Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.0% → 46.0%

Time: 15.6s

Alternatives: 10

Speedup: 156.0×

• could not determine a ground truth

  1. c0 = -8.745675878738694e+216
  2. w = -1.0125536082155218e-287
  3. h = -1.2455547325985975e-206
  4. D = 6.222659099008243e-305
  5. d = 1.5639586415976535e+300
  6. M = -1.8380683906545774e+257

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: 24.0% 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: 46.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ \mathbf{if}\;M\_m \leq 6.2 \cdot 10^{-188}:\\ \;\;\;\;t\_0 \cdot \left(d \cdot \frac{\frac{c0}{D}}{\frac{w}{2 \cdot d} \cdot \left(D \cdot h\right)}\right)\\ \mathbf{elif}\;M\_m \leq 7.6 \cdot 10^{-150}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \left(c0 \cdot 2\right)}{D \cdot \left(h \cdot \left(w \cdot D\right)\right)} \cdot \left(t\_0 \cdot d\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w))))
   (if (<= M_m 6.2e-188)
     (* t_0 (* d (/ (/ c0 D) (* (/ w (* 2.0 d)) (* D h)))))
     (if (<= M_m 7.6e-150)
       0.0
       (* (/ (* d (* c0 2.0)) (* D (* h (* w D)))) (* t_0 d))))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = c0 / (2.0 * w);
	double tmp;
	if (M_m <= 6.2e-188) {
		tmp = t_0 * (d * ((c0 / D) / ((w / (2.0 * d)) * (D * h))));
	} else if (M_m <= 7.6e-150) {
		tmp = 0.0;
	} else {
		tmp = ((d * (c0 * 2.0)) / (D * (h * (w * D)))) * (t_0 * d);
	}
	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 / (2.0d0 * w)
    if (m_m <= 6.2d-188) then
        tmp = t_0 * (d_1 * ((c0 / d) / ((w / (2.0d0 * d_1)) * (d * h))))
    else if (m_m <= 7.6d-150) then
        tmp = 0.0d0
    else
        tmp = ((d_1 * (c0 * 2.0d0)) / (d * (h * (w * d)))) * (t_0 * d_1)
    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 / (2.0 * w);
	double tmp;
	if (M_m <= 6.2e-188) {
		tmp = t_0 * (d * ((c0 / D) / ((w / (2.0 * d)) * (D * h))));
	} else if (M_m <= 7.6e-150) {
		tmp = 0.0;
	} else {
		tmp = ((d * (c0 * 2.0)) / (D * (h * (w * D)))) * (t_0 * d);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	t_0 = c0 / (2.0 * w)
	tmp = 0
	if M_m <= 6.2e-188:
		tmp = t_0 * (d * ((c0 / D) / ((w / (2.0 * d)) * (D * h))))
	elif M_m <= 7.6e-150:
		tmp = 0.0
	else:
		tmp = ((d * (c0 * 2.0)) / (D * (h * (w * D)))) * (t_0 * d)
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	t_0 = Float64(c0 / Float64(2.0 * w))
	tmp = 0.0
	if (M_m <= 6.2e-188)
		tmp = Float64(t_0 * Float64(d * Float64(Float64(c0 / D) / Float64(Float64(w / Float64(2.0 * d)) * Float64(D * h)))));
	elseif (M_m <= 7.6e-150)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(d * Float64(c0 * 2.0)) / Float64(D * Float64(h * Float64(w * D)))) * Float64(t_0 * d));
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	t_0 = c0 / (2.0 * w);
	tmp = 0.0;
	if (M_m <= 6.2e-188)
		tmp = t_0 * (d * ((c0 / D) / ((w / (2.0 * d)) * (D * h))));
	elseif (M_m <= 7.6e-150)
		tmp = 0.0;
	else
		tmp = ((d * (c0 * 2.0)) / (D * (h * (w * D)))) * (t_0 * d);
	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[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M$95$m, 6.2e-188], N[(t$95$0 * N[(d * N[(N[(c0 / D), $MachinePrecision] / N[(N[(w / N[(2.0 * d), $MachinePrecision]), $MachinePrecision] * N[(D * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[M$95$m, 7.6e-150], 0.0, N[(N[(N[(d * N[(c0 * 2.0), $MachinePrecision]), $MachinePrecision] / N[(D * N[(h * N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * d), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if M < 6.2000000000000004e-188

   1. Initial program 19.4%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 27.3

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

   5. Simplified 27.3%

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

      1. lift-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   7. Applied egg-rr 41.7%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 44.4

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

   9. Applied egg-rr 44.4%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. associate-*l/ N/A

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

       6. *-commutative N/A

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

       7. associate-*r/ N/A

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

       8. clear-num N/A

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

       9. un-div-inv N/A

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

       10. frac-2neg N/A

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

       11. lift-/.f64 N/A

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

       12. distribute-frac-neg N/A

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

       13. lower-/.f64 N/A

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

       14. remove-double-neg N/A

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

       15. frac-2neg N/A

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

       16. lower-/.f64 N/A

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

       17. lower-neg.f64 N/A

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

   11. Applied egg-rr 47.0%

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

   if 6.2000000000000004e-188 < M < 7.5999999999999997e-150

   1. Initial program 8.9%

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

   3. Taylor expanded in c0 around -inf

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. mul-1-neg N/A

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

      7. distribute-lft-neg-in N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 70.2

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

   5. Simplified 70.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mul0-rgt N/A

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

      4. mul0-rgt 70.2

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

   7. Applied egg-rr 70.2%

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

   if 7.5999999999999997e-150 < M

   1. Initial program 24.4%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 47.0

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

   5. Simplified 47.0%

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

      1. lift-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   7. Applied egg-rr 57.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 57.3

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

   9. Applied egg-rr 57.3%

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

   11. Applied egg-rr 58.6%

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

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 6.2 \cdot 10^{-188}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(d \cdot \frac{\frac{c0}{D}}{\frac{w}{2 \cdot d} \cdot \left(D \cdot h\right)}\right)\\ \mathbf{elif}\;M \leq 7.6 \cdot 10^{-150}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \left(c0 \cdot 2\right)}{D \cdot \left(h \cdot \left(w \cdot D\right)\right)} \cdot \left(\frac{c0}{2 \cdot w} \cdot d\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 50.7% accurate, 0.7× speedup

Math

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

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 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M_m M_m)))))
        INFINITY)
     (* (* c0 c0) (/ (* d d) (* w (* D (* h (* w D))))))
     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 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M_m * M_m))))) <= ((double) INFINITY)) {
		tmp = (c0 * c0) * ((d * d) / (w * (D * (h * (w * D)))));
	} 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 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M_m * M_m))))) <= Double.POSITIVE_INFINITY) {
		tmp = (c0 * c0) * ((d * d) / (w * (D * (h * (w * D)))));
	} 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 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M_m * M_m))))) <= math.inf:
		tmp = (c0 * c0) * ((d * d) / (w * (D * (h * (w * D)))))
	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(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M_m * M_m))))) <= Inf)
		tmp = Float64(Float64(c0 * c0) * Float64(Float64(d * d) / Float64(w * Float64(D * Float64(h * Float64(w * D))))));
	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 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M_m * M_m))))) <= Inf)
		tmp = (c0 * c0) * ((d * d) / (w * (D * (h * (w * D)))));
	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[(2.0 * w), $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[(N[(c0 * c0), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / N[(w * N[(D * N[(h * N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) 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 65.0%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 67.8

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

   5. Simplified 67.8%

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

      1. lift-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   7. Applied egg-rr 73.5%

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

   8. Taylor expanded in c0 around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 58.9

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

   10. Simplified 58.9%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*r* N/A

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

       9. associate-/l* N/A

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

       10. lower-*.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. lower-/.f64 56.2

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

       13. lift-*.f64 N/A

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

       14. *-commutative N/A

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

       15. lift-*.f64 N/A

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

       16. lift-*.f64 N/A

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

       17. associate-*l* N/A

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

       18. lift-*.f64 N/A

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

       19. associate-*l* N/A

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

   12. Applied egg-rr 59.7%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) 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. Add Preprocessing

