

       8. *-commutative 48.1%

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

       9. associate-/l* 48.1%

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

       10. metadata-eval 48.1%

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

   11. Simplified 48.1%

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

       1. add-sqr-sqrt 41.9%

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

       2. pow1/2 41.9%

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

       3. pow1/2 45.6%

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

       4. pow-prod-down 47.8%

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

       5. pow2 47.8%

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

       6. *-commutative 47.8%

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

   13. Applied egg-rr 47.8%

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

       1. unpow1/2 47.8%

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

       2. unpow2 47.8%

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

       3. rem-sqrt-square 52.1%

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

       4. *-commutative 52.1%

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

   15. Simplified 52.1%

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

   if 1.05e-217 < M < 4.0000000000000001e-99

   1. Initial program 15.7%

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

      1. +-commutative 15.7%

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

      2. +-commutative 15.7%

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

      3. times-frac 15.7%

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

      4. fma-neg 15.7%

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

   3. Simplified 15.8%

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

   5. Taylor expanded in c0 around -inf 4.1%

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

      1. associate-*r* 4.1%

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

      2. neg-mul-1 4.1%

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

      3. distribute-lft1-in 4.1%

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

      4. metadata-eval 4.1%

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

      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%

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

      9. mul0-lft 4.1%

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

      10. metadata-eval 4.1%

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

      11. distribute-lft1-in 4.1%

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

      12. distribute-lft-in 4.1%

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

   7. Simplified 45.4%

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

   8. Taylor expanded in c0 around 0 53.1%

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

   if 2.10000000000000018e-40 < M

   1. Initial program 18.5%

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

      1. +-commutative 18.5%

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

      2. +-commutative 18.5%

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

      3. times-frac 18.5%

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

      4. fma-neg 18.5%

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

   3. Simplified 20.0%

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

   5. Taylor expanded in c0 around inf 43.8%

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

      1. *-commutative 43.8%

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

      2. *-commutative 43.8%

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

      3. associate-*r* 43.8%

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

      4. associate-*r/ 44.0%

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

      5. associate-*r* 44.0%

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

      6. *-commutative 44.0%

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

      7. *-commutative 44.0%

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

      8. associate-/r* 45.6%

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

      9. unpow2 45.6%

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

      10. associate-*r/ 47.3%

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

      11. unpow2 47.3%

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

      12. associate-/l/ 53.5%

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

      13. associate-*r/ 53.6%

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

      14. associate-*l/ 53.6%

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

      15. unpow2 53.6%

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

   7. Simplified 53.6%

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

   9. Applied egg-rr 22.1%

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

       1. expm1-def 22.1%

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

       2. expm1-log1p 53.6%

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

       3. associate-*l* 53.6%

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

       4. associate-*l/ 53.6%

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

       5. associate-/l* 53.6%

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

       6. *-commutative 53.6%

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

       7. associate-/l* 53.6%

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

       8. *-commutative 53.6%

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

       9. associate-/l* 53.6%

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

       10. metadata-eval 53.6%

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

   11. Simplified 53.6%

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

       1. clear-num 53.6%

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

       2. inv-pow 53.6%

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

       3. /-rgt-identity 53.6%

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

       4. associate-/r* 50.4%

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

       5. associate-/l/ 53.6%

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

   13. Applied egg-rr 53.6%

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

       1. unpow-1 53.6%

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

       2. associate-/r* 54.9%

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

       3. *-commutative 54.9%

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

   15. Simplified 54.9%

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

       1. associate-/r/ 55.0%

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

       2. div-inv 55.0%

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

       3. pow-flip 55.0%

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

       4. metadata-eval 55.0%

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

       5. *-commutative 55.0%

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

   17. Applied egg-rr 55.0%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.05 \cdot 10^{-217}:\\ \;\;\;\;\frac{c0}{\frac{w}{{\left(\frac{d}{D}\right)}^{2} \cdot \left|\frac{c0}{w \cdot h}\right|}}\\ \mathbf{elif}\;M \leq 4 \cdot 10^{-99}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 2.1 \cdot 10^{-40}:\\ \;\;\;\;\frac{c0}{\frac{w}{{\left(\frac{d}{D}\right)}^{2} \cdot \left|\frac{c0}{w \cdot h}\right|}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{w \cdot {\left(\frac{d}{D}\right)}^{-2}}{c0} \cdot \left(w \cdot h\right)}{c0}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 49.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{w \cdot 2} \cdot \left(t_0 + \sqrt{t_0 \cdot t_0 - M_m \cdot M_m}\right) \leq \infty:\\ \;\;\;\;\frac{1}{\frac{\frac{w \cdot {\left(\frac{d}{D}\right)}^{-2}}{c0} \cdot \left(w \cdot h\right)}{c0}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* w 2.0)) (+ t_0 (sqrt (- (* t_0 t_0) (* M_m M_m)))))
        INFINITY)
     (/ 1.0 (/ (* (/ (* w (pow (/ d D) -2.0)) c0) (* w h)) c0))
     0.0)))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (w * 2.0)) * (t_0 + sqrt(((t_0 * t_0) - (M_m * M_m))))) <= ((double) INFINITY)) {
		tmp = 1.0 / ((((w * pow((d / D), -2.0)) / c0) * (w * h)) / c0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (w * 2.0)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M_m * M_m))))) <= Double.POSITIVE_INFINITY) {
		tmp = 1.0 / ((((w * Math.pow((d / D), -2.0)) / c0) * (w * h)) / c0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if ((c0 / (w * 2.0)) * (t_0 + math.sqrt(((t_0 * t_0) - (M_m * M_m))))) <= math.inf:
		tmp = 1.0 / ((((w * math.pow((d / D), -2.0)) / c0) * (w * h)) / c0)
	else:
		tmp = 0.0
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(w * 2.0)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M_m * M_m))))) <= Inf)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(w * (Float64(d / D) ^ -2.0)) / c0) * Float64(w * h)) / c0));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if (((c0 / (w * 2.0)) * (t_0 + sqrt(((t_0 * t_0) - (M_m * M_m))))) <= Inf)
		tmp = 1.0 / ((((w * ((d / D) ^ -2.0)) / c0) * (w * h)) / c0);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(w * 2.0), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(1.0 / N[(N[(N[(N[(w * N[Power[N[(d / D), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / c0), $MachinePrecision] * N[(w * h), $MachinePrecision]), $MachinePrecision] / c0), $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 80.1%

