Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.6% → 69.6%

Time: 20.2s

Alternatives: 16

Speedup: 156.0×

• could not determine a ground truth

  1. c0 = -3.242811731230253e+190
  2. w = -7.99156477163313e-7
  3. h = -1.0409192860836356e-286
  4. D = -1.2022659032549689e-264
  5. d = 2.0110561218284678e+182
  6. M = 2.2911020646406024e+145

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 24.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 69.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{\left(2 \cdot d\right) \cdot \frac{c0}{w \cdot \left(2 \cdot D\right)}}{\left(w \cdot h\right) \cdot \frac{D}{c0 \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(D \cdot \frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d \cdot 4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
        INFINITY)
     (/ (* (* 2.0 d) (/ c0 (* w (* 2.0 D)))) (* (* w h) (/ D (* c0 d))))
     (* (/ 1.0 d) (* D (/ (* h (* D (* M M))) (* d 4.0)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= ((double) INFINITY)) {
		tmp = ((2.0 * d) * (c0 / (w * (2.0 * D)))) / ((w * h) * (D / (c0 * d)));
	} else {
		tmp = (1.0 / d) * (D * ((h * (D * (M * M))) / (d * 4.0)));
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = ((2.0 * d) * (c0 / (w * (2.0 * D)))) / ((w * h) * (D / (c0 * d)));
	} else {
		tmp = (1.0 / d) * (D * ((h * (D * (M * M))) / (d * 4.0)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, 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))))) <= math.inf:
		tmp = ((2.0 * d) * (c0 / (w * (2.0 * D)))) / ((w * h) * (D / (c0 * d)))
	else:
		tmp = (1.0 / d) * (D * ((h * (D * (M * M))) / (d * 4.0)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M))))) <= Inf)
		tmp = Float64(Float64(Float64(2.0 * d) * Float64(c0 / Float64(w * Float64(2.0 * D)))) / Float64(Float64(w * h) * Float64(D / Float64(c0 * d))));
	else
		tmp = Float64(Float64(1.0 / d) * Float64(D * Float64(Float64(h * Float64(D * Float64(M * M))) / Float64(d * 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, 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))))) <= Inf)
		tmp = ((2.0 * d) * (c0 / (w * (2.0 * D)))) / ((w * h) * (D / (c0 * d)));
	else
		tmp = (1.0 / d) * (D * ((h * (D * (M * M))) / (d * 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(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 * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(2.0 * d), $MachinePrecision] * N[(c0 / N[(w * N[(2.0 * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D / N[(c0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / d), $MachinePrecision] * N[(D * N[(N[(h * N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

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 67.9%

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

   3. Taylor expanded in c0 around inf

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)} \]
   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. /-lowering-/.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 71.1

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

   5. Simplified 71.1%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. /-lowering-/.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

   7. Applied egg-rr 76.7%

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 77.8

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

   9. Applied egg-rr 77.8%

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

       1. associate-*r* N/A

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

       2. clear-num N/A

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

       3. un-div-inv N/A

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

       4. associate-/r* N/A

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

       5. associate-*l/ N/A

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

       6. associate-/l/ N/A

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

       7. /-lowering-/.f64 N/A

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

   11. Applied egg-rr 85.9%

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

   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. Step-by-step derivation

      1. +-commutative N/A

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

      2. frac-2neg N/A

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

      3. distribute-frac-neg N/A

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

      4. unsub-neg N/A

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

      5. distribute-frac-neg2 N/A

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

      6. --lowering--.f64 N/A

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

   4. Applied egg-rr 0.1%

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

   5. Taylor expanded in c0 around -inf

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{{d}^{2}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{{d}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)}{{d}^{2}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)}{{d}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\color{blue}{d \cdot d}} \]

      12. *-lowering-*.f64 41.9

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

   7. Simplified 41.9%

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

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{d \cdot d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}}} \]

      2. inv-pow N/A

         \[\leadsto \color{blue}{{\left(\frac{d \cdot d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\right)}^{-1}} \]

      3. associate-/l* N/A

         \[\leadsto {\color{blue}{\left(d \cdot \frac{d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\right)}}^{-1} \]

      4. unpow-prod-down N/A

         \[\leadsto \color{blue}{{d}^{-1} \cdot {\left(\frac{d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\right)}^{-1}} \]

      5. inv-pow N/A

         \[\leadsto \color{blue}{\frac{1}{d}} \cdot {\left(\frac{d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\right)}^{-1} \]

      6. inv-pow N/A

         \[\leadsto \frac{1}{d} \cdot \color{blue}{\frac{1}{\frac{d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}}} \]

      7. clear-num N/A

         \[\leadsto \frac{1}{d} \cdot \color{blue}{\frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{d} \cdot \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d}} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{d}} \cdot \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d} \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{d} \cdot \frac{\color{blue}{\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{1}{4}}}{d} \]