   3. Taylor expanded in c0 around -inf

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. mul-1-neg N/A

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

      7. distribute-lft-neg-in N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 33.7

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

   5. Simplified 33.7%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mul0-rgt N/A

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

      4. mul0-rgt 37.2

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

   7. Applied egg-rr 37.2%

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

4. Final simplification 44.4%

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

Alternative 3: 43.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} t_0 := d \cdot \left(c0 \cdot 2\right)\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := t\_1 \cdot \left(d \cdot \frac{t\_0}{D \cdot \left(h \cdot \left(w \cdot D\right)\right)}\right)\\ \mathbf{if}\;M\_m \leq 6.2 \cdot 10^{-188}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;M\_m \leq 8.2 \cdot 10^{-150}:\\ \;\;\;\;0\\ \mathbf{elif}\;M\_m \leq 7.8 \cdot 10^{-20}:\\ \;\;\;\;\left(t\_1 \cdot d\right) \cdot \frac{t\_0}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}\\ \mathbf{elif}\;M\_m \leq 6.4 \cdot 10^{+38}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (* d (* c0 2.0)))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (* t_1 (* d (/ t_0 (* D (* h (* w D))))))))
   (if (<= M_m 6.2e-188)
     t_2
     (if (<= M_m 8.2e-150)
       0.0
       (if (<= M_m 7.8e-20)
         (* (* t_1 d) (/ t_0 (* D (* D (* w h)))))
         (if (<= M_m 6.4e+38) 0.0 t_2))))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = d * (c0 * 2.0);
	double t_1 = c0 / (2.0 * w);
	double t_2 = t_1 * (d * (t_0 / (D * (h * (w * D)))));
	double tmp;
	if (M_m <= 6.2e-188) {
		tmp = t_2;
	} else if (M_m <= 8.2e-150) {
		tmp = 0.0;
	} else if (M_m <= 7.8e-20) {
		tmp = (t_1 * d) * (t_0 / (D * (D * (w * h))));
	} else if (M_m <= 6.4e+38) {
		tmp = 0.0;
	} else {
		tmp = t_2;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = d_1 * (c0 * 2.0d0)
    t_1 = c0 / (2.0d0 * w)
    t_2 = t_1 * (d_1 * (t_0 / (d * (h * (w * d)))))
    if (m_m <= 6.2d-188) then
        tmp = t_2
    else if (m_m <= 8.2d-150) then
        tmp = 0.0d0
    else if (m_m <= 7.8d-20) then
        tmp = (t_1 * d_1) * (t_0 / (d * (d * (w * h))))
    else if (m_m <= 6.4d+38) then
        tmp = 0.0d0
    else
        tmp = t_2
    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 = d * (c0 * 2.0);
	double t_1 = c0 / (2.0 * w);
	double t_2 = t_1 * (d * (t_0 / (D * (h * (w * D)))));
	double tmp;
	if (M_m <= 6.2e-188) {
		tmp = t_2;
	} else if (M_m <= 8.2e-150) {
		tmp = 0.0;
	} else if (M_m <= 7.8e-20) {
		tmp = (t_1 * d) * (t_0 / (D * (D * (w * h))));
	} else if (M_m <= 6.4e+38) {
		tmp = 0.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	t_0 = d * (c0 * 2.0)
	t_1 = c0 / (2.0 * w)
	t_2 = t_1 * (d * (t_0 / (D * (h * (w * D)))))
	tmp = 0
	if M_m <= 6.2e-188:
		tmp = t_2
	elif M_m <= 8.2e-150:
		tmp = 0.0
	elif M_m <= 7.8e-20:
		tmp = (t_1 * d) * (t_0 / (D * (D * (w * h))))
	elif M_m <= 6.4e+38:
		tmp = 0.0
	else:
		tmp = t_2
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	t_0 = Float64(d * Float64(c0 * 2.0))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(t_1 * Float64(d * Float64(t_0 / Float64(D * Float64(h * Float64(w * D))))))
	tmp = 0.0
	if (M_m <= 6.2e-188)
		tmp = t_2;
	elseif (M_m <= 8.2e-150)
		tmp = 0.0;
	elseif (M_m <= 7.8e-20)
		tmp = Float64(Float64(t_1 * d) * Float64(t_0 / Float64(D * Float64(D * Float64(w * h)))));
	elseif (M_m <= 6.4e+38)
		tmp = 0.0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	t_0 = d * (c0 * 2.0);
	t_1 = c0 / (2.0 * w);
	t_2 = t_1 * (d * (t_0 / (D * (h * (w * D)))));
	tmp = 0.0;
	if (M_m <= 6.2e-188)
		tmp = t_2;
	elseif (M_m <= 8.2e-150)
		tmp = 0.0;
	elseif (M_m <= 7.8e-20)
		tmp = (t_1 * d) * (t_0 / (D * (D * (w * h))));
	elseif (M_m <= 6.4e+38)
		tmp = 0.0;
	else
		tmp = t_2;
	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[(d * N[(c0 * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(d * N[(t$95$0 / N[(D * N[(h * N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M$95$m, 6.2e-188], t$95$2, If[LessEqual[M$95$m, 8.2e-150], 0.0, If[LessEqual[M$95$m, 7.8e-20], N[(N[(t$95$1 * d), $MachinePrecision] * N[(t$95$0 / N[(D * N[(D * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[M$95$m, 6.4e+38], 0.0, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if M < 6.2000000000000004e-188 or 6.3999999999999997e38 < M

   1. Initial program 17.6%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 33.7

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

   5. Simplified 33.7%

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

      1. lift-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   7. Applied egg-rr 46.9%

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

   if 6.2000000000000004e-188 < M < 8.1999999999999997e-150 or 7.80000000000000014e-20 < M < 6.3999999999999997e38

   1. Initial program 21.5%

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

   3. Taylor expanded in c0 around -inf

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. mul-1-neg N/A

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

      7. distribute-lft-neg-in N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 59.2

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

   5. Simplified 59.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mul0-rgt N/A

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

      4. mul0-rgt 59.4

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

   7. Applied egg-rr 59.4%

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

   if 8.1999999999999997e-150 < M < 7.80000000000000014e-20

   1. Initial program 39.2%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 45.0

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

   5. Simplified 45.0%

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

      1. lift-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   7. Applied egg-rr 56.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   9. Applied egg-rr 59.6%