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

      1. +-commutative 80.1%

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

      2. +-commutative 80.1%

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

      3. times-frac 78.8%

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

      4. fma-neg 78.8%

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

   3. Simplified 78.8%

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

   5. Taylor expanded in c0 around inf 83.6%

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

      1. *-commutative 83.6%

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

      2. *-commutative 83.6%

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

      3. associate-*r* 83.3%

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

      4. associate-*r/ 80.8%

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

      5. associate-*r* 82.4%

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

      6. *-commutative 82.4%

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

      7. *-commutative 82.4%

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

      8. associate-/r* 82.3%

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

      9. unpow2 82.3%

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

      10. associate-*r/ 87.3%

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

      11. unpow2 87.3%

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

      12. associate-/l/ 87.3%

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

      13. associate-*r/ 86.0%

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

      14. associate-*l/ 87.3%

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

      15. unpow2 87.3%

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

   7. Simplified 87.3%

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

   9. Applied egg-rr 39.0%

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

       1. expm1-def 42.7%

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

       2. expm1-log1p 86.4%

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

       3. associate-*l* 86.4%

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

       4. associate-*l/ 83.9%

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

       5. associate-/l* 87.6%

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

       6. *-commutative 87.6%

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

       7. associate-/l* 87.6%

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

       8. *-commutative 87.6%

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

       9. associate-/l* 87.6%

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

       10. metadata-eval 87.6%

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

   11. Simplified 87.6%

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

       1. clear-num 87.6%

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

       2. inv-pow 87.6%

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

       3. /-rgt-identity 87.6%

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

       4. associate-/r* 81.1%

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

       5. associate-/l/ 87.6%

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

   13. Applied egg-rr 87.6%

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

       1. unpow-1 87.6%

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

       2. associate-/r* 90.0%

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

       3. *-commutative 90.0%

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

   15. Simplified 90.0%

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

       1. associate-/r/ 91.3%

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

       2. div-inv 90.9%

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

       3. pow-flip 91.2%

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

       4. metadata-eval 91.2%

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

       5. *-commutative 91.2%

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

   17. Applied egg-rr 91.2%

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

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

   1. Initial program 0.0%

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

      1. +-commutative 0.0%

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

      2. +-commutative 0.0%

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

      3. times-frac 0.0%

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

      4. fma-neg 0.0%

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

   3. Simplified 1.2%

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

   5. Taylor expanded in c0 around -inf 1.2%

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

      1. associate-*r* 1.2%

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

      2. neg-mul-1 1.2%

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

      3. distribute-lft1-in 1.2%

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

      4. metadata-eval 1.2%

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

      5. mul0-lft 31.8%

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

      6. distribute-lft-neg-in 31.8%

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

      7. distribute-rgt-neg-in 31.8%

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

      8. metadata-eval 31.8%

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

      9. mul0-lft 1.2%

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

      10. metadata-eval 1.2%

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

      11. distribute-lft1-in 1.2%

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

      12. distribute-lft-in 1.2%

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

   7. Simplified 31.8%

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

   8. Taylor expanded in c0 around 0 36.0%

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

4. Final simplification 52.4%

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

Alternative 3: 47.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} \mathbf{if}\;w \leq -1.6 \cdot 10^{+49}:\\ \;\;\;\;0\\ \mathbf{elif}\;w \leq 7.2 \cdot 10^{-173} \lor \neg \left(w \leq 3.8 \cdot 10^{-154}\right) \land w \leq 4.4 \cdot 10^{+180}:\\ \;\;\;\;\frac{c0}{w} \cdot \left({\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (if (<= w -1.6e+49)
   0.0
   (if (or (<= w 7.2e-173) (and (not (<= w 3.8e-154)) (<= w 4.4e+180)))
     (* (/ c0 w) (* (pow (/ d D) 2.0) (/ c0 (* w h))))
     0.0)))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double tmp;
	if (w <= -1.6e+49) {
		tmp = 0.0;
	} else if ((w <= 7.2e-173) || (!(w <= 3.8e-154) && (w <= 4.4e+180))) {
		tmp = (c0 / w) * (pow((d / D), 2.0) * (c0 / (w * h)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

M_m = abs(M)
real(8) function code(c0, w, h, d, d_1, m_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_m
    real(8) :: tmp
    if (w <= (-1.6d+49)) then
        tmp = 0.0d0
    else if ((w <= 7.2d-173) .or. (.not. (w <= 3.8d-154)) .and. (w <= 4.4d+180)) then
        tmp = (c0 / w) * (((d_1 / d) ** 2.0d0) * (c0 / (w * h)))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	double tmp;
	if (w <= -1.6e+49) {
		tmp = 0.0;
	} else if ((w <= 7.2e-173) || (!(w <= 3.8e-154) && (w <= 4.4e+180))) {
		tmp = (c0 / w) * (Math.pow((d / D), 2.0) * (c0 / (w * h)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	tmp = 0
	if w <= -1.6e+49:
		tmp = 0.0
	elif (w <= 7.2e-173) or (not (w <= 3.8e-154) and (w <= 4.4e+180)):
		tmp = (c0 / w) * (math.pow((d / D), 2.0) * (c0 / (w * h)))
	else:
		tmp = 0.0
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	tmp = 0.0
	if (w <= -1.6e+49)
		tmp = 0.0;
	elseif ((w <= 7.2e-173) || (!(w <= 3.8e-154) && (w <= 4.4e+180)))
		tmp = Float64(Float64(c0 / w) * Float64((Float64(d / D) ^ 2.0) * Float64(c0 / Float64(w * h))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	tmp = 0.0;
	if (w <= -1.6e+49)
		tmp = 0.0;
	elseif ((w <= 7.2e-173) || (~((w <= 3.8e-154)) && (w <= 4.4e+180)))
		tmp = (c0 / w) * (((d / D) ^ 2.0) * (c0 / (w * h)));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := If[LessEqual[w, -1.6e+49], 0.0, If[Or[LessEqual[w, 7.2e-173], And[N[Not[LessEqual[w, 3.8e-154]], $MachinePrecision], LessEqual[w, 4.4e+180]]], N[(N[(c0 / w), $MachinePrecision] * N[(N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision] * N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if w < -1.60000000000000007e49 or 7.19999999999999943e-173 < w < 3.8000000000000001e-154 or 4.3999999999999999e180 < w

   1. Initial program 7.4%

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

      1. +-commutative 7.4%

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

      2. +-commutative 7.4%

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

      3. times-frac 5.6%

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

      4. fma-neg 5.6%

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

   3. Simplified 7.5%

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

   5. Taylor expanded in c0 around -inf 5.7%

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

      1. associate-*r* 5.7%

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

      2. neg-mul-1 5.7%

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

      3. distribute-lft1-in 5.7%

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

      4. metadata-eval 5.7%

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

      5. mul0-lft 49.5%

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

      6. distribute-lft-neg-in 49.5%

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

      7. distribute-rgt-neg-in 49.5%

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

      8. metadata-eval 49.5%

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

      9. mul0-lft 5.7%

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

      10. metadata-eval 5.7%

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

      11. distribute-lft1-in 5.7%

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

      12. distribute-lft-in 5.7%

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

   7. Simplified 49.5%

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

   8. Taylor expanded in c0 around 0 51.3%

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

   if -1.60000000000000007e49 < w < 7.19999999999999943e-173 or 3.8000000000000001e-154 < w < 4.3999999999999999e180