      11. associate-/l* N/A

          \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \left(\color{blue}{\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)} \cdot \frac{\frac{1}{4}}{d}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \left(\left(\color{blue}{\left(D \cdot D\right)} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \left(\left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot \left(M \cdot M\right)\right)}\right) \cdot \frac{\frac{1}{4}}{d}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \left(\left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right) \cdot \frac{\frac{1}{4}}{d}\right) \]

      17. /-lowering-/.f64 53.9

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

   9. Applied egg-rr 53.9%

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

       1. associate-*l* N/A

          \[\leadsto \frac{1}{d} \cdot \left(\color{blue}{\left(D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)} \cdot \frac{\frac{1}{4}}{d}\right) \]

       2. associate-*l* N/A

          \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(D \cdot \left(\left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right)\right)} \]

       3. *-commutative N/A

          \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(\left(\left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot D\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(\left(\left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot D\right)} \]

       5. clear-num N/A

          \[\leadsto \frac{1}{d} \cdot \left(\left(\left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \color{blue}{\frac{1}{\frac{d}{\frac{1}{4}}}}\right) \cdot D\right) \]

       6. un-div-inv N/A

          \[\leadsto \frac{1}{d} \cdot \left(\color{blue}{\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d}{\frac{1}{4}}}} \cdot D\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \left(\color{blue}{\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d}{\frac{1}{4}}}} \cdot D\right) \]

       8. *-commutative N/A

          \[\leadsto \frac{1}{d} \cdot \left(\frac{D \cdot \color{blue}{\left(\left(M \cdot M\right) \cdot h\right)}}{\frac{d}{\frac{1}{4}}} \cdot D\right) \]

       9. associate-*r* N/A

          \[\leadsto \frac{1}{d} \cdot \left(\frac{\color{blue}{\left(D \cdot \left(M \cdot M\right)\right) \cdot h}}{\frac{d}{\frac{1}{4}}} \cdot D\right) \]

       10. *-commutative N/A

           \[\leadsto \frac{1}{d} \cdot \left(\frac{\color{blue}{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}}{\frac{d}{\frac{1}{4}}} \cdot D\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \frac{1}{d} \cdot \left(\frac{\color{blue}{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}}{\frac{d}{\frac{1}{4}}} \cdot D\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \frac{1}{d} \cdot \left(\frac{h \cdot \color{blue}{\left(D \cdot \left(M \cdot M\right)\right)}}{\frac{d}{\frac{1}{4}}} \cdot D\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \frac{1}{d} \cdot \left(\frac{h \cdot \left(D \cdot \color{blue}{\left(M \cdot M\right)}\right)}{\frac{d}{\frac{1}{4}}} \cdot D\right) \]

       14. div-inv N/A

           \[\leadsto \frac{1}{d} \cdot \left(\frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{\color{blue}{d \cdot \frac{1}{\frac{1}{4}}}} \cdot D\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \frac{1}{d} \cdot \left(\frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{\color{blue}{d \cdot \frac{1}{\frac{1}{4}}}} \cdot D\right) \]

       16. metadata-eval 64.2

           \[\leadsto \frac{1}{d} \cdot \left(\frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d \cdot \color{blue}{4}} \cdot D\right) \]

   11. Applied egg-rr 64.2%

       \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(\frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d \cdot 4} \cdot D\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{\left(2 \cdot d\right) \cdot \frac{c0}{w \cdot \left(2 \cdot D\right)}}{\left(w \cdot h\right) \cdot \frac{D}{c0 \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(D \cdot \frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d \cdot 4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 68.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{-2 \cdot \left(d \cdot \left(c0 \cdot d\right)\right)}{D \cdot \left(\left(w \cdot h\right) \cdot \frac{w \cdot \left(2 \cdot D\right)}{-c0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(D \cdot \frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d \cdot 4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
        INFINITY)
     (/ (* -2.0 (* d (* c0 d))) (* D (* (* w h) (/ (* w (* 2.0 D)) (- c0)))))
     (* (/ 1.0 d) (* D (/ (* h (* D (* M M))) (* d 4.0)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= ((double) INFINITY)) {
		tmp = (-2.0 * (d * (c0 * d))) / (D * ((w * h) * ((w * (2.0 * D)) / -c0)));
	} else {
		tmp = (1.0 / d) * (D * ((h * (D * (M * M))) / (d * 4.0)));
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = (-2.0 * (d * (c0 * d))) / (D * ((w * h) * ((w * (2.0 * D)) / -c0)));
	} else {
		tmp = (1.0 / d) * (D * ((h * (D * (M * M))) / (d * 4.0)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, 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))))) <= math.inf:
		tmp = (-2.0 * (d * (c0 * d))) / (D * ((w * h) * ((w * (2.0 * D)) / -c0)))
	else:
		tmp = (1.0 / d) * (D * ((h * (D * (M * M))) / (d * 4.0)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M))))) <= Inf)
		tmp = Float64(Float64(-2.0 * Float64(d * Float64(c0 * d))) / Float64(D * Float64(Float64(w * h) * Float64(Float64(w * Float64(2.0 * D)) / Float64(-c0)))));
	else
		tmp = Float64(Float64(1.0 / d) * Float64(D * Float64(Float64(h * Float64(D * Float64(M * M))) / Float64(d * 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, 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))))) <= Inf)
		tmp = (-2.0 * (d * (c0 * d))) / (D * ((w * h) * ((w * (2.0 * D)) / -c0)));
	else
		tmp = (1.0 / d) * (D * ((h * (D * (M * M))) / (d * 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(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 * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(-2.0 * N[(d * N[(c0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(D * N[(N[(w * h), $MachinePrecision] * N[(N[(w * N[(2.0 * D), $MachinePrecision]), $MachinePrecision] / (-c0)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / d), $MachinePrecision] * N[(D * N[(N[(h * N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