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

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 6.2 \cdot 10^{-188}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(d \cdot \frac{d \cdot \left(c0 \cdot 2\right)}{D \cdot \left(h \cdot \left(w \cdot D\right)\right)}\right)\\ \mathbf{elif}\;M \leq 8.2 \cdot 10^{-150}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 7.8 \cdot 10^{-20}:\\ \;\;\;\;\left(\frac{c0}{2 \cdot w} \cdot d\right) \cdot \frac{d \cdot \left(c0 \cdot 2\right)}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}\\ \mathbf{elif}\;M \leq 6.4 \cdot 10^{+38}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(d \cdot \frac{d \cdot \left(c0 \cdot 2\right)}{D \cdot \left(h \cdot \left(w \cdot D\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 43.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w} \cdot \left(d \cdot \frac{d \cdot \left(c0 \cdot 2\right)}{D \cdot \left(h \cdot \left(w \cdot D\right)\right)}\right)\\ \mathbf{if}\;M\_m \leq 6.2 \cdot 10^{-188}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M\_m \leq 1.15 \cdot 10^{-149}:\\ \;\;\;\;0\\ \mathbf{elif}\;M\_m \leq 1.5 \cdot 10^{-19}:\\ \;\;\;\;\frac{c0 \cdot \left(d \cdot \left(c0 \cdot -2\right)\right)}{\left(w \cdot -2\right) \cdot \left(D \cdot h\right)} \cdot \frac{d}{w \cdot D}\\ \mathbf{elif}\;M\_m \leq 6.4 \cdot 10^{+38}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0
         (* (/ c0 (* 2.0 w)) (* d (/ (* d (* c0 2.0)) (* D (* h (* w D))))))))
   (if (<= M_m 6.2e-188)
     t_0
     (if (<= M_m 1.15e-149)
       0.0
       (if (<= M_m 1.5e-19)
         (* (/ (* c0 (* d (* c0 -2.0))) (* (* w -2.0) (* D h))) (/ d (* w D)))
         (if (<= M_m 6.4e+38) 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 / (2.0 * w)) * (d * ((d * (c0 * 2.0)) / (D * (h * (w * D)))));
	double tmp;
	if (M_m <= 6.2e-188) {
		tmp = t_0;
	} else if (M_m <= 1.15e-149) {
		tmp = 0.0;
	} else if (M_m <= 1.5e-19) {
		tmp = ((c0 * (d * (c0 * -2.0))) / ((w * -2.0) * (D * h))) * (d / (w * D));
	} else if (M_m <= 6.4e+38) {
		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 / (2.0d0 * w)) * (d_1 * ((d_1 * (c0 * 2.0d0)) / (d * (h * (w * d)))))
    if (m_m <= 6.2d-188) then
        tmp = t_0
    else if (m_m <= 1.15d-149) then
        tmp = 0.0d0
    else if (m_m <= 1.5d-19) then
        tmp = ((c0 * (d_1 * (c0 * (-2.0d0)))) / ((w * (-2.0d0)) * (d * h))) * (d_1 / (w * d))
    else if (m_m <= 6.4d+38) 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 / (2.0 * w)) * (d * ((d * (c0 * 2.0)) / (D * (h * (w * D)))));
	double tmp;
	if (M_m <= 6.2e-188) {
		tmp = t_0;
	} else if (M_m <= 1.15e-149) {
		tmp = 0.0;
	} else if (M_m <= 1.5e-19) {
		tmp = ((c0 * (d * (c0 * -2.0))) / ((w * -2.0) * (D * h))) * (d / (w * D));
	} else if (M_m <= 6.4e+38) {
		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 / (2.0 * w)) * (d * ((d * (c0 * 2.0)) / (D * (h * (w * D)))))
	tmp = 0
	if M_m <= 6.2e-188:
		tmp = t_0
	elif M_m <= 1.15e-149:
		tmp = 0.0
	elif M_m <= 1.5e-19:
		tmp = ((c0 * (d * (c0 * -2.0))) / ((w * -2.0) * (D * h))) * (d / (w * D))
	elif M_m <= 6.4e+38:
		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(Float64(c0 / Float64(2.0 * w)) * Float64(d * Float64(Float64(d * Float64(c0 * 2.0)) / Float64(D * Float64(h * Float64(w * D))))))
	tmp = 0.0
	if (M_m <= 6.2e-188)
		tmp = t_0;
	elseif (M_m <= 1.15e-149)
		tmp = 0.0;
	elseif (M_m <= 1.5e-19)
		tmp = Float64(Float64(Float64(c0 * Float64(d * Float64(c0 * -2.0))) / Float64(Float64(w * -2.0) * Float64(D * h))) * Float64(d / Float64(w * D)));
	elseif (M_m <= 6.4e+38)
		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 / (2.0 * w)) * (d * ((d * (c0 * 2.0)) / (D * (h * (w * D)))));
	tmp = 0.0;
	if (M_m <= 6.2e-188)
		tmp = t_0;
	elseif (M_m <= 1.15e-149)
		tmp = 0.0;
	elseif (M_m <= 1.5e-19)
		tmp = ((c0 * (d * (c0 * -2.0))) / ((w * -2.0) * (D * h))) * (d / (w * D));
	elseif (M_m <= 6.4e+38)
		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[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(d * N[(N[(d * N[(c0 * 2.0), $MachinePrecision]), $MachinePrecision] / N[(D * N[(h * N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M$95$m, 6.2e-188], t$95$0, If[LessEqual[M$95$m, 1.15e-149], 0.0, If[LessEqual[M$95$m, 1.5e-19], N[(N[(N[(c0 * N[(d * N[(c0 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(w * -2.0), $MachinePrecision] * N[(D * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[M$95$m, 6.4e+38], 0.0, t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if M < 6.2000000000000004e-188 or 6.3999999999999997e38 < M

   1. Initial program 17.6%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 33.7

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

   5. Simplified 33.7%

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

      1. lift-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   7. Applied egg-rr 46.9%

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

   if 6.2000000000000004e-188 < M < 1.15e-149 or 1.49999999999999996e-19 < M < 6.3999999999999997e38

   1. Initial program 21.5%

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

   3. Taylor expanded in c0 around -inf

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. mul-1-neg N/A

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

      7. distribute-lft-neg-in N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 59.2

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

   5. Simplified 59.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mul0-rgt N/A

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

      4. mul0-rgt 59.4

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

   7. Applied egg-rr 59.4%

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

   if 1.15e-149 < M < 1.49999999999999996e-19

   1. Initial program 39.2%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 45.0

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

   5. Simplified 45.0%

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

      1. lift-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   7. Applied egg-rr 56.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 56.8

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

   9. Applied egg-rr 56.8%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-/.f64 N/A

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

       8. lift-/.f64 N/A

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

       9. associate-*r* N/A

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

       10. lower-*.f64 N/A

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

   11. Applied egg-rr 54.0%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 6.2 \cdot 10^{-188}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(d \cdot \frac{d \cdot \left(c0 \cdot 2\right)}{D \cdot \left(h \cdot \left(w \cdot D\right)\right)}\right)\\ \mathbf{elif}\;M \leq 1.15 \cdot 10^{-149}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 1.5 \cdot 10^{-19}:\\ \;\;\;\;\frac{c0 \cdot \left(d \cdot \left(c0 \cdot -2\right)\right)}{\left(w \cdot -2\right) \cdot \left(D \cdot h\right)} \cdot \frac{d}{w \cdot D}\\ \mathbf{elif}\;M \leq 6.4 \cdot 10^{+38}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(d \cdot \frac{d \cdot \left(c0 \cdot 2\right)}{D \cdot \left(h \cdot \left(w \cdot D\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 40.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} t_0 := D \cdot \left(h \cdot \left(w \cdot D\right)\right)\\ t_1 := d \cdot \left(c0 \cdot d\right)\\ t_2 := \frac{t\_1}{t\_0} \cdot \frac{c0}{w}\\ \mathbf{if}\;M\_m \leq 6.2 \cdot 10^{-188}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;M\_m \leq 3.05 \cdot 10^{-151}:\\ \;\;\;\;0\\ \mathbf{elif}\;M\_m \leq 1.5 \cdot 10^{-19}:\\ \;\;\;\;t\_1 \cdot \frac{c0}{w \cdot t\_0}\\ \mathbf{elif}\;M\_m \leq 9 \cdot 10^{+44}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (* D (* h (* w D))))
        (t_1 (* d (* c0 d)))
        (t_2 (* (/ t_1 t_0) (/ c0 w))))
   (if (<= M_m 6.2e-188)
     t_2
     (if (<= M_m 3.05e-151)
       0.0
       (if (<= M_m 1.5e-19)
         (* t_1 (/ c0 (* w t_0)))
         (if (<= M_m 9e+44) 0.0 t_2))))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = D * (h * (w * D));
	double t_1 = d * (c0 * d);
	double t_2 = (t_1 / t_0) * (c0 / w);
	double tmp;
	if (M_m <= 6.2e-188) {
		tmp = t_2;
	} else if (M_m <= 3.05e-151) {
		tmp = 0.0;
	} else if (M_m <= 1.5e-19) {
		tmp = t_1 * (c0 / (w * t_0));
	} else if (M_m <= 9e+44) {
		tmp = 0.0;
	} else {
		tmp = t_2;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = d * (h * (w * d))
    t_1 = d_1 * (c0 * d_1)
    t_2 = (t_1 / t_0) * (c0 / w)
    if (m_m <= 6.2d-188) then
        tmp = t_2
    else if (m_m <= 3.05d-151) then
        tmp = 0.0d0
    else if (m_m <= 1.5d-19) then
        tmp = t_1 * (c0 / (w * t_0))
    else if (m_m <= 9d+44) then
        tmp = 0.0d0
    else
        tmp = t_2
    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 = D * (h * (w * D));
	double t_1 = d * (c0 * d);
	double t_2 = (t_1 / t_0) * (c0 / w);
	double tmp;
	if (M_m <= 6.2e-188) {
		tmp = t_2;
	} else if (M_m <= 3.05e-151) {
		tmp = 0.0;
	} else if (M_m <= 1.5e-19) {
		tmp = t_1 * (c0 / (w * t_0));
	} else if (M_m <= 9e+44) {
		tmp = 0.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	t_0 = D * (h * (w * D))
	t_1 = d * (c0 * d)
	t_2 = (t_1 / t_0) * (c0 / w)
	tmp = 0
	if M_m <= 6.2e-188:
		tmp = t_2
	elif M_m <= 3.05e-151:
		tmp = 0.0
	elif M_m <= 1.5e-19:
		tmp = t_1 * (c0 / (w * t_0))
	elif M_m <= 9e+44:
		tmp = 0.0
	else:
		tmp = t_2
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	t_0 = Float64(D * Float64(h * Float64(w * D)))
	t_1 = Float64(d * Float64(c0 * d))
	t_2 = Float64(Float64(t_1 / t_0) * Float64(c0 / w))
	tmp = 0.0
	if (M_m <= 6.2e-188)
		tmp = t_2;
	elseif (M_m <= 3.05e-151)
		tmp = 0.0;
	elseif (M_m <= 1.5e-19)
		tmp = Float64(t_1 * Float64(c0 / Float64(w * t_0)));
	elseif (M_m <= 9e+44)
		tmp = 0.0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	t_0 = D * (h * (w * D));
	t_1 = d * (c0 * d);
	t_2 = (t_1 / t_0) * (c0 / w);
	tmp = 0.0;
	if (M_m <= 6.2e-188)
		tmp = t_2;
	elseif (M_m <= 3.05e-151)
		tmp = 0.0;
	elseif (M_m <= 1.5e-19)
		tmp = t_1 * (c0 / (w * t_0));
	elseif (M_m <= 9e+44)
		tmp = 0.0;
	else
		tmp = t_2;
	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[(D * N[(h * N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(d * N[(c0 * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 / t$95$0), $MachinePrecision] * N[(c0 / w), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M$95$m, 6.2e-188], t$95$2, If[LessEqual[M$95$m, 3.05e-151], 0.0, If[LessEqual[M$95$m, 1.5e-19], N[(t$95$1 * N[(c0 / N[(w * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[M$95$m, 9e+44], 0.0, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if M < 6.2000000000000004e-188 or 9e44 < M