   1. Initial program 28.3%

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

      1. +-commutative 28.3%

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

      2. +-commutative 28.3%

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

      3. times-frac 28.2%

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

      4. fma-neg 28.2%

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

   3. Simplified 28.8%

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

   5. Taylor expanded in c0 around inf 44.5%

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

      1. *-commutative 44.5%

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

      2. *-commutative 44.5%

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

      3. associate-*r* 44.9%

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

      4. associate-*r/ 44.6%

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

      5. associate-*r* 44.2%

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

      6. *-commutative 44.2%

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

      7. *-commutative 44.2%

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

      8. associate-/r* 44.7%

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

      9. unpow2 44.7%

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

      10. associate-*r/ 51.1%

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

      11. unpow2 51.1%

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

      12. associate-/l/ 55.7%

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

      13. associate-*r/ 55.7%

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

      14. associate-*l/ 56.2%

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

      15. unpow2 56.2%

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

   7. Simplified 56.2%

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

   9. Applied egg-rr 25.0%

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

       1. expm1-def 26.6%

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

       2. expm1-log1p 55.9%

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

       3. associate-*l* 55.9%

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

       4. associate-*l/ 54.5%

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

       5. associate-/l* 56.0%

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

       6. *-commutative 56.0%

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

       7. associate-/l* 56.0%

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

       8. *-commutative 56.0%

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

       9. associate-/l* 56.0%

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

       10. metadata-eval 56.0%

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

   11. Simplified 56.0%

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

       1. associate-/r/ 55.9%

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

       2. /-rgt-identity 55.9%

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

       3. associate-/r* 54.3%

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

       4. associate-/l/ 55.9%

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

   13. Applied egg-rr 55.9%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -1.6 \cdot 10^{+49}:\\ \;\;\;\;0\\ \mathbf{elif}\;w \leq 7.2 \cdot 10^{-173} \lor \neg \left(w \leq 3.8 \cdot 10^{-154}\right) \land w \leq 4.4 \cdot 10^{+180}:\\ \;\;\;\;\frac{c0}{w} \cdot \left({\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 44.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} t_0 := \frac{c0}{\frac{w \cdot {\left(\frac{d}{D}\right)}^{-2}}{c0} \cdot \left(w \cdot h\right)}\\ \mathbf{if}\;M_m \leq 10^{-217}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;M_m \leq 1.56 \cdot 10^{-97}:\\ \;\;\;\;0\\ \mathbf{elif}\;M_m \leq 5.2 \cdot 10^{-76}:\\ \;\;\;\;\frac{c0}{w \cdot \frac{h \cdot \frac{w}{c0}}{{\left(\frac{d}{D}\right)}^{2}}}\\ \mathbf{elif}\;M_m \leq 1.3 \cdot 10^{-65}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (/ c0 (* (/ (* w (pow (/ d D) -2.0)) c0) (* w h)))))
   (if (<= M_m 1e-217)
     t_0
     (if (<= M_m 1.56e-97)
       0.0
       (if (<= M_m 5.2e-76)
         (/ c0 (* w (/ (* h (/ w c0)) (pow (/ d D) 2.0))))
         (if (<= M_m 1.3e-65) 0.0 t_0))))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = c0 / (((w * pow((d / D), -2.0)) / c0) * (w * h));
	double tmp;
	if (M_m <= 1e-217) {
		tmp = t_0;
	} else if (M_m <= 1.56e-97) {
		tmp = 0.0;
	} else if (M_m <= 5.2e-76) {
		tmp = c0 / (w * ((h * (w / c0)) / pow((d / D), 2.0)));
	} else if (M_m <= 1.3e-65) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

M_m = abs(M)
real(8) function code(c0, w, h, d, d_1, m_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_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 / (((w * ((d_1 / d) ** (-2.0d0))) / c0) * (w * h))
    if (m_m <= 1d-217) then
        tmp = t_0
    else if (m_m <= 1.56d-97) then
        tmp = 0.0d0
    else if (m_m <= 5.2d-76) then
        tmp = c0 / (w * ((h * (w / c0)) / ((d_1 / d) ** 2.0d0)))
    else if (m_m <= 1.3d-65) then
        tmp = 0.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = c0 / (((w * Math.pow((d / D), -2.0)) / c0) * (w * h));
	double tmp;
	if (M_m <= 1e-217) {
		tmp = t_0;
	} else if (M_m <= 1.56e-97) {
		tmp = 0.0;
	} else if (M_m <= 5.2e-76) {
		tmp = c0 / (w * ((h * (w / c0)) / Math.pow((d / D), 2.0)));
	} else if (M_m <= 1.3e-65) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	t_0 = c0 / (((w * math.pow((d / D), -2.0)) / c0) * (w * h))
	tmp = 0
	if M_m <= 1e-217:
		tmp = t_0
	elif M_m <= 1.56e-97:
		tmp = 0.0
	elif M_m <= 5.2e-76:
		tmp = c0 / (w * ((h * (w / c0)) / math.pow((d / D), 2.0)))
	elif M_m <= 1.3e-65:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	t_0 = Float64(c0 / Float64(Float64(Float64(w * (Float64(d / D) ^ -2.0)) / c0) * Float64(w * h)))
	tmp = 0.0
	if (M_m <= 1e-217)
		tmp = t_0;
	elseif (M_m <= 1.56e-97)
		tmp = 0.0;
	elseif (M_m <= 5.2e-76)
		tmp = Float64(c0 / Float64(w * Float64(Float64(h * Float64(w / c0)) / (Float64(d / D) ^ 2.0))));
	elseif (M_m <= 1.3e-65)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	t_0 = c0 / (((w * ((d / D) ^ -2.0)) / c0) * (w * h));
	tmp = 0.0;
	if (M_m <= 1e-217)
		tmp = t_0;
	elseif (M_m <= 1.56e-97)
		tmp = 0.0;
	elseif (M_m <= 5.2e-76)
		tmp = c0 / (w * ((h * (w / c0)) / ((d / D) ^ 2.0)));
	elseif (M_m <= 1.3e-65)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := Block[{t$95$0 = N[(c0 / N[(N[(N[(w * N[Power[N[(d / D), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / c0), $MachinePrecision] * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M$95$m, 1e-217], t$95$0, If[LessEqual[M$95$m, 1.56e-97], 0.0, If[LessEqual[M$95$m, 5.2e-76], N[(c0 / N[(w * N[(N[(h * N[(w / c0), $MachinePrecision]), $MachinePrecision] / N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[M$95$m, 1.3e-65], 0.0, t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if M < 1.00000000000000008e-217 or 1.30000000000000005e-65 < M

   1. Initial program 25.0%

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

      1. +-commutative 25.0%

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

      2. +-commutative 25.0%

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

      3. times-frac 24.6%

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

      4. fma-neg 24.6%

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

   3. Simplified 25.5%

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

   5. Taylor expanded in c0 around inf 39.5%

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

      1. *-commutative 39.5%

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

      2. *-commutative 39.5%

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

      3. associate-*r* 39.4%

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

      4. associate-*r/ 39.5%

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

      5. associate-*r* 39.6%

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

      6. *-commutative 39.6%

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

      7. *-commutative 39.6%

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

      8. associate-/r* 40.1%

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

      9. unpow2 40.1%

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

      10. associate-*r/ 45.5%

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

      11. unpow2 45.5%

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

      12. associate-/l/ 49.6%

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

      13. associate-*r/ 49.2%

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

      14. associate-*l/ 50.1%

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

      15. unpow2 50.1%

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

   7. Simplified 50.1%

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

   9. Applied egg-rr 21.8%

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

       1. expm1-def 23.6%

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

       2. expm1-log1p 49.3%

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

       3. associate-*l* 49.3%

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

       4. associate-*l/ 48.5%

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

       5. associate-/l* 49.8%

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

       6. *-commutative 49.8%

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

       7. associate-/l* 49.9%

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

       8. *-commutative 49.9%

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

       9. associate-/l* 49.9%

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

       10. metadata-eval 49.9%

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

   11. Simplified 49.9%

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

       1. div-inv 49.9%

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

       2. /-rgt-identity 49.9%

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

       3. associate-/r* 47.5%

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

       4. associate-/l/ 49.9%

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

   13. Applied egg-rr 49.9%

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

       1. un-div-inv 49.9%

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

       2. associate-/r* 51.0%

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

       3. *-commutative 51.0%

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

       4. associate-/r/ 51.9%

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

       5. div-inv 51.8%

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

       6. pow-flip 51.9%

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

       7. metadata-eval 51.9%

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

       8. *-commutative 51.9%

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

   15. Applied egg-rr 51.9%

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

   if 1.00000000000000008e-217 < M < 1.55999999999999989e-97 or 5.1999999999999999e-76 < M < 1.30000000000000005e-65