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 67.9%

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

   3. Taylor expanded in c0 around inf

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)} \]
   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. /-lowering-/.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 71.1

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

   5. Simplified 71.1%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. /-lowering-/.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

   7. Applied egg-rr 76.7%

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 77.8

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

   9. Applied egg-rr 77.8%

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

       1. *-commutative N/A

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

       2. clear-num N/A

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

       3. un-div-inv N/A

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

       4. associate-*r/ N/A

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

       5. frac-2neg N/A

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

       6. associate-/l/ N/A

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

       7. /-lowering-/.f64 N/A

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

       8. associate-*l* N/A

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

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

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

       10. *-lowering-*.f64 N/A

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

       11. metadata-eval N/A

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

       12. *-lowering-*.f64 N/A

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

       13. *-lowering-*.f64 N/A

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

       14. *-lowering-*.f64 N/A

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

   11. Applied egg-rr 78.9%

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

   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. Step-by-step derivation

      1. +-commutative N/A

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

      2. frac-2neg N/A

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

      3. distribute-frac-neg N/A

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

      4. unsub-neg N/A

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

      5. distribute-frac-neg2 N/A

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

      6. --lowering--.f64 N/A

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

   4. Applied egg-rr 0.1%

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

   5. Taylor expanded in c0 around -inf

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{{d}^{2}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{{d}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)}{{d}^{2}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)}{{d}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\color{blue}{d \cdot d}} \]

      12. *-lowering-*.f64 41.9

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

   7. Simplified 41.9%

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

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{d \cdot d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}}} \]

      2. inv-pow N/A

         \[\leadsto \color{blue}{{\left(\frac{d \cdot d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\right)}^{-1}} \]

      3. associate-/l* N/A

         \[\leadsto {\color{blue}{\left(d \cdot \frac{d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\right)}}^{-1} \]

      4. unpow-prod-down N/A

         \[\leadsto \color{blue}{{d}^{-1} \cdot {\left(\frac{d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\right)}^{-1}} \]

      5. inv-pow N/A

         \[\leadsto \color{blue}{\frac{1}{d}} \cdot {\left(\frac{d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\right)}^{-1} \]

      6. inv-pow N/A

         \[\leadsto \frac{1}{d} \cdot \color{blue}{\frac{1}{\frac{d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}}} \]

      7. clear-num N/A

         \[\leadsto \frac{1}{d} \cdot \color{blue}{\frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{d} \cdot \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d}} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{d}} \cdot \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d} \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{d} \cdot \frac{\color{blue}{\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{1}{4}}}{d} \]

      11. associate-/l* N/A

          \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \left(\color{blue}{\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)} \cdot \frac{\frac{1}{4}}{d}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \left(\left(\color{blue}{\left(D \cdot D\right)} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \left(\left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot \left(M \cdot M\right)\right)}\right) \cdot \frac{\frac{1}{4}}{d}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \left(\left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right) \cdot \frac{\frac{1}{4}}{d}\right) \]

      17. /-lowering-/.f64 53.9

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

   9. Applied egg-rr 53.9%

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

       1. associate-*l* N/A

          \[\leadsto \frac{1}{d} \cdot \left(\color{blue}{\left(D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)} \cdot \frac{\frac{1}{4}}{d}\right) \]

       2. associate-*l* N/A

          \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(D \cdot \left(\left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right)\right)} \]

       3. *-commutative N/A

          \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(\left(\left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot D\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(\left(\left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot D\right)} \]

       5. clear-num N/A

          \[\leadsto \frac{1}{d} \cdot \left(\left(\left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \color{blue}{\frac{1}{\frac{d}{\frac{1}{4}}}}\right) \cdot D\right) \]