   1. Initial program 17.7%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 33.9

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

   5. Simplified 33.9%

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

      1. lift-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   7. Applied egg-rr 47.1%

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

   8. Taylor expanded in c0 around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 30.1

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

   10. Simplified 30.1%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. times-frac N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

   12. Applied egg-rr 45.1%

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

   if 6.2000000000000004e-188 < M < 3.05e-151 or 1.49999999999999996e-19 < M < 9e44

   1. Initial program 20.6%

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

   3. Taylor expanded in c0 around -inf

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. mul-1-neg N/A

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

      7. distribute-lft-neg-in N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 56.9

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

   5. Simplified 56.9%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mul0-rgt N/A

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

      4. mul0-rgt 57.1

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

   7. Applied egg-rr 57.1%

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

   if 3.05e-151 < M < 1.49999999999999996e-19

   1. Initial program 39.2%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 45.0

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

   5. Simplified 45.0%

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

      1. lift-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   7. Applied egg-rr 56.7%

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

   8. Taylor expanded in c0 around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 44.9

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

   10. Simplified 44.9%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. associate-/l* N/A

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

       9. lower-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. associate-*r* N/A

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

       13. *-commutative N/A

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

       14. lower-*.f64 N/A

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

       15. *-commutative N/A

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

       16. lower-*.f64 N/A

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

       17. lower-/.f64 48.2

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

       18. lift-*.f64 N/A

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

       19. *-commutative N/A

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

       20. lift-*.f64 N/A

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

       21. lift-*.f64 N/A

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

       22. associate-*l* N/A

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

       23. lift-*.f64 N/A

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

   12. Applied egg-rr 54.0%

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

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 6.2 \cdot 10^{-188}:\\ \;\;\;\;\frac{d \cdot \left(c0 \cdot d\right)}{D \cdot \left(h \cdot \left(w \cdot D\right)\right)} \cdot \frac{c0}{w}\\ \mathbf{elif}\;M \leq 3.05 \cdot 10^{-151}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 1.5 \cdot 10^{-19}:\\ \;\;\;\;\left(d \cdot \left(c0 \cdot d\right)\right) \cdot \frac{c0}{w \cdot \left(D \cdot \left(h \cdot \left(w \cdot D\right)\right)\right)}\\ \mathbf{elif}\;M \leq 9 \cdot 10^{+44}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \left(c0 \cdot d\right)}{D \cdot \left(h \cdot \left(w \cdot D\right)\right)} \cdot \frac{c0}{w}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 46.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ \mathbf{if}\;M\_m \leq 6.2 \cdot 10^{-188}:\\ \;\;\;\;t\_0 \cdot \left(d \cdot \left(\frac{2 \cdot d}{D \cdot \left(w \cdot h\right)} \cdot \frac{c0}{D}\right)\right)\\ \mathbf{elif}\;M\_m \leq 7.6 \cdot 10^{-150}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \left(c0 \cdot 2\right)}{D \cdot \left(h \cdot \left(w \cdot D\right)\right)} \cdot \left(t\_0 \cdot d\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w))))
   (if (<= M_m 6.2e-188)
     (* t_0 (* d (* (/ (* 2.0 d) (* D (* w h))) (/ c0 D))))
     (if (<= M_m 7.6e-150)
       0.0
       (* (/ (* d (* c0 2.0)) (* D (* h (* w D)))) (* t_0 d))))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = c0 / (2.0 * w);
	double tmp;
	if (M_m <= 6.2e-188) {
		tmp = t_0 * (d * (((2.0 * d) / (D * (w * h))) * (c0 / D)));
	} else if (M_m <= 7.6e-150) {
		tmp = 0.0;
	} else {
		tmp = ((d * (c0 * 2.0)) / (D * (h * (w * D)))) * (t_0 * d);
	}
	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 / (2.0d0 * w)
    if (m_m <= 6.2d-188) then
        tmp = t_0 * (d_1 * (((2.0d0 * d_1) / (d * (w * h))) * (c0 / d)))
    else if (m_m <= 7.6d-150) then
        tmp = 0.0d0
    else
        tmp = ((d_1 * (c0 * 2.0d0)) / (d * (h * (w * d)))) * (t_0 * d_1)
    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 / (2.0 * w);
	double tmp;
	if (M_m <= 6.2e-188) {
		tmp = t_0 * (d * (((2.0 * d) / (D * (w * h))) * (c0 / D)));
	} else if (M_m <= 7.6e-150) {
		tmp = 0.0;
	} else {
		tmp = ((d * (c0 * 2.0)) / (D * (h * (w * D)))) * (t_0 * d);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	t_0 = c0 / (2.0 * w)
	tmp = 0
	if M_m <= 6.2e-188:
		tmp = t_0 * (d * (((2.0 * d) / (D * (w * h))) * (c0 / D)))
	elif M_m <= 7.6e-150:
		tmp = 0.0
	else:
		tmp = ((d * (c0 * 2.0)) / (D * (h * (w * D)))) * (t_0 * d)
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	t_0 = Float64(c0 / Float64(2.0 * w))
	tmp = 0.0
	if (M_m <= 6.2e-188)
		tmp = Float64(t_0 * Float64(d * Float64(Float64(Float64(2.0 * d) / Float64(D * Float64(w * h))) * Float64(c0 / D))));
	elseif (M_m <= 7.6e-150)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(d * Float64(c0 * 2.0)) / Float64(D * Float64(h * Float64(w * D)))) * Float64(t_0 * d));
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	t_0 = c0 / (2.0 * w);
	tmp = 0.0;
	if (M_m <= 6.2e-188)
		tmp = t_0 * (d * (((2.0 * d) / (D * (w * h))) * (c0 / D)));
	elseif (M_m <= 7.6e-150)
		tmp = 0.0;
	else
		tmp = ((d * (c0 * 2.0)) / (D * (h * (w * D)))) * (t_0 * d);
	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[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M$95$m, 6.2e-188], N[(t$95$0 * N[(d * N[(N[(N[(2.0 * d), $MachinePrecision] / N[(D * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c0 / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[M$95$m, 7.6e-150], 0.0, N[(N[(N[(d * N[(c0 * 2.0), $MachinePrecision]), $MachinePrecision] / N[(D * N[(h * N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * d), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if M < 6.2000000000000004e-188