   1. Initial program 14.1%

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

      1. +-commutative 14.1%

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

      2. +-commutative 14.1%

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

      3. times-frac 14.1%

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

      4. fma-neg 14.1%

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

   3. Simplified 14.2%

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

   5. Taylor expanded in c0 around -inf 3.7%

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

      1. associate-*r* 3.7%

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

      2. neg-mul-1 3.7%

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

      3. distribute-lft1-in 3.7%

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

      4. metadata-eval 3.7%

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

      5. mul0-lft 50.9%

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

      6. distribute-lft-neg-in 50.9%

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

      7. distribute-rgt-neg-in 50.9%

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

      8. metadata-eval 50.9%

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

      9. mul0-lft 3.7%

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

      10. metadata-eval 3.7%

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

      11. distribute-lft1-in 3.7%

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

      12. distribute-lft-in 3.7%

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

   7. Simplified 50.9%

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

   8. Taylor expanded in c0 around 0 57.7%

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

   if 1.55999999999999989e-97 < M < 5.1999999999999999e-76

   1. Initial program 27.2%

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

      1. +-commutative 27.2%

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

      2. +-commutative 27.2%

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

      3. times-frac 27.2%

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

      4. fma-neg 27.2%

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

   3. Simplified 28.8%

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

   5. Taylor expanded in c0 around inf 50.0%

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

      1. *-commutative 50.0%

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

      2. *-commutative 50.0%

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

      3. associate-*r* 50.0%

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

      4. associate-*r/ 50.0%

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

      5. associate-*r* 50.0%

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

      6. *-commutative 50.0%

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

      7. *-commutative 50.0%

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

      8. associate-/r* 51.2%

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

      9. unpow2 51.2%

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

      10. associate-*r/ 74.0%

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

      11. unpow2 74.0%

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

      12. associate-/l/ 74.2%

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

      13. associate-*r/ 73.6%

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

      14. associate-*l/ 73.6%

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

      15. unpow2 73.6%

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

   7. Simplified 73.6%

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

      1. associate-*r/ 74.0%

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

      2. *-commutative 74.0%

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

      3. associate-*l/ 74.0%

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

      4. *-commutative 74.0%

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

      5. unpow2 74.0%

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

      6. times-frac 51.2%

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

      7. associate-/r* 51.2%

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

      8. associate-/r* 51.2%

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

      9. frac-times 27.2%

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

      10. pow2 27.2%

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

   9. Applied egg-rr 27.2%

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

       1. expm1-log1p-u 2.2%

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

       2. expm1-udef 2.2%

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

       3. associate-*r* 2.2%

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

       4. associate-/r* 2.2%

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

       5. times-frac 24.2%

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

       6. associate-/r* 24.2%

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

       7. *-commutative 24.2%

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

       8. associate-/r* 24.2%

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

   11. Applied egg-rr 24.2%

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

       1. expm1-def 24.2%

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

       2. expm1-log1p 51.2%

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

       3. associate-*l* 51.2%

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

       4. associate-/l/ 51.2%

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

       5. associate-/r/ 51.6%

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

       6. associate-/l* 51.6%

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

       7. *-commutative 51.6%

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

       8. associate-/l* 51.6%

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

       9. associate-/l/ 51.2%

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

       10. unpow2 51.2%

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

       11. times-frac 74.6%

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

       12. unpow2 74.6%

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

       13. metadata-eval 74.6%

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

   13. Simplified 74.0%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 10^{-217}:\\ \;\;\;\;\frac{c0}{\frac{w \cdot {\left(\frac{d}{D}\right)}^{-2}}{c0} \cdot \left(w \cdot h\right)}\\ \mathbf{elif}\;M \leq 1.56 \cdot 10^{-97}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 5.2 \cdot 10^{-76}:\\ \;\;\;\;\frac{c0}{w \cdot \frac{h \cdot \frac{w}{c0}}{{\left(\frac{d}{D}\right)}^{2}}}\\ \mathbf{elif}\;M \leq 1.3 \cdot 10^{-65}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{w \cdot {\left(\frac{d}{D}\right)}^{-2}}{c0} \cdot \left(w \cdot h\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 44.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} t_0 := {\left(\frac{d}{D}\right)}^{2}\\ \mathbf{if}\;M_m \leq 1.3 \cdot 10^{-217}:\\ \;\;\;\;\frac{c0}{\frac{\frac{w}{t_0}}{\frac{c0}{w \cdot h}}}\\ \mathbf{elif}\;M_m \leq 2.05 \cdot 10^{-97}:\\ \;\;\;\;0\\ \mathbf{elif}\;M_m \leq 3.6 \cdot 10^{-76}:\\ \;\;\;\;\frac{c0}{w \cdot \frac{h \cdot \frac{w}{c0}}{t_0}}\\ \mathbf{elif}\;M_m \leq 1.22 \cdot 10^{-65}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{w \cdot {\left(\frac{d}{D}\right)}^{-2}}{c0} \cdot \left(w \cdot h\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (pow (/ d D) 2.0)))
   (if (<= M_m 1.3e-217)
     (/ c0 (/ (/ w t_0) (/ c0 (* w h))))
     (if (<= M_m 2.05e-97)
       0.0
       (if (<= M_m 3.6e-76)
         (/ c0 (* w (/ (* h (/ w c0)) t_0)))
         (if (<= M_m 1.22e-65)
           0.0
           (/ c0 (* (/ (* w (pow (/ d D) -2.0)) c0) (* w h)))))))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = pow((d / D), 2.0);
	double tmp;
	if (M_m <= 1.3e-217) {
		tmp = c0 / ((w / t_0) / (c0 / (w * h)));
	} else if (M_m <= 2.05e-97) {
		tmp = 0.0;
	} else if (M_m <= 3.6e-76) {
		tmp = c0 / (w * ((h * (w / c0)) / t_0));
	} else if (M_m <= 1.22e-65) {
		tmp = 0.0;
	} else {
		tmp = c0 / (((w * pow((d / D), -2.0)) / c0) * (w * h));
	}
	return tmp;
}