       6. un-div-inv N/A

          \[\leadsto \frac{1}{d} \cdot \left(\color{blue}{\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d}{\frac{1}{4}}}} \cdot D\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \left(\color{blue}{\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d}{\frac{1}{4}}}} \cdot D\right) \]

       8. *-commutative N/A

          \[\leadsto \frac{1}{d} \cdot \left(\frac{D \cdot \color{blue}{\left(\left(M \cdot M\right) \cdot h\right)}}{\frac{d}{\frac{1}{4}}} \cdot D\right) \]

       9. associate-*r* N/A

          \[\leadsto \frac{1}{d} \cdot \left(\frac{\color{blue}{\left(D \cdot \left(M \cdot M\right)\right) \cdot h}}{\frac{d}{\frac{1}{4}}} \cdot D\right) \]

       10. *-commutative N/A

           \[\leadsto \frac{1}{d} \cdot \left(\frac{\color{blue}{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}}{\frac{d}{\frac{1}{4}}} \cdot D\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \frac{1}{d} \cdot \left(\frac{\color{blue}{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}}{\frac{d}{\frac{1}{4}}} \cdot D\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \frac{1}{d} \cdot \left(\frac{h \cdot \color{blue}{\left(D \cdot \left(M \cdot M\right)\right)}}{\frac{d}{\frac{1}{4}}} \cdot D\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \frac{1}{d} \cdot \left(\frac{h \cdot \left(D \cdot \color{blue}{\left(M \cdot M\right)}\right)}{\frac{d}{\frac{1}{4}}} \cdot D\right) \]

       14. div-inv N/A

           \[\leadsto \frac{1}{d} \cdot \left(\frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{\color{blue}{d \cdot \frac{1}{\frac{1}{4}}}} \cdot D\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \frac{1}{d} \cdot \left(\frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{\color{blue}{d \cdot \frac{1}{\frac{1}{4}}}} \cdot D\right) \]

       16. metadata-eval 64.2

           \[\leadsto \frac{1}{d} \cdot \left(\frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d \cdot \color{blue}{4}} \cdot D\right) \]

   11. Applied egg-rr 64.2%

       \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(\frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d \cdot 4} \cdot D\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{-2 \cdot \left(d \cdot \left(c0 \cdot d\right)\right)}{D \cdot \left(\left(w \cdot h\right) \cdot \frac{w \cdot \left(2 \cdot D\right)}{-c0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(D \cdot \frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d \cdot 4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;d \cdot \left(t\_0 \cdot \frac{c0 \cdot \left(2 \cdot d\right)}{h \cdot \left(D \cdot \left(w \cdot D\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(D \cdot \frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d \cdot 4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w))) (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M))))) INFINITY)
     (* d (* t_0 (/ (* c0 (* 2.0 d)) (* h (* D (* w D))))))
     (* (/ 1.0 d) (* D (/ (* h (* D (* M M))) (* d 4.0)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = d * (t_0 * ((c0 * (2.0 * d)) / (h * (D * (w * D)))));
	} else {
		tmp = (1.0 / d) * (D * ((h * (D * (M * M))) / (d * 4.0)));
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = d * (t_0 * ((c0 * (2.0 * d)) / (h * (D * (w * D)))));
	} else {
		tmp = (1.0 / d) * (D * ((h * (D * (M * M))) / (d * 4.0)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if (t_0 * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))) <= math.inf:
		tmp = d * (t_0 * ((c0 * (2.0 * d)) / (h * (D * (w * D)))))
	else:
		tmp = (1.0 / d) * (D * ((h * (D * (M * M))) / (d * 4.0)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))))) <= Inf)
		tmp = Float64(d * Float64(t_0 * Float64(Float64(c0 * Float64(2.0 * d)) / Float64(h * Float64(D * Float64(w * D))))));
	else
		tmp = Float64(Float64(1.0 / d) * Float64(D * Float64(Float64(h * Float64(D * Float64(M * M))) / Float64(d * 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= Inf)
		tmp = d * (t_0 * ((c0 * (2.0 * d)) / (h * (D * (w * D)))));
	else
		tmp = (1.0 / d) * (D * ((h * (D * (M * M))) / (d * 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(d * N[(t$95$0 * N[(N[(c0 * N[(2.0 * d), $MachinePrecision]), $MachinePrecision] / N[(h * N[(D * N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / d), $MachinePrecision] * N[(D * N[(N[(h * N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

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 67.9%

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

   3. Taylor expanded in c0 around inf

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)} \]
   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. /-lowering-/.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 71.1

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

   5. Simplified 71.1%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. /-lowering-/.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

   7. Applied egg-rr 76.7%

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 77.8

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

   9. Applied egg-rr 77.8%

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

       1. associate-*l/ N/A

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

       2. *-commutative N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. associate-/l* N/A