   1. Initial program 19.4%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 27.3

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

   5. Simplified 27.3%

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

      1. lift-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   7. Applied egg-rr 41.7%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 44.4

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

   9. Applied egg-rr 44.4%

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

   if 6.2000000000000004e-188 < M < 7.5999999999999997e-150

   1. Initial program 8.9%

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

   3. Taylor expanded in c0 around -inf

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. mul-1-neg N/A

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

      7. distribute-lft-neg-in N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 70.2

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

   5. Simplified 70.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mul0-rgt N/A

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

      4. mul0-rgt 70.2

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

   7. Applied egg-rr 70.2%

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

   if 7.5999999999999997e-150 < M

   1. Initial program 24.4%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 47.0

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

   5. Simplified 47.0%

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

      1. lift-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   7. Applied egg-rr 57.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 57.3

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

   9. Applied egg-rr 57.3%

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

   11. Applied egg-rr 58.6%

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

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 6.2 \cdot 10^{-188}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(d \cdot \left(\frac{2 \cdot d}{D \cdot \left(w \cdot h\right)} \cdot \frac{c0}{D}\right)\right)\\ \mathbf{elif}\;M \leq 7.6 \cdot 10^{-150}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \left(c0 \cdot 2\right)}{D \cdot \left(h \cdot \left(w \cdot D\right)\right)} \cdot \left(\frac{c0}{2 \cdot w} \cdot d\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 45.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} t_0 := \frac{d \cdot \left(c0 \cdot 2\right)}{D \cdot \left(h \cdot \left(w \cdot D\right)\right)}\\ t_1 := \frac{c0}{2 \cdot w}\\ \mathbf{if}\;M\_m \leq 6.2 \cdot 10^{-188}:\\ \;\;\;\;t\_1 \cdot \left(d \cdot t\_0\right)\\ \mathbf{elif}\;M\_m \leq 7.6 \cdot 10^{-150}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 \cdot d\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (/ (* d (* c0 2.0)) (* D (* h (* w D))))) (t_1 (/ c0 (* 2.0 w))))
   (if (<= M_m 6.2e-188)
     (* t_1 (* d t_0))
     (if (<= M_m 7.6e-150) 0.0 (* t_0 (* t_1 d))))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = (d * (c0 * 2.0)) / (D * (h * (w * D)));
	double t_1 = c0 / (2.0 * w);
	double tmp;
	if (M_m <= 6.2e-188) {
		tmp = t_1 * (d * t_0);
	} else if (M_m <= 7.6e-150) {
		tmp = 0.0;
	} else {
		tmp = t_0 * (t_1 * d);
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (d_1 * (c0 * 2.0d0)) / (d * (h * (w * d)))
    t_1 = c0 / (2.0d0 * w)
    if (m_m <= 6.2d-188) then
        tmp = t_1 * (d_1 * t_0)
    else if (m_m <= 7.6d-150) then
        tmp = 0.0d0
    else
        tmp = t_0 * (t_1 * d_1)
    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 = (d * (c0 * 2.0)) / (D * (h * (w * D)));
	double t_1 = c0 / (2.0 * w);
	double tmp;
	if (M_m <= 6.2e-188) {
		tmp = t_1 * (d * t_0);
	} else if (M_m <= 7.6e-150) {
		tmp = 0.0;
	} else {
		tmp = t_0 * (t_1 * d);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	t_0 = (d * (c0 * 2.0)) / (D * (h * (w * D)))
	t_1 = c0 / (2.0 * w)
	tmp = 0
	if M_m <= 6.2e-188:
		tmp = t_1 * (d * t_0)
	elif M_m <= 7.6e-150:
		tmp = 0.0
	else:
		tmp = t_0 * (t_1 * d)
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	t_0 = Float64(Float64(d * Float64(c0 * 2.0)) / Float64(D * Float64(h * Float64(w * D))))
	t_1 = Float64(c0 / Float64(2.0 * w))
	tmp = 0.0
	if (M_m <= 6.2e-188)
		tmp = Float64(t_1 * Float64(d * t_0));
	elseif (M_m <= 7.6e-150)
		tmp = 0.0;
	else
		tmp = Float64(t_0 * Float64(t_1 * d));
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	t_0 = (d * (c0 * 2.0)) / (D * (h * (w * D)));
	t_1 = c0 / (2.0 * w);
	tmp = 0.0;
	if (M_m <= 6.2e-188)
		tmp = t_1 * (d * t_0);
	elseif (M_m <= 7.6e-150)
		tmp = 0.0;
	else
		tmp = t_0 * (t_1 * d);
	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[(d * N[(c0 * 2.0), $MachinePrecision]), $MachinePrecision] / N[(D * N[(h * N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M$95$m, 6.2e-188], N[(t$95$1 * N[(d * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[M$95$m, 7.6e-150], 0.0, N[(t$95$0 * N[(t$95$1 * d), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if M < 6.2000000000000004e-188

   1. Initial program 19.4%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 27.3

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

   5. Simplified 27.3%

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

      1. lift-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   7. Applied egg-rr 41.7%

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

   if 6.2000000000000004e-188 < M < 7.5999999999999997e-150

   1. Initial program 8.9%

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

   3. Taylor expanded in c0 around -inf

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. mul-1-neg N/A

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

      7. distribute-lft-neg-in N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 70.2

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

   5. Simplified 70.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mul0-rgt N/A

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

      4. mul0-rgt 70.2

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

   7. Applied egg-rr 70.2%

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

   if 7.5999999999999997e-150 < M

   1. Initial program 24.4%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 47.0

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

   5. Simplified 47.0%

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

      1. lift-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   7. Applied egg-rr 57.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot \left(2 \cdot c0\right)}{\color{blue}{D \cdot h}} \cdot \frac{d}{D \cdot w}\right) \]

      16. lower-/.f64 57.3

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot \left(2 \cdot c0\right)}{D \cdot h} \cdot \color{blue}{\frac{d}{D \cdot w}}\right) \]

   9. Applied egg-rr 57.3%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\frac{d \cdot \left(2 \cdot c0\right)}{D \cdot h} \cdot \frac{d}{D \cdot w}\right)} \]