Fortran

M_m = abs(M)
real(8) function code(c0, w, h, d, d_1, m_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_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (d_1 / d) ** 2.0d0
    if (m_m <= 1.3d-217) then
        tmp = c0 / ((w / t_0) / (c0 / (w * h)))
    else if (m_m <= 2.05d-97) then
        tmp = 0.0d0
    else if (m_m <= 3.6d-76) then
        tmp = c0 / (w * ((h * (w / c0)) / t_0))
    else if (m_m <= 1.22d-65) then
        tmp = 0.0d0
    else
        tmp = c0 / (((w * ((d_1 / d) ** (-2.0d0))) / c0) * (w * h))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = Math.pow((d / D), 2.0);
	double tmp;
	if (M_m <= 1.3e-217) {
		tmp = c0 / ((w / t_0) / (c0 / (w * h)));
	} else if (M_m <= 2.05e-97) {
		tmp = 0.0;
	} else if (M_m <= 3.6e-76) {
		tmp = c0 / (w * ((h * (w / c0)) / t_0));
	} else if (M_m <= 1.22e-65) {
		tmp = 0.0;
	} else {
		tmp = c0 / (((w * Math.pow((d / D), -2.0)) / c0) * (w * h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	t_0 = math.pow((d / D), 2.0)
	tmp = 0
	if M_m <= 1.3e-217:
		tmp = c0 / ((w / t_0) / (c0 / (w * h)))
	elif M_m <= 2.05e-97:
		tmp = 0.0
	elif M_m <= 3.6e-76:
		tmp = c0 / (w * ((h * (w / c0)) / t_0))
	elif M_m <= 1.22e-65:
		tmp = 0.0
	else:
		tmp = c0 / (((w * math.pow((d / D), -2.0)) / c0) * (w * h))
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	t_0 = Float64(d / D) ^ 2.0
	tmp = 0.0
	if (M_m <= 1.3e-217)
		tmp = Float64(c0 / Float64(Float64(w / t_0) / Float64(c0 / Float64(w * h))));
	elseif (M_m <= 2.05e-97)
		tmp = 0.0;
	elseif (M_m <= 3.6e-76)
		tmp = Float64(c0 / Float64(w * Float64(Float64(h * Float64(w / c0)) / t_0)));
	elseif (M_m <= 1.22e-65)
		tmp = 0.0;
	else
		tmp = Float64(c0 / Float64(Float64(Float64(w * (Float64(d / D) ^ -2.0)) / c0) * Float64(w * h)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	t_0 = (d / D) ^ 2.0;
	tmp = 0.0;
	if (M_m <= 1.3e-217)
		tmp = c0 / ((w / t_0) / (c0 / (w * h)));
	elseif (M_m <= 2.05e-97)
		tmp = 0.0;
	elseif (M_m <= 3.6e-76)
		tmp = c0 / (w * ((h * (w / c0)) / t_0));
	elseif (M_m <= 1.22e-65)
		tmp = 0.0;
	else
		tmp = c0 / (((w * ((d / D) ^ -2.0)) / c0) * (w * h));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := Block[{t$95$0 = N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[M$95$m, 1.3e-217], N[(c0 / N[(N[(w / t$95$0), $MachinePrecision] / N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[M$95$m, 2.05e-97], 0.0, If[LessEqual[M$95$m, 3.6e-76], N[(c0 / N[(w * N[(N[(h * N[(w / c0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[M$95$m, 1.22e-65], 0.0, N[(c0 / N[(N[(N[(w * N[Power[N[(d / D), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / c0), $MachinePrecision] * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if M < 1.29999999999999997e-217

   1. Initial program 28.5%

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

      1. +-commutative 28.5%

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

      2. +-commutative 28.5%

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

      3. times-frac 27.8%

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

      4. fma-neg 27.8%

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

   3. Simplified 28.6%

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

   5. Taylor expanded in c0 around inf 38.6%

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

      1. *-commutative 38.6%

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

      2. *-commutative 38.6%

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

      3. associate-*r* 38.5%

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

      4. associate-*r/ 38.6%

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

      5. associate-*r* 38.7%

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

      6. *-commutative 38.7%

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

      7. *-commutative 38.7%

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

      8. associate-/r* 38.8%

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

      9. unpow2 38.8%

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

      10. associate-*r/ 45.9%

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

      11. unpow2 45.9%

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

      12. associate-/l/ 49.2%

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

      13. associate-*r/ 48.6%

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

      14. associate-*l/ 49.8%

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

      15. unpow2 49.8%

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

   7. Simplified 49.8%

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

   9. Applied egg-rr 22.2%

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

       1. expm1-def 24.8%

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

       2. expm1-log1p 48.8%

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

       3. associate-*l* 48.8%

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

       4. associate-*l/ 47.6%

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

       5. associate-/l* 49.5%

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

       6. *-commutative 49.5%

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

       7. associate-/l* 49.6%

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

       8. *-commutative 49.6%

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

       9. associate-/l* 49.6%

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

       10. metadata-eval 49.6%

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

   11. Simplified 49.6%

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

       1. expm1-log1p-u 46.8%

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

       2. expm1-udef 27.3%

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

       3. /-rgt-identity 27.3%

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

       4. associate-/r* 27.7%

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

       5. associate-/l/ 27.3%

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

   13. Applied egg-rr 27.3%

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

       1. expm1-def 46.8%

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

       2. expm1-log1p 49.6%

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

       3. associate-/r* 50.7%

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

       4. *-commutative 50.7%

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

   15. Simplified 50.7%

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

   if 1.29999999999999997e-217 < M < 2.04999999999999996e-97 or 3.6e-76 < M < 1.21999999999999999e-65