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

       6. associate-*r/ N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. associate-*r* N/A

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

       10. *-commutative N/A

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

       11. associate-/l/ N/A

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

       12. associate-*r* N/A

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

       13. associate-*r* N/A

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

       14. *-commutative N/A

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

       15. *-commutative N/A

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

       16. associate-*l/ N/A

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

       17. associate-*l/ N/A

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

   11. Applied egg-rr 78.1%

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

   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. Step-by-step derivation

      1. +-commutative N/A

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

      2. frac-2neg N/A

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

      3. distribute-frac-neg N/A

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

      4. unsub-neg N/A

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

      5. distribute-frac-neg2 N/A

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

      6. --lowering--.f64 N/A

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

   4. Applied egg-rr 0.1%

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

   5. Taylor expanded in c0 around -inf

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{{d}^{2}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{{d}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)}{{d}^{2}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)}{{d}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\color{blue}{d \cdot d}} \]

      12. *-lowering-*.f64 41.9

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

   7. Simplified 41.9%

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

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{d \cdot d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}}} \]

      2. inv-pow N/A

         \[\leadsto \color{blue}{{\left(\frac{d \cdot d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\right)}^{-1}} \]

      3. associate-/l* N/A

         \[\leadsto {\color{blue}{\left(d \cdot \frac{d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\right)}}^{-1} \]

      4. unpow-prod-down N/A

         \[\leadsto \color{blue}{{d}^{-1} \cdot {\left(\frac{d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\right)}^{-1}} \]

      5. inv-pow N/A

         \[\leadsto \color{blue}{\frac{1}{d}} \cdot {\left(\frac{d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\right)}^{-1} \]

      6. inv-pow N/A

         \[\leadsto \frac{1}{d} \cdot \color{blue}{\frac{1}{\frac{d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}}} \]

      7. clear-num N/A

         \[\leadsto \frac{1}{d} \cdot \color{blue}{\frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{d} \cdot \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d}} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{d}} \cdot \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d} \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{d} \cdot \frac{\color{blue}{\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{1}{4}}}{d} \]

      11. associate-/l* N/A

          \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \left(\color{blue}{\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)} \cdot \frac{\frac{1}{4}}{d}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \left(\left(\color{blue}{\left(D \cdot D\right)} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \left(\left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot \left(M \cdot M\right)\right)}\right) \cdot \frac{\frac{1}{4}}{d}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \left(\left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right) \cdot \frac{\frac{1}{4}}{d}\right) \]

      17. /-lowering-/.f64 53.9

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

   9. Applied egg-rr 53.9%

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

       1. associate-*l* N/A

          \[\leadsto \frac{1}{d} \cdot \left(\color{blue}{\left(D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)} \cdot \frac{\frac{1}{4}}{d}\right) \]

       2. associate-*l* N/A

          \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(D \cdot \left(\left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right)\right)} \]

       3. *-commutative N/A

          \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(\left(\left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot D\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(\left(\left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot D\right)} \]

       5. clear-num N/A

          \[\leadsto \frac{1}{d} \cdot \left(\left(\left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \color{blue}{\frac{1}{\frac{d}{\frac{1}{4}}}}\right) \cdot D\right) \]

       6. un-div-inv N/A

          \[\leadsto \frac{1}{d} \cdot \left(\color{blue}{\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d}{\frac{1}{4}}}} \cdot D\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \left(\color{blue}{\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d}{\frac{1}{4}}}} \cdot D\right) \]

       8. *-commutative N/A

          \[\leadsto \frac{1}{d} \cdot \left(\frac{D \cdot \color{blue}{\left(\left(M \cdot M\right) \cdot h\right)}}{\frac{d}{\frac{1}{4}}} \cdot D\right) \]

       9. associate-*r* N/A

          \[\leadsto \frac{1}{d} \cdot \left(\frac{\color{blue}{\left(D \cdot \left(M \cdot M\right)\right) \cdot h}}{\frac{d}{\frac{1}{4}}} \cdot D\right) \]

       10. *-commutative N/A

           \[\leadsto \frac{1}{d} \cdot \left(\frac{\color{blue}{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}}{\frac{d}{\frac{1}{4}}} \cdot D\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \frac{1}{d} \cdot \left(\frac{\color{blue}{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}}{\frac{d}{\frac{1}{4}}} \cdot D\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \frac{1}{d} \cdot \left(\frac{h \cdot \color{blue}{\left(D \cdot \left(M \cdot M\right)\right)}}{\frac{d}{\frac{1}{4}}} \cdot D\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \frac{1}{d} \cdot \left(\frac{h \cdot \left(D \cdot \color{blue}{\left(M \cdot M\right)}\right)}{\frac{d}{\frac{1}{4}}} \cdot D\right) \]