   11. Applied egg-rr 58.6%

       \[\leadsto \color{blue}{\frac{d \cdot \left(2 \cdot c0\right)}{D \cdot \left(h \cdot \left(D \cdot w\right)\right)} \cdot \left(d \cdot \frac{c0}{2 \cdot w}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 6.2 \cdot 10^{-188}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(d \cdot \frac{d \cdot \left(c0 \cdot 2\right)}{D \cdot \left(h \cdot \left(w \cdot D\right)\right)}\right)\\ \mathbf{elif}\;M \leq 7.6 \cdot 10^{-150}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \left(c0 \cdot 2\right)}{D \cdot \left(h \cdot \left(w \cdot D\right)\right)} \cdot \left(\frac{c0}{2 \cdot w} \cdot d\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 44.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w} \cdot \left(d \cdot \frac{d \cdot \left(c0 \cdot 2\right)}{D \cdot \left(h \cdot \left(w \cdot D\right)\right)}\right)\\ \mathbf{if}\;M\_m \leq 6.2 \cdot 10^{-188}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M\_m \leq 7.6 \cdot 10^{-150}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0
         (* (/ c0 (* 2.0 w)) (* d (/ (* d (* c0 2.0)) (* D (* h (* w D))))))))
   (if (<= M_m 6.2e-188) t_0 (if (<= M_m 7.6e-150) 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 / (2.0 * w)) * (d * ((d * (c0 * 2.0)) / (D * (h * (w * D)))));
	double tmp;
	if (M_m <= 6.2e-188) {
		tmp = t_0;
	} else if (M_m <= 7.6e-150) {
		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 / (2.0d0 * w)) * (d_1 * ((d_1 * (c0 * 2.0d0)) / (d * (h * (w * d)))))
    if (m_m <= 6.2d-188) then
        tmp = t_0
    else if (m_m <= 7.6d-150) 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 / (2.0 * w)) * (d * ((d * (c0 * 2.0)) / (D * (h * (w * D)))));
	double tmp;
	if (M_m <= 6.2e-188) {
		tmp = t_0;
	} else if (M_m <= 7.6e-150) {
		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 / (2.0 * w)) * (d * ((d * (c0 * 2.0)) / (D * (h * (w * D)))))
	tmp = 0
	if M_m <= 6.2e-188:
		tmp = t_0
	elif M_m <= 7.6e-150:
		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(Float64(c0 / Float64(2.0 * w)) * Float64(d * Float64(Float64(d * Float64(c0 * 2.0)) / Float64(D * Float64(h * Float64(w * D))))))
	tmp = 0.0
	if (M_m <= 6.2e-188)
		tmp = t_0;
	elseif (M_m <= 7.6e-150)
		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 / (2.0 * w)) * (d * ((d * (c0 * 2.0)) / (D * (h * (w * D)))));
	tmp = 0.0;
	if (M_m <= 6.2e-188)
		tmp = t_0;
	elseif (M_m <= 7.6e-150)
		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[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(d * N[(N[(d * N[(c0 * 2.0), $MachinePrecision]), $MachinePrecision] / N[(D * N[(h * N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M$95$m, 6.2e-188], t$95$0, If[LessEqual[M$95$m, 7.6e-150], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if M < 6.2000000000000004e-188 or 7.5999999999999997e-150 < M

   1. Initial program 21.4%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot \left(h \cdot w\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot \left(h \cdot w\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\color{blue}{2 \cdot \left(c0 \cdot {d}^{2}\right)}}{{D}^{2} \cdot \left(h \cdot w\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \color{blue}{\left(c0 \cdot {d}^{2}\right)}}{{D}^{2} \cdot \left(h \cdot w\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \color{blue}{\left(d \cdot d\right)}\right)}{{D}^{2} \cdot \left(h \cdot w\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \color{blue}{\left(d \cdot d\right)}\right)}{{D}^{2} \cdot \left(h \cdot w\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{\left(h \cdot w\right) \cdot {D}^{2}}} \]

      8. associate-*l* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{h \cdot \left(w \cdot {D}^{2}\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h \cdot \color{blue}{\left({D}^{2} \cdot w\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{h \cdot \left({D}^{2} \cdot w\right)}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h \cdot \color{blue}{\left({D}^{2} \cdot w\right)}} \]

      12. unpow2 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot w\right)} \]

      13. lower-*.f64 35.4

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot w\right)} \]

   5. Simplified 35.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h \cdot \left(\left(D \cdot D\right) \cdot w\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \color{blue}{\left(d \cdot d\right)}\right)}{h \cdot \left(\left(D \cdot D\right) \cdot w\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\color{blue}{\left(2 \cdot c0\right) \cdot \left(d \cdot d\right)}}{h \cdot \left(\left(D \cdot D\right) \cdot w\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\color{blue}{\left(c0 \cdot 2\right)} \cdot \left(d \cdot d\right)}{h \cdot \left(\left(D \cdot D\right) \cdot w\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\color{blue}{\left(c0 \cdot 2\right)} \cdot \left(d \cdot d\right)}{h \cdot \left(\left(D \cdot D\right) \cdot w\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\color{blue}{\left(d \cdot d\right) \cdot \left(c0 \cdot 2\right)}}{h \cdot \left(\left(D \cdot D\right) \cdot w\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\left(d \cdot d\right) \cdot \left(c0 \cdot 2\right)}{h \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot w\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\left(d \cdot d\right) \cdot \left(c0 \cdot 2\right)}{h \cdot \color{blue}{\left(\left(D \cdot D\right) \cdot w\right)}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\left(d \cdot d\right) \cdot \left(c0 \cdot 2\right)}{\color{blue}{h \cdot \left(\left(D \cdot D\right) \cdot w\right)}} \]

      9. associate-*r/ N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\left(d \cdot d\right) \cdot \frac{c0 \cdot 2}{h \cdot \left(\left(D \cdot D\right) \cdot w\right)}\right)} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \color{blue}{\frac{c0 \cdot 2}{h \cdot \left(\left(D \cdot D\right) \cdot w\right)}}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(d \cdot d\right)} \cdot \frac{c0 \cdot 2}{h \cdot \left(\left(D \cdot D\right) \cdot w\right)}\right) \]

      12. associate-*l* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(d \cdot \left(d \cdot \frac{c0 \cdot 2}{h \cdot \left(\left(D \cdot D\right) \cdot w\right)}\right)\right)} \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\left(d \cdot \frac{c0 \cdot 2}{h \cdot \left(\left(D \cdot D\right) \cdot w\right)}\right) \cdot d\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\left(d \cdot \frac{c0 \cdot 2}{h \cdot \left(\left(D \cdot D\right) \cdot w\right)}\right) \cdot d\right)} \]

   7. Applied egg-rr 48.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\frac{d \cdot \left(2 \cdot c0\right)}{D \cdot \left(h \cdot \left(D \cdot w\right)\right)} \cdot d\right)} \]

   if 6.2000000000000004e-188 < M < 7.5999999999999997e-150

   1. Initial program 8.9%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around -inf

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\left(-1 \cdot c0\right) \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)} \]

      2. distribute-lft1-in N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot c0\right) \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right) \]

      3. metadata-eval N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot c0\right) \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \]

      4. mul0-lft N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot c0\right) \cdot \color{blue}{0}\right) \]

      5. metadata-eval N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot c0\right) \cdot \color{blue}{\left(-1 + 1\right)}\right) \]

      6. mul-1-neg N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(c0\right)\right)} \cdot \left(-1 + 1\right)\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\mathsf{neg}\left(c0 \cdot \left(-1 + 1\right)\right)\right)} \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot \left(\mathsf{neg}\left(\left(-1 + 1\right)\right)\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \left(\mathsf{neg}\left(\color{blue}{0}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{0}\right) \]

      11. metadata-eval N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{\left(-1 + 1\right)}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot \left(-1 + 1\right)\right)} \]

      13. metadata-eval 70.2

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{0}\right) \]

   5. Simplified 70.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{c0}{\color{blue}{2 \cdot w}} \cdot \left(c0 \cdot 0\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{c0}{2 \cdot w}} \cdot \left(c0 \cdot 0\right) \]

      3. mul0-rgt N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{0} \]

      4. mul0-rgt 70.2

         \[\leadsto \color{blue}{0} \]

   7. Applied egg-rr 70.2%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 6.2 \cdot 10^{-188}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(d \cdot \frac{d \cdot \left(c0 \cdot 2\right)}{D \cdot \left(h \cdot \left(w \cdot D\right)\right)}\right)\\ \mathbf{elif}\;M \leq 7.6 \cdot 10^{-150}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(d \cdot \frac{d \cdot \left(c0 \cdot 2\right)}{D \cdot \left(h \cdot \left(w \cdot D\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 41.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} t_0 := \left(d \cdot \left(c0 \cdot d\right)\right) \cdot \frac{c0}{w \cdot \left(D \cdot \left(h \cdot \left(w \cdot D\right)\right)\right)}\\ \mathbf{if}\;M\_m \leq 6.2 \cdot 10^{-188}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M\_m \leq 3.05 \cdot 10^{-151}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (* (* d (* c0 d)) (/ c0 (* w (* D (* h (* w D))))))))
   (if (<= M_m 6.2e-188) t_0 (if (<= M_m 3.05e-151) 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 = (d * (c0 * d)) * (c0 / (w * (D * (h * (w * D)))));
	double tmp;
	if (M_m <= 6.2e-188) {
		tmp = t_0;
	} else if (M_m <= 3.05e-151) {
		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 = (d_1 * (c0 * d_1)) * (c0 / (w * (d * (h * (w * d)))))
    if (m_m <= 6.2d-188) then
        tmp = t_0
    else if (m_m <= 3.05d-151) 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 = (d * (c0 * d)) * (c0 / (w * (D * (h * (w * D)))));
	double tmp;
	if (M_m <= 6.2e-188) {
		tmp = t_0;
	} else if (M_m <= 3.05e-151) {
		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 = (d * (c0 * d)) * (c0 / (w * (D * (h * (w * D)))))
	tmp = 0
	if M_m <= 6.2e-188:
		tmp = t_0
	elif M_m <= 3.05e-151:
		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(Float64(d * Float64(c0 * d)) * Float64(c0 / Float64(w * Float64(D * Float64(h * Float64(w * D))))))
	tmp = 0.0
	if (M_m <= 6.2e-188)
		tmp = t_0;
	elseif (M_m <= 3.05e-151)
		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 = (d * (c0 * d)) * (c0 / (w * (D * (h * (w * D)))));
	tmp = 0.0;
	if (M_m <= 6.2e-188)
		tmp = t_0;
	elseif (M_m <= 3.05e-151)
		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[(N[(d * N[(c0 * d), $MachinePrecision]), $MachinePrecision] * N[(c0 / N[(w * N[(D * N[(h * N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M$95$m, 6.2e-188], t$95$0, If[LessEqual[M$95$m, 3.05e-151], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if M < 6.2000000000000004e-188 or 3.05e-151 < M