   1. Initial program 14.1%

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

      1. +-commutative 14.1%

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

      2. +-commutative 14.1%

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

      3. times-frac 14.1%

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

      4. fma-neg 14.1%

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

   3. Simplified 14.2%

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

   5. Taylor expanded in c0 around -inf 3.7%

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

      1. associate-*r* 3.7%

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

      2. neg-mul-1 3.7%

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

      3. distribute-lft1-in 3.7%

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

      4. metadata-eval 3.7%

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

      5. mul0-lft 50.9%

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

      6. distribute-lft-neg-in 50.9%

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

      7. distribute-rgt-neg-in 50.9%

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

      8. metadata-eval 50.9%

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

      9. mul0-lft 3.7%

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

      10. metadata-eval 3.7%

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

      11. distribute-lft1-in 3.7%

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

      12. distribute-lft-in 3.7%

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

   7. Simplified 50.9%

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

   8. Taylor expanded in c0 around 0 57.7%

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

   if 2.04999999999999996e-97 < M < 3.6e-76

   1. Initial program 27.2%

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

      1. +-commutative 27.2%

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

      2. +-commutative 27.2%

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

      3. times-frac 27.2%

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

      4. fma-neg 27.2%

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

   3. Simplified 28.8%

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

   5. Taylor expanded in c0 around inf 50.0%

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

      1. *-commutative 50.0%

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

      2. *-commutative 50.0%

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

      3. associate-*r* 50.0%

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

      4. associate-*r/ 50.0%

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

      5. associate-*r* 50.0%

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

      6. *-commutative 50.0%

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

      7. *-commutative 50.0%

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

      8. associate-/r* 51.2%

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

      9. unpow2 51.2%

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

      10. associate-*r/ 74.0%

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

      11. unpow2 74.0%

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

      12. associate-/l/ 74.2%

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

      13. associate-*r/ 73.6%

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

      14. associate-*l/ 73.6%

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

      15. unpow2 73.6%

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

   7. Simplified 73.6%

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

      1. associate-*r/ 74.0%

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

      2. *-commutative 74.0%

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

      3. associate-*l/ 74.0%

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

      4. *-commutative 74.0%

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

      5. unpow2 74.0%

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

      6. times-frac 51.2%

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

      7. associate-/r* 51.2%

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

      8. associate-/r* 51.2%

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

      9. frac-times 27.2%

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

      10. pow2 27.2%

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

   9. Applied egg-rr 27.2%

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

       1. expm1-log1p-u 2.2%

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

       2. expm1-udef 2.2%

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

       3. associate-*r* 2.2%

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

       4. associate-/r* 2.2%

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

       5. times-frac 24.2%

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

       6. associate-/r* 24.2%

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

       7. *-commutative 24.2%

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

       8. associate-/r* 24.2%

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

   11. Applied egg-rr 24.2%

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

       1. expm1-def 24.2%

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

       2. expm1-log1p 51.2%

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

       3. associate-*l* 51.2%

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

       4. associate-/l/ 51.2%

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

       5. associate-/r/ 51.6%

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

       6. associate-/l* 51.6%

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

       7. *-commutative 51.6%

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

       8. associate-/l* 51.6%

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

       9. associate-/l/ 51.2%

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

       10. unpow2 51.2%

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

       11. times-frac 74.6%

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

       12. unpow2 74.6%

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

       13. metadata-eval 74.6%

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

   13. Simplified 74.0%

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

   if 1.21999999999999999e-65 < M

   1. Initial program 17.4%

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

      1. +-commutative 17.4%

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

      2. +-commutative 17.4%

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

      3. times-frac 17.4%

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

      4. fma-neg 17.4%

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

   3. Simplified 18.9%

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

   5. Taylor expanded in c0 around inf 41.4%

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

      1. *-commutative 41.4%

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

      2. *-commutative 41.4%

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

      3. associate-*r* 41.4%

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

      4. associate-*r/ 41.6%

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

      5. associate-*r* 41.6%

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

      6. *-commutative 41.6%

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

      7. *-commutative 41.6%

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

      8. associate-/r* 43.0%

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

      9. unpow2 43.0%

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

      10. associate-*r/ 44.6%

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

      11. unpow2 44.6%

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

      12. associate-/l/ 50.5%

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

      13. associate-*r/ 50.6%

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

      14. associate-*l/ 50.6%

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

      15. unpow2 50.6%

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

   7. Simplified 50.6%

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

   9. Applied egg-rr 20.8%

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

       1. expm1-def 20.8%

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

       2. expm1-log1p 50.6%

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

       3. associate-*l* 50.6%

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

       4. associate-*l/ 50.6%

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

       5. associate-/l* 50.6%

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

       6. *-commutative 50.6%

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

       7. associate-/l* 50.6%

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

       8. *-commutative 50.6%

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

       9. associate-/l* 50.6%

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

       10. metadata-eval 50.6%

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

   11. Simplified 50.6%

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

       1. div-inv 50.6%

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

       2. /-rgt-identity 50.6%

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

       3. associate-/r* 47.6%

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

       4. associate-/l/ 50.6%

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

   13. Applied egg-rr 50.6%

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

       1. un-div-inv 50.6%

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

       2. associate-/r* 51.8%

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

       3. *-commutative 51.8%

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

       4. associate-/r/ 51.9%

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

       5. div-inv 51.9%

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

       6. pow-flip 51.9%

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

       7. metadata-eval 51.9%

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

       8. *-commutative 51.9%

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

   15. Applied egg-rr 51.9%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.3 \cdot 10^{-217}:\\ \;\;\;\;\frac{c0}{\frac{\frac{w}{{\left(\frac{d}{D}\right)}^{2}}}{\frac{c0}{w \cdot h}}}\\ \mathbf{elif}\;M \leq 2.05 \cdot 10^{-97}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 3.6 \cdot 10^{-76}:\\ \;\;\;\;\frac{c0}{w \cdot \frac{h \cdot \frac{w}{c0}}{{\left(\frac{d}{D}\right)}^{2}}}\\ \mathbf{elif}\;M \leq 1.22 \cdot 10^{-65}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{w \cdot {\left(\frac{d}{D}\right)}^{-2}}{c0} \cdot \left(w \cdot h\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 45.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} \mathbf{if}\;M_m \leq 8.5 \cdot 10^{-210} \lor \neg \left(M_m \leq 1.65 \cdot 10^{-97}\right):\\ \;\;\;\;\frac{c0}{w \cdot \frac{h \cdot \frac{w}{c0}}{{\left(\frac{d}{D}\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (if (or (<= M_m 8.5e-210) (not (<= M_m 1.65e-97)))
   (/ c0 (* w (/ (* h (/ w c0)) (pow (/ d D) 2.0))))
   0.0))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double tmp;
	if ((M_m <= 8.5e-210) || !(M_m <= 1.65e-97)) {
		tmp = c0 / (w * ((h * (w / c0)) / pow((d / D), 2.0)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

M_m = abs(M)
real(8) function code(c0, w, h, d, d_1, m_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_m
    real(8) :: tmp
    if ((m_m <= 8.5d-210) .or. (.not. (m_m <= 1.65d-97))) then
        tmp = c0 / (w * ((h * (w / c0)) / ((d_1 / d) ** 2.0d0)))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	double tmp;
	if ((M_m <= 8.5e-210) || !(M_m <= 1.65e-97)) {
		tmp = c0 / (w * ((h * (w / c0)) / Math.pow((d / D), 2.0)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	tmp = 0
	if (M_m <= 8.5e-210) or not (M_m <= 1.65e-97):
		tmp = c0 / (w * ((h * (w / c0)) / math.pow((d / D), 2.0)))
	else:
		tmp = 0.0
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	tmp = 0.0
	if ((M_m <= 8.5e-210) || !(M_m <= 1.65e-97))
		tmp = Float64(c0 / Float64(w * Float64(Float64(h * Float64(w / c0)) / (Float64(d / D) ^ 2.0))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	tmp = 0.0;
	if ((M_m <= 8.5e-210) || ~((M_m <= 1.65e-97)))
		tmp = c0 / (w * ((h * (w / c0)) / ((d / D) ^ 2.0)));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := If[Or[LessEqual[M$95$m, 8.5e-210], N[Not[LessEqual[M$95$m, 1.65e-97]], $MachinePrecision]], N[(c0 / N[(w * N[(N[(h * N[(w / c0), $MachinePrecision]), $MachinePrecision] / N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if M < 8.4999999999999997e-210 or 1.6500000000000001e-97 < M