       14. div-inv N/A

           \[\leadsto \frac{1}{d} \cdot \left(\frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{\color{blue}{d \cdot \frac{1}{\frac{1}{4}}}} \cdot D\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \frac{1}{d} \cdot \left(\frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{\color{blue}{d \cdot \frac{1}{\frac{1}{4}}}} \cdot D\right) \]

       16. metadata-eval 64.2

           \[\leadsto \frac{1}{d} \cdot \left(\frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d \cdot \color{blue}{4}} \cdot D\right) \]

   11. Applied egg-rr 64.2%

       \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(\frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d \cdot 4} \cdot D\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;d \cdot \left(\frac{c0}{2 \cdot w} \cdot \frac{c0 \cdot \left(2 \cdot d\right)}{h \cdot \left(D \cdot \left(w \cdot D\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(D \cdot \frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d \cdot 4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\left(t\_0 \cdot \left(d \cdot \frac{d}{D \cdot \left(h \cdot \left(w \cdot D\right)\right)}\right)\right) \cdot \left(c0 \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(D \cdot \frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d \cdot 4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w))) (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M))))) INFINITY)
     (* (* t_0 (* d (/ d (* D (* h (* w D)))))) (* c0 2.0))
     (* (/ 1.0 d) (* D (/ (* h (* D (* M M))) (* d 4.0)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = (t_0 * (d * (d / (D * (h * (w * D)))))) * (c0 * 2.0);
	} else {
		tmp = (1.0 / d) * (D * ((h * (D * (M * M))) / (d * 4.0)));
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = (t_0 * (d * (d / (D * (h * (w * D)))))) * (c0 * 2.0);
	} else {
		tmp = (1.0 / d) * (D * ((h * (D * (M * M))) / (d * 4.0)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if (t_0 * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))) <= math.inf:
		tmp = (t_0 * (d * (d / (D * (h * (w * D)))))) * (c0 * 2.0)
	else:
		tmp = (1.0 / d) * (D * ((h * (D * (M * M))) / (d * 4.0)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))))) <= Inf)
		tmp = Float64(Float64(t_0 * Float64(d * Float64(d / Float64(D * Float64(h * Float64(w * D)))))) * Float64(c0 * 2.0));
	else
		tmp = Float64(Float64(1.0 / d) * Float64(D * Float64(Float64(h * Float64(D * Float64(M * M))) / Float64(d * 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= Inf)
		tmp = (t_0 * (d * (d / (D * (h * (w * D)))))) * (c0 * 2.0);
	else
		tmp = (1.0 / d) * (D * ((h * (D * (M * M))) / (d * 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(t$95$0 * N[(d * N[(d / N[(D * N[(h * N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c0 * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / d), $MachinePrecision] * N[(D * N[(N[(h * N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

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 67.9%

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

   3. Taylor expanded in c0 around inf

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)} \]
   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. /-lowering-/.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 71.1

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

   5. Simplified 71.1%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. /-lowering-/.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

   7. Applied egg-rr 76.7%

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

      1. frac-times N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*l/ N/A

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

      10. associate-*r* N/A

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

      11. *-lowering-*.f64 N/A

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

   9. Applied egg-rr 71.4%

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

   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. Step-by-step derivation

      1. +-commutative N/A

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

      2. frac-2neg N/A

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

      3. distribute-frac-neg N/A

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

      4. unsub-neg N/A

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

      5. distribute-frac-neg2 N/A

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

      6. --lowering--.f64 N/A

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

   4. Applied egg-rr 0.1%

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

   5. Taylor expanded in c0 around -inf

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{{d}^{2}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{{d}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)}{{d}^{2}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)}{{d}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\color{blue}{d \cdot d}} \]

      12. *-lowering-*.f64 41.9

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

   7. Simplified 41.9%

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

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{d \cdot d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}}} \]

      2. inv-pow N/A

         \[\leadsto \color{blue}{{\left(\frac{d \cdot d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\right)}^{-1}} \]

      3. associate-/l* N/A

         \[\leadsto {\color{blue}{\left(d \cdot \frac{d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\right)}}^{-1} \]

      4. unpow-prod-down N/A

         \[\leadsto \color{blue}{{d}^{-1} \cdot {\left(\frac{d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\right)}^{-1}} \]

      5. inv-pow N/A

         \[\leadsto \color{blue}{\frac{1}{d}} \cdot {\left(\frac{d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\right)}^{-1} \]

      6. inv-pow N/A

         \[\leadsto \frac{1}{d} \cdot \color{blue}{\frac{1}{\frac{d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}}} \]

      7. clear-num N/A

         \[\leadsto \frac{1}{d} \cdot \color{blue}{\frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{d} \cdot \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d}} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{d}} \cdot \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d} \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{d} \cdot \frac{\color{blue}{\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{1}{4}}}{d} \]