   1. Initial program 21.4%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot \left(h \cdot w\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot \left(h \cdot w\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\color{blue}{2 \cdot \left(c0 \cdot {d}^{2}\right)}}{{D}^{2} \cdot \left(h \cdot w\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \color{blue}{\left(c0 \cdot {d}^{2}\right)}}{{D}^{2} \cdot \left(h \cdot w\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \color{blue}{\left(d \cdot d\right)}\right)}{{D}^{2} \cdot \left(h \cdot w\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \color{blue}{\left(d \cdot d\right)}\right)}{{D}^{2} \cdot \left(h \cdot w\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{\left(h \cdot w\right) \cdot {D}^{2}}} \]

      8. associate-*l* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{h \cdot \left(w \cdot {D}^{2}\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h \cdot \color{blue}{\left({D}^{2} \cdot w\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{h \cdot \left({D}^{2} \cdot w\right)}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h \cdot \color{blue}{\left({D}^{2} \cdot w\right)}} \]

      12. unpow2 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot w\right)} \]

      13. lower-*.f64 35.4

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot w\right)} \]

   5. Simplified 35.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h \cdot \left(\left(D \cdot D\right) \cdot w\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \color{blue}{\left(d \cdot d\right)}\right)}{h \cdot \left(\left(D \cdot D\right) \cdot w\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\color{blue}{\left(2 \cdot c0\right) \cdot \left(d \cdot d\right)}}{h \cdot \left(\left(D \cdot D\right) \cdot w\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\color{blue}{\left(c0 \cdot 2\right)} \cdot \left(d \cdot d\right)}{h \cdot \left(\left(D \cdot D\right) \cdot w\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\color{blue}{\left(c0 \cdot 2\right)} \cdot \left(d \cdot d\right)}{h \cdot \left(\left(D \cdot D\right) \cdot w\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\color{blue}{\left(d \cdot d\right) \cdot \left(c0 \cdot 2\right)}}{h \cdot \left(\left(D \cdot D\right) \cdot w\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\left(d \cdot d\right) \cdot \left(c0 \cdot 2\right)}{h \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot w\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\left(d \cdot d\right) \cdot \left(c0 \cdot 2\right)}{h \cdot \color{blue}{\left(\left(D \cdot D\right) \cdot w\right)}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\left(d \cdot d\right) \cdot \left(c0 \cdot 2\right)}{\color{blue}{h \cdot \left(\left(D \cdot D\right) \cdot w\right)}} \]

      9. associate-*r/ N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\left(d \cdot d\right) \cdot \frac{c0 \cdot 2}{h \cdot \left(\left(D \cdot D\right) \cdot w\right)}\right)} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \color{blue}{\frac{c0 \cdot 2}{h \cdot \left(\left(D \cdot D\right) \cdot w\right)}}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(d \cdot d\right)} \cdot \frac{c0 \cdot 2}{h \cdot \left(\left(D \cdot D\right) \cdot w\right)}\right) \]

      12. associate-*l* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(d \cdot \left(d \cdot \frac{c0 \cdot 2}{h \cdot \left(\left(D \cdot D\right) \cdot w\right)}\right)\right)} \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\left(d \cdot \frac{c0 \cdot 2}{h \cdot \left(\left(D \cdot D\right) \cdot w\right)}\right) \cdot d\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\left(d \cdot \frac{c0 \cdot 2}{h \cdot \left(\left(D \cdot D\right) \cdot w\right)}\right) \cdot d\right)} \]

   7. Applied egg-rr 48.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\frac{d \cdot \left(2 \cdot c0\right)}{D \cdot \left(h \cdot \left(D \cdot w\right)\right)} \cdot d\right)} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\left(c0 \cdot c0\right)} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{c0 \cdot \left(c0 \cdot {d}^{2}\right)}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{c0 \cdot \left(c0 \cdot {d}^{2}\right)}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{c0 \cdot \color{blue}{\left(c0 \cdot {d}^{2}\right)}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \]

      6. unpow2 N/A

         \[\leadsto \frac{c0 \cdot \left(c0 \cdot \color{blue}{\left(d \cdot d\right)}\right)}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{c0 \cdot \left(c0 \cdot \color{blue}{\left(d \cdot d\right)}\right)}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \]

      8. unpow2 N/A

         \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{{D}^{2} \cdot \left(h \cdot \color{blue}{\left(w \cdot w\right)}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{{D}^{2} \cdot \color{blue}{\left(\left(h \cdot w\right) \cdot w\right)}} \]

      10. associate-*r* N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{\left({D}^{2} \cdot \left(h \cdot w\right)\right) \cdot w}} \]

      11. *-commutative N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{w \cdot \left({D}^{2} \cdot \left(h \cdot w\right)\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{w \cdot \left({D}^{2} \cdot \left(h \cdot w\right)\right)}} \]

      13. associate-*r* N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \color{blue}{\left(\left({D}^{2} \cdot h\right) \cdot w\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \color{blue}{\left(w \cdot \left({D}^{2} \cdot h\right)\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \color{blue}{\left(w \cdot \left({D}^{2} \cdot h\right)\right)}} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \left(w \cdot \color{blue}{\left({D}^{2} \cdot h\right)}\right)} \]

      17. unpow2 N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \left(w \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot h\right)\right)} \]

      18. lower-*.f64 31.9

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \left(w \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot h\right)\right)} \]

   10. Simplified 31.9%

       \[\leadsto \color{blue}{\frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \left(w \cdot \left(\left(D \cdot D\right) \cdot h\right)\right)}} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \color{blue}{\left(d \cdot d\right)}\right)}{w \cdot \left(w \cdot \left(\left(D \cdot D\right) \cdot h\right)\right)} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{c0 \cdot \color{blue}{\left(c0 \cdot \left(d \cdot d\right)\right)}}{w \cdot \left(w \cdot \left(\left(D \cdot D\right) \cdot h\right)\right)} \]

       3. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(c0 \cdot \left(d \cdot d\right)\right) \cdot c0}}{w \cdot \left(w \cdot \left(\left(D \cdot D\right) \cdot h\right)\right)} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{\left(c0 \cdot \left(d \cdot d\right)\right) \cdot c0}{w \cdot \left(w \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot h\right)\right)} \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{\left(c0 \cdot \left(d \cdot d\right)\right) \cdot c0}{w \cdot \left(w \cdot \color{blue}{\left(\left(D \cdot D\right) \cdot h\right)}\right)} \]

       6. lift-*.f64 N/A

          \[\leadsto \frac{\left(c0 \cdot \left(d \cdot d\right)\right) \cdot c0}{w \cdot \color{blue}{\left(w \cdot \left(\left(D \cdot D\right) \cdot h\right)\right)}} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{\left(c0 \cdot \left(d \cdot d\right)\right) \cdot c0}{\color{blue}{w \cdot \left(w \cdot \left(\left(D \cdot D\right) \cdot h\right)\right)}} \]