   1. Initial program 24.7%

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

      1. +-commutative 24.7%

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

      2. +-commutative 24.7%

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

      3. times-frac 24.3%

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

      4. fma-neg 24.3%

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

   3. Simplified 25.3%

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

   5. Taylor expanded in c0 around inf 38.9%

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

      1. *-commutative 38.9%

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

      2. *-commutative 38.9%

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

      3. associate-*r* 39.3%

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

      4. associate-*r/ 39.4%

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

      5. associate-*r* 39.0%

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

      6. *-commutative 39.0%

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

      7. *-commutative 39.0%

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

      8. associate-/r* 39.5%

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

      9. unpow2 39.5%

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

      10. associate-*r/ 45.1%

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

      11. unpow2 45.1%

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

      12. associate-/l/ 49.0%

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

      13. associate-*r/ 48.6%

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

      14. associate-*l/ 49.5%

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

      15. unpow2 49.5%

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

   7. Simplified 49.5%

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

      1. associate-*r/ 49.2%

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

      2. *-commutative 49.2%

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

      3. associate-*l/ 48.8%

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

      4. *-commutative 48.8%

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

      5. unpow2 48.8%

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

      6. times-frac 39.0%

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

      7. associate-/r* 44.6%

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

      8. associate-/r* 45.9%

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

      9. frac-times 46.6%

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

      10. pow2 46.6%

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

   9. Applied egg-rr 46.6%

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

       1. expm1-log1p-u 21.1%

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

       2. expm1-udef 21.4%

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

       3. associate-*r* 21.4%

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

       4. associate-/r* 21.4%

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

       5. times-frac 21.2%

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

       6. associate-/r* 19.9%

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

       7. *-commutative 19.9%

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

       8. associate-/r* 19.4%

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

   11. Applied egg-rr 19.4%

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

       1. expm1-def 20.2%

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

       2. expm1-log1p 43.5%

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

       3. associate-*l* 43.5%

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

       4. associate-/l/ 43.5%

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

       5. associate-/r/ 44.0%

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

       6. associate-/l* 44.0%

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

       7. *-commutative 44.0%

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

       8. associate-/l* 44.0%

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

       9. associate-/l/ 38.1%

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

       10. unpow2 38.1%

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

       11. times-frac 47.5%

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

       12. unpow2 47.5%

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

       13. metadata-eval 47.5%

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

   13. Simplified 50.5%

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

   if 8.4999999999999997e-210 < M < 1.6500000000000001e-97

   1. Initial program 14.1%

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

      1. +-commutative 14.1%

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

      2. +-commutative 14.1%

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

      3. times-frac 14.1%

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

      4. fma-neg 14.1%

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

   3. Simplified 14.2%

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

   5. Taylor expanded in c0 around -inf 4.9%

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

      1. associate-*r* 4.9%

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

      2. neg-mul-1 4.9%

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

      3. distribute-lft1-in 4.9%

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

      4. metadata-eval 4.9%

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

      5. mul0-lft 44.6%

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

      6. distribute-lft-neg-in 44.6%

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

      7. distribute-rgt-neg-in 44.6%

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

      8. metadata-eval 44.6%

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

      9. mul0-lft 4.9%

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

      10. metadata-eval 4.9%

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

      11. distribute-lft1-in 4.9%

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

      12. distribute-lft-in 4.9%

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

   7. Simplified 44.6%

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

   8. Taylor expanded in c0 around 0 53.4%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 8.5 \cdot 10^{-210} \lor \neg \left(M \leq 1.65 \cdot 10^{-97}\right):\\ \;\;\;\;\frac{c0}{w \cdot \frac{h \cdot \frac{w}{c0}}{{\left(\frac{d}{D}\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 28.7% accurate, 12.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} \mathbf{if}\;M_m \leq 1.55 \cdot 10^{+241}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(M_m \cdot \frac{c0}{w}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (if (<= M_m 1.55e+241) 0.0 (* -0.5 (* M_m (/ c0 w)))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double tmp;
	if (M_m <= 1.55e+241) {
		tmp = 0.0;
	} else {
		tmp = -0.5 * (M_m * (c0 / w));
	}
	return tmp;
}

Fortran

M_m = abs(M)
real(8) function code(c0, w, h, d, d_1, m_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_m
    real(8) :: tmp
    if (m_m <= 1.55d+241) then
        tmp = 0.0d0
    else
        tmp = (-0.5d0) * (m_m * (c0 / w))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	double tmp;
	if (M_m <= 1.55e+241) {
		tmp = 0.0;
	} else {
		tmp = -0.5 * (M_m * (c0 / w));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	tmp = 0
	if M_m <= 1.55e+241:
		tmp = 0.0
	else:
		tmp = -0.5 * (M_m * (c0 / w))
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	tmp = 0.0
	if (M_m <= 1.55e+241)
		tmp = 0.0;
	else
		tmp = Float64(-0.5 * Float64(M_m * Float64(c0 / w)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	tmp = 0.0;
	if (M_m <= 1.55e+241)
		tmp = 0.0;
	else
		tmp = -0.5 * (M_m * (c0 / w));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := If[LessEqual[M$95$m, 1.55e+241], 0.0, N[(-0.5 * N[(M$95$m * N[(c0 / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.55e241

   1. Initial program 25.0%

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

      1. +-commutative 25.0%

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

      2. +-commutative 25.0%

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

      3. times-frac 24.5%

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

      4. fma-neg 24.5%

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

   3. Simplified 25.5%

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

   5. Taylor expanded in c0 around -inf 2.7%

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

      1. associate-*r* 2.7%

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

      2. neg-mul-1 2.7%

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

      3. distribute-lft1-in 2.7%

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

      4. metadata-eval 2.7%

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

      5. mul0-lft 25.9%

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

      6. distribute-lft-neg-in 25.9%

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

      7. distribute-rgt-neg-in 25.9%

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

      8. metadata-eval 25.9%

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

      9. mul0-lft 2.7%

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

      10. metadata-eval 2.7%

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

      11. distribute-lft1-in 2.7%

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

      12. distribute-lft-in 2.7%

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

   7. Simplified 25.9%

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

   8. Taylor expanded in c0 around 0 29.1%

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

   if 1.55e241 < M

   1. Initial program 0.0%

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

      1. +-commutative 0.0%

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

      2. +-commutative 0.0%

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

      3. times-frac 0.0%

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

      4. fma-neg 0.0%

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

   3. Simplified 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. fma-udef 0.0%

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

      2. *-commutative 0.0%

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

      3. fma-udef 0.0%

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

   8. Applied egg-rr 25.0%

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

      1. fma-def 25.0%

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

      2. associate-/r* 16.7%

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

      3. associate-/r* 16.7%

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

   10. Simplified 16.7%

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

   11. Taylor expanded in M around -inf 43.1%

       \[\leadsto \color{blue}{-0.5 \cdot \frac{M \cdot c0}{w}} \]
   12. Step-by-step derivation

       1. associate-/l* 43.1%

          \[\leadsto -0.5 \cdot \color{blue}{\frac{M}{\frac{w}{c0}}} \]

   13. Simplified 43.1%

       \[\leadsto \color{blue}{-0.5 \cdot \frac{M}{\frac{w}{c0}}} \]
   14. Step-by-step derivation

       1. div-inv 43.1%

          \[\leadsto -0.5 \cdot \color{blue}{\left(M \cdot \frac{1}{\frac{w}{c0}}\right)} \]

       2. clear-num 43.1%

          \[\leadsto -0.5 \cdot \left(M \cdot \color{blue}{\frac{c0}{w}}\right) \]

   15. Applied egg-rr 43.1%

       \[\leadsto -0.5 \cdot \color{blue}{\left(M \cdot \frac{c0}{w}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.55 \cdot 10^{+241}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(M \cdot \frac{c0}{w}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 32.9% accurate, 12.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} \mathbf{if}\;M_m \leq 4.8 \cdot 10^{+116}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{M_m}{\frac{w}{c0}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (if (<= M_m 4.8e+116) 0.0 (* 0.5 (/ M_m (/ w c0)))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double tmp;
	if (M_m <= 4.8e+116) {
		tmp = 0.0;
	} else {
		tmp = 0.5 * (M_m / (w / c0));
	}
	return tmp;
}

Fortran

M_m = abs(M)
real(8) function code(c0, w, h, d, d_1, m_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_m
    real(8) :: tmp
    if (m_m <= 4.8d+116) then
        tmp = 0.0d0
    else
        tmp = 0.5d0 * (m_m / (w / c0))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	double tmp;
	if (M_m <= 4.8e+116) {
		tmp = 0.0;
	} else {
		tmp = 0.5 * (M_m / (w / c0));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	tmp = 0
	if M_m <= 4.8e+116:
		tmp = 0.0
	else:
		tmp = 0.5 * (M_m / (w / c0))
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	tmp = 0.0
	if (M_m <= 4.8e+116)
		tmp = 0.0;
	else
		tmp = Float64(0.5 * Float64(M_m / Float64(w / c0)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	tmp = 0.0;
	if (M_m <= 4.8e+116)
		tmp = 0.0;
	else
		tmp = 0.5 * (M_m / (w / c0));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := If[LessEqual[M$95$m, 4.8e+116], 0.0, N[(0.5 * N[(M$95$m / N[(w / c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4.8000000000000001e116