      11. associate-/l* N/A

          \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \left(\color{blue}{\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)} \cdot \frac{\frac{1}{4}}{d}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \left(\left(\color{blue}{\left(D \cdot D\right)} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \left(\left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot \left(M \cdot M\right)\right)}\right) \cdot \frac{\frac{1}{4}}{d}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \left(\left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right) \cdot \frac{\frac{1}{4}}{d}\right) \]

      17. /-lowering-/.f64 53.9

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

   9. Applied egg-rr 53.9%

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

       1. associate-*l* N/A

          \[\leadsto \frac{1}{d} \cdot \left(\color{blue}{\left(D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)} \cdot \frac{\frac{1}{4}}{d}\right) \]

       2. associate-*l* N/A

          \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(D \cdot \left(\left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right)\right)} \]

       3. *-commutative N/A

          \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(\left(\left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot D\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(\left(\left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot D\right)} \]

       5. clear-num N/A

          \[\leadsto \frac{1}{d} \cdot \left(\left(\left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \color{blue}{\frac{1}{\frac{d}{\frac{1}{4}}}}\right) \cdot D\right) \]

       6. un-div-inv N/A

          \[\leadsto \frac{1}{d} \cdot \left(\color{blue}{\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d}{\frac{1}{4}}}} \cdot D\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \left(\color{blue}{\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d}{\frac{1}{4}}}} \cdot D\right) \]

       8. *-commutative N/A

          \[\leadsto \frac{1}{d} \cdot \left(\frac{D \cdot \color{blue}{\left(\left(M \cdot M\right) \cdot h\right)}}{\frac{d}{\frac{1}{4}}} \cdot D\right) \]

       9. associate-*r* N/A

          \[\leadsto \frac{1}{d} \cdot \left(\frac{\color{blue}{\left(D \cdot \left(M \cdot M\right)\right) \cdot h}}{\frac{d}{\frac{1}{4}}} \cdot D\right) \]

       10. *-commutative N/A

           \[\leadsto \frac{1}{d} \cdot \left(\frac{\color{blue}{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}}{\frac{d}{\frac{1}{4}}} \cdot D\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \frac{1}{d} \cdot \left(\frac{\color{blue}{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}}{\frac{d}{\frac{1}{4}}} \cdot D\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \frac{1}{d} \cdot \left(\frac{h \cdot \color{blue}{\left(D \cdot \left(M \cdot M\right)\right)}}{\frac{d}{\frac{1}{4}}} \cdot D\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \frac{1}{d} \cdot \left(\frac{h \cdot \left(D \cdot \color{blue}{\left(M \cdot M\right)}\right)}{\frac{d}{\frac{1}{4}}} \cdot D\right) \]

       14. div-inv N/A

           \[\leadsto \frac{1}{d} \cdot \left(\frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{\color{blue}{d \cdot \frac{1}{\frac{1}{4}}}} \cdot D\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \frac{1}{d} \cdot \left(\frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{\color{blue}{d \cdot \frac{1}{\frac{1}{4}}}} \cdot D\right) \]

       16. metadata-eval 64.2

           \[\leadsto \frac{1}{d} \cdot \left(\frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d \cdot \color{blue}{4}} \cdot D\right) \]

   11. Applied egg-rr 64.2%

       \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(\frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d \cdot 4} \cdot D\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\left(\frac{c0}{2 \cdot w} \cdot \left(d \cdot \frac{d}{D \cdot \left(h \cdot \left(w \cdot D\right)\right)}\right)\right) \cdot \left(c0 \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(D \cdot \frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d \cdot 4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{\left(c0 \cdot d\right) \cdot \left(\left(c0 \cdot d\right) \cdot -2\right)}{\left(2 \cdot w\right) \cdot \left(D \cdot \left(h \cdot \left(w \cdot \left(-D\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(D \cdot \frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d \cdot 4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
        INFINITY)
     (/ (* (* c0 d) (* (* c0 d) -2.0)) (* (* 2.0 w) (* D (* h (* w (- D))))))
     (* (/ 1.0 d) (* D (/ (* h (* D (* M M))) (* d 4.0)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= ((double) INFINITY)) {
		tmp = ((c0 * d) * ((c0 * d) * -2.0)) / ((2.0 * w) * (D * (h * (w * -D))));
	} else {
		tmp = (1.0 / d) * (D * ((h * (D * (M * M))) / (d * 4.0)));
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = ((c0 * d) * ((c0 * d) * -2.0)) / ((2.0 * w) * (D * (h * (w * -D))));
	} else {
		tmp = (1.0 / d) * (D * ((h * (D * (M * M))) / (d * 4.0)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, 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))))) <= math.inf:
		tmp = ((c0 * d) * ((c0 * d) * -2.0)) / ((2.0 * w) * (D * (h * (w * -D))))
	else:
		tmp = (1.0 / d) * (D * ((h * (D * (M * M))) / (d * 4.0)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M))))) <= Inf)
		tmp = Float64(Float64(Float64(c0 * d) * Float64(Float64(c0 * d) * -2.0)) / Float64(Float64(2.0 * w) * Float64(D * Float64(h * Float64(w * Float64(-D))))));
	else
		tmp = Float64(Float64(1.0 / d) * Float64(D * Float64(Float64(h * Float64(D * Float64(M * M))) / Float64(d * 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, 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))))) <= Inf)
		tmp = ((c0 * d) * ((c0 * d) * -2.0)) / ((2.0 * w) * (D * (h * (w * -D))));
	else
		tmp = (1.0 / d) * (D * ((h * (D * (M * M))) / (d * 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(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 * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(c0 * d), $MachinePrecision] * N[(N[(c0 * d), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * w), $MachinePrecision] * N[(D * N[(h * N[(w * (-D)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / d), $MachinePrecision] * N[(D * N[(N[(h * N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