       8. associate-/l* N/A

          \[\leadsto \color{blue}{\left(c0 \cdot \left(d \cdot d\right)\right) \cdot \frac{c0}{w \cdot \left(w \cdot \left(\left(D \cdot D\right) \cdot h\right)\right)}} \]

       9. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(c0 \cdot \left(d \cdot d\right)\right) \cdot \frac{c0}{w \cdot \left(w \cdot \left(\left(D \cdot D\right) \cdot h\right)\right)}} \]

       10. lift-*.f64 N/A

           \[\leadsto \color{blue}{\left(c0 \cdot \left(d \cdot d\right)\right)} \cdot \frac{c0}{w \cdot \left(w \cdot \left(\left(D \cdot D\right) \cdot h\right)\right)} \]

       11. lift-*.f64 N/A

           \[\leadsto \left(c0 \cdot \color{blue}{\left(d \cdot d\right)}\right) \cdot \frac{c0}{w \cdot \left(w \cdot \left(\left(D \cdot D\right) \cdot h\right)\right)} \]

       12. associate-*r* N/A

           \[\leadsto \color{blue}{\left(\left(c0 \cdot d\right) \cdot d\right)} \cdot \frac{c0}{w \cdot \left(w \cdot \left(\left(D \cdot D\right) \cdot h\right)\right)} \]

       13. *-commutative N/A

           \[\leadsto \color{blue}{\left(d \cdot \left(c0 \cdot d\right)\right)} \cdot \frac{c0}{w \cdot \left(w \cdot \left(\left(D \cdot D\right) \cdot h\right)\right)} \]

       14. lower-*.f64 N/A

           \[\leadsto \color{blue}{\left(d \cdot \left(c0 \cdot d\right)\right)} \cdot \frac{c0}{w \cdot \left(w \cdot \left(\left(D \cdot D\right) \cdot h\right)\right)} \]

       15. *-commutative N/A

           \[\leadsto \left(d \cdot \color{blue}{\left(d \cdot c0\right)}\right) \cdot \frac{c0}{w \cdot \left(w \cdot \left(\left(D \cdot D\right) \cdot h\right)\right)} \]

       16. lower-*.f64 N/A

           \[\leadsto \left(d \cdot \color{blue}{\left(d \cdot c0\right)}\right) \cdot \frac{c0}{w \cdot \left(w \cdot \left(\left(D \cdot D\right) \cdot h\right)\right)} \]

       17. lower-/.f64 37.9

           \[\leadsto \left(d \cdot \left(d \cdot c0\right)\right) \cdot \color{blue}{\frac{c0}{w \cdot \left(w \cdot \left(\left(D \cdot D\right) \cdot h\right)\right)}} \]

       18. lift-*.f64 N/A

           \[\leadsto \left(d \cdot \left(d \cdot c0\right)\right) \cdot \frac{c0}{w \cdot \color{blue}{\left(w \cdot \left(\left(D \cdot D\right) \cdot h\right)\right)}} \]

       19. *-commutative N/A

           \[\leadsto \left(d \cdot \left(d \cdot c0\right)\right) \cdot \frac{c0}{w \cdot \color{blue}{\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot w\right)}} \]

       20. lift-*.f64 N/A

           \[\leadsto \left(d \cdot \left(d \cdot c0\right)\right) \cdot \frac{c0}{w \cdot \left(\color{blue}{\left(\left(D \cdot D\right) \cdot h\right)} \cdot w\right)} \]

       21. lift-*.f64 N/A

           \[\leadsto \left(d \cdot \left(d \cdot c0\right)\right) \cdot \frac{c0}{w \cdot \left(\left(\color{blue}{\left(D \cdot D\right)} \cdot h\right) \cdot w\right)} \]

       22. associate-*l* N/A

           \[\leadsto \left(d \cdot \left(d \cdot c0\right)\right) \cdot \frac{c0}{w \cdot \left(\color{blue}{\left(D \cdot \left(D \cdot h\right)\right)} \cdot w\right)} \]

       23. lift-*.f64 N/A

           \[\leadsto \left(d \cdot \left(d \cdot c0\right)\right) \cdot \frac{c0}{w \cdot \left(\left(D \cdot \color{blue}{\left(D \cdot h\right)}\right) \cdot w\right)} \]

   12. Applied egg-rr 46.1%

       \[\leadsto \color{blue}{\left(d \cdot \left(d \cdot c0\right)\right) \cdot \frac{c0}{w \cdot \left(D \cdot \left(h \cdot \left(D \cdot w\right)\right)\right)}} \]

   if 6.2000000000000004e-188 < M < 3.05e-151

   1. Initial program 8.9%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around -inf

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\left(-1 \cdot c0\right) \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)} \]

      2. distribute-lft1-in N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot c0\right) \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right) \]

      3. metadata-eval N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot c0\right) \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \]

      4. mul0-lft N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot c0\right) \cdot \color{blue}{0}\right) \]

      5. metadata-eval N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot c0\right) \cdot \color{blue}{\left(-1 + 1\right)}\right) \]

      6. mul-1-neg N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(c0\right)\right)} \cdot \left(-1 + 1\right)\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\mathsf{neg}\left(c0 \cdot \left(-1 + 1\right)\right)\right)} \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot \left(\mathsf{neg}\left(\left(-1 + 1\right)\right)\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \left(\mathsf{neg}\left(\color{blue}{0}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{0}\right) \]

      11. metadata-eval N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{\left(-1 + 1\right)}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot \left(-1 + 1\right)\right)} \]

      13. metadata-eval 70.2

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{0}\right) \]

   5. Simplified 70.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{c0}{\color{blue}{2 \cdot w}} \cdot \left(c0 \cdot 0\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{c0}{2 \cdot w}} \cdot \left(c0 \cdot 0\right) \]

      3. mul0-rgt N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{0} \]

      4. mul0-rgt 70.2

         \[\leadsto \color{blue}{0} \]

   7. Applied egg-rr 70.2%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 6.2 \cdot 10^{-188}:\\ \;\;\;\;\left(d \cdot \left(c0 \cdot d\right)\right) \cdot \frac{c0}{w \cdot \left(D \cdot \left(h \cdot \left(w \cdot D\right)\right)\right)}\\ \mathbf{elif}\;M \leq 3.05 \cdot 10^{-151}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(d \cdot \left(c0 \cdot d\right)\right) \cdot \frac{c0}{w \cdot \left(D \cdot \left(h \cdot \left(w \cdot D\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 33.7% accurate, 156.0× 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 20.8%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
2. Add Preprocessing

3. Taylor expanded in c0 around -inf

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\left(-1 \cdot c0\right) \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)} \]

   2. distribute-lft1-in N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot c0\right) \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right) \]

   3. metadata-eval N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot c0\right) \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \]

   4. mul0-lft N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot c0\right) \cdot \color{blue}{0}\right) \]

   5. metadata-eval N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot c0\right) \cdot \color{blue}{\left(-1 + 1\right)}\right) \]

   6. mul-1-neg N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(c0\right)\right)} \cdot \left(-1 + 1\right)\right) \]

   7. distribute-lft-neg-in N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\mathsf{neg}\left(c0 \cdot \left(-1 + 1\right)\right)\right)} \]

   8. distribute-rgt-neg-in N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot \left(\mathsf{neg}\left(\left(-1 + 1\right)\right)\right)\right)} \]

   9. metadata-eval N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \left(\mathsf{neg}\left(\color{blue}{0}\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{0}\right) \]

   11. metadata-eval N/A

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{\left(-1 + 1\right)}\right) \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot \left(-1 + 1\right)\right)} \]

   13. metadata-eval 26.6

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{0}\right) \]

5. Simplified 26.6%

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{c0}{\color{blue}{2 \cdot w}} \cdot \left(c0 \cdot 0\right) \]

   2. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{c0}{2 \cdot w}} \cdot \left(c0 \cdot 0\right) \]

   3. mul0-rgt N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{0} \]

   4. mul0-rgt 29.1

      \[\leadsto \color{blue}{0} \]

7. Applied egg-rr 29.1%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(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))))))