   1. Initial program 25.7%

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

      1. +-commutative 25.7%

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

      2. +-commutative 25.7%

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

      3. times-frac 25.3%

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

      4. fma-neg 25.3%

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

   3. Simplified 26.3%

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

   5. Taylor expanded in c0 around -inf 3.0%

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

      1. associate-*r* 3.0%

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

      2. neg-mul-1 3.0%

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

      3. distribute-lft1-in 3.0%

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

      4. metadata-eval 3.0%

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

      5. mul0-lft 27.8%

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

      6. distribute-lft-neg-in 27.8%

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

      7. distribute-rgt-neg-in 27.8%

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

      8. metadata-eval 27.8%

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

      9. mul0-lft 3.0%

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

      10. metadata-eval 3.0%

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

      11. distribute-lft1-in 3.0%

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

      12. distribute-lft-in 3.0%

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

   7. Simplified 27.8%

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

   8. Taylor expanded in c0 around 0 31.3%

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

   if 4.8000000000000001e116 < M

   1. Initial program 11.4%

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

      1. +-commutative 11.4%

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

      2. +-commutative 11.4%

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

      3. times-frac 11.4%

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

      4. fma-neg 11.4%

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

   3. Simplified 11.4%

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

   6. Applied egg-rr 8.6%

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

      1. fma-udef 8.6%

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

      2. *-commutative 8.6%

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

      3. fma-udef 8.6%

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

   8. Applied egg-rr 43.0%

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

      1. fma-def 43.1%

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

      2. associate-/r* 40.2%

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

      3. associate-/r* 40.2%

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

   10. Simplified 40.2%

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

   11. Taylor expanded in c0 around 0 38.2%

       \[\leadsto \color{blue}{0.5 \cdot \frac{M \cdot c0}{w}} \]
   12. Step-by-step derivation

       1. associate-/l* 32.9%

          \[\leadsto 0.5 \cdot \color{blue}{\frac{M}{\frac{w}{c0}}} \]

   13. Simplified 32.9%

       \[\leadsto \color{blue}{0.5 \cdot \frac{M}{\frac{w}{c0}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 4.8 \cdot 10^{+116}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{M}{\frac{w}{c0}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 33.6% accurate, 12.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} \mathbf{if}\;M_m \leq 9.2 \cdot 10^{+115}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(M_m \cdot c0\right) \cdot 0.5}{w}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (if (<= M_m 9.2e+115) 0.0 (/ (* (* M_m c0) 0.5) w)))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double tmp;
	if (M_m <= 9.2e+115) {
		tmp = 0.0;
	} else {
		tmp = ((M_m * c0) * 0.5) / w;
	}
	return tmp;
}

Fortran

M_m = abs(M)
real(8) function code(c0, w, h, d, d_1, m_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_m
    real(8) :: tmp
    if (m_m <= 9.2d+115) then
        tmp = 0.0d0
    else
        tmp = ((m_m * c0) * 0.5d0) / w
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	double tmp;
	if (M_m <= 9.2e+115) {
		tmp = 0.0;
	} else {
		tmp = ((M_m * c0) * 0.5) / w;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	tmp = 0
	if M_m <= 9.2e+115:
		tmp = 0.0
	else:
		tmp = ((M_m * c0) * 0.5) / w
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	tmp = 0.0
	if (M_m <= 9.2e+115)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(M_m * c0) * 0.5) / w);
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	tmp = 0.0;
	if (M_m <= 9.2e+115)
		tmp = 0.0;
	else
		tmp = ((M_m * c0) * 0.5) / w;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := If[LessEqual[M$95$m, 9.2e+115], 0.0, N[(N[(N[(M$95$m * c0), $MachinePrecision] * 0.5), $MachinePrecision] / w), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 9.20000000000000014e115

   1. Initial program 25.7%

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

      1. +-commutative 25.7%

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

      2. +-commutative 25.7%

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

      3. times-frac 25.3%

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

      4. fma-neg 25.3%

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

   3. Simplified 26.3%

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

   5. Taylor expanded in c0 around -inf 3.0%

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

      1. associate-*r* 3.0%

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

      2. neg-mul-1 3.0%

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

      3. distribute-lft1-in 3.0%

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

      4. metadata-eval 3.0%

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

      5. mul0-lft 27.8%

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

      6. distribute-lft-neg-in 27.8%

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

      7. distribute-rgt-neg-in 27.8%

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

      8. metadata-eval 27.8%

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

      9. mul0-lft 3.0%

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

      10. metadata-eval 3.0%

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

      11. distribute-lft1-in 3.0%

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

      12. distribute-lft-in 3.0%

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

   7. Simplified 27.8%

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

   8. Taylor expanded in c0 around 0 31.3%

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

   if 9.20000000000000014e115 < M

   1. Initial program 11.4%

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

      1. +-commutative 11.4%

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

      2. +-commutative 11.4%

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

      3. times-frac 11.4%

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

      4. fma-neg 11.4%

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

   3. Simplified 11.4%

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

   6. Applied egg-rr 8.6%

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

      1. fma-udef 8.6%

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

      2. *-commutative 8.6%

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

      3. fma-udef 8.6%

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

   8. Applied egg-rr 43.0%

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

      1. fma-def 43.1%

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

      2. associate-/r* 40.2%

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

      3. associate-/r* 40.2%

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

   10. Simplified 40.2%

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

   11. Taylor expanded in c0 around 0 38.2%

       \[\leadsto \color{blue}{0.5 \cdot \frac{M \cdot c0}{w}} \]
   12. Step-by-step derivation

       1. *-commutative 38.2%

          \[\leadsto \color{blue}{\frac{M \cdot c0}{w} \cdot 0.5} \]

       2. associate-*l/ 38.2%

          \[\leadsto \color{blue}{\frac{\left(M \cdot c0\right) \cdot 0.5}{w}} \]

       3. *-commutative 38.2%

          \[\leadsto \frac{\color{blue}{\left(c0 \cdot M\right)} \cdot 0.5}{w} \]

   13. Simplified 38.2%

       \[\leadsto \color{blue}{\frac{\left(c0 \cdot M\right) \cdot 0.5}{w}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 9.2 \cdot 10^{+115}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(M \cdot c0\right) \cdot 0.5}{w}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 28.2% accurate, 151.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ 0 \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m) :precision binary64 0.0)

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	return 0.0;
}

Fortran

M_m = abs(M)
real(8) function code(c0, w, h, d, d_1, m_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_m
    code = 0.0d0
end function

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	return 0.0;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	return 0.0

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	return 0.0
end

MATLAB

M_m = abs(M);
function tmp = code(c0, w, h, D, d, M_m)
	tmp = 0.0;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := 0.0

Derivation

1. Initial program 23.8%

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

   1. +-commutative 23.8%

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

   2. +-commutative 23.8%

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

   3. times-frac 23.4%

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

   4. fma-neg 23.4%

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

3. Simplified 24.3%

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

5. Taylor expanded in c0 around -inf 2.6%

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

   1. associate-*r* 2.6%

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

   2. neg-mul-1 2.6%

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

   3. distribute-lft1-in 2.6%

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

   4. metadata-eval 2.6%

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

   5. mul0-lft 24.8%

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

   6. distribute-lft-neg-in 24.8%

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

   7. distribute-rgt-neg-in 24.8%

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

   8. metadata-eval 24.8%

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

   9. mul0-lft 2.6%

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

   10. metadata-eval 2.6%

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

   11. distribute-lft1-in 2.6%

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

   12. distribute-lft-in 2.6%

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

7. Simplified 24.8%

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

8. Taylor expanded in c0 around 0 27.8%

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

9. Final simplification 27.8%

   \[\leadsto 0 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024024 
(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))))))