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 67.9%

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

   3. Taylor expanded in c0 around inf

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)} \]
   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. /-lowering-/.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 71.1

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

   5. Simplified 71.1%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. /-lowering-/.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

   7. Applied egg-rr 76.7%

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

      1. frac-2neg N/A

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

      2. frac-times N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. frac-times N/A

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

      10. /-lowering-/.f64 N/A

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

   9. Applied egg-rr 70.7%

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

   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. Step-by-step derivation

      1. +-commutative N/A

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

      2. frac-2neg N/A

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

      3. distribute-frac-neg N/A

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

      4. unsub-neg N/A

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

      5. distribute-frac-neg2 N/A

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

      6. --lowering--.f64 N/A

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

   4. Applied egg-rr 0.1%

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

   5. Taylor expanded in c0 around -inf

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{{d}^{2}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{{d}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)}{{d}^{2}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)}{{d}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\color{blue}{d \cdot d}} \]

      12. *-lowering-*.f64 41.9

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

   7. Simplified 41.9%

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

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{d \cdot d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}}} \]

      2. inv-pow N/A

         \[\leadsto \color{blue}{{\left(\frac{d \cdot d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\right)}^{-1}} \]

      3. associate-/l* N/A

         \[\leadsto {\color{blue}{\left(d \cdot \frac{d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\right)}}^{-1} \]

      4. unpow-prod-down N/A

         \[\leadsto \color{blue}{{d}^{-1} \cdot {\left(\frac{d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\right)}^{-1}} \]

      5. inv-pow N/A

         \[\leadsto \color{blue}{\frac{1}{d}} \cdot {\left(\frac{d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\right)}^{-1} \]

      6. inv-pow N/A

         \[\leadsto \frac{1}{d} \cdot \color{blue}{\frac{1}{\frac{d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}}} \]

      7. clear-num N/A

         \[\leadsto \frac{1}{d} \cdot \color{blue}{\frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{d} \cdot \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d}} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{d}} \cdot \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d} \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{d} \cdot \frac{\color{blue}{\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{1}{4}}}{d} \]

      11. associate-/l* N/A

          \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \left(\color{blue}{\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)} \cdot \frac{\frac{1}{4}}{d}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \left(\left(\color{blue}{\left(D \cdot D\right)} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \left(\left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot \left(M \cdot M\right)\right)}\right) \cdot \frac{\frac{1}{4}}{d}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \left(\left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right) \cdot \frac{\frac{1}{4}}{d}\right) \]

      17. /-lowering-/.f64 53.9

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

   9. Applied egg-rr 53.9%

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

       1. associate-*l* N/A

          \[\leadsto \frac{1}{d} \cdot \left(\color{blue}{\left(D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)} \cdot \frac{\frac{1}{4}}{d}\right) \]

       2. associate-*l* N/A

          \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(D \cdot \left(\left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right)\right)} \]

       3. *-commutative N/A

          \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(\left(\left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot D\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(\left(\left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot D\right)} \]

       5. clear-num N/A

          \[\leadsto \frac{1}{d} \cdot \left(\left(\left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \color{blue}{\frac{1}{\frac{d}{\frac{1}{4}}}}\right) \cdot D\right) \]

       6. un-div-inv N/A

          \[\leadsto \frac{1}{d} \cdot \left(\color{blue}{\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d}{\frac{1}{4}}}} \cdot D\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \left(\color{blue}{\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d}{\frac{1}{4}}}} \cdot D\right) \]

       8. *-commutative N/A

          \[\leadsto \frac{1}{d} \cdot \left(\frac{D \cdot \color{blue}{\left(\left(M \cdot M\right) \cdot h\right)}}{\frac{d}{\frac{1}{4}}} \cdot D\right) \]

       9. associate-*r* N/A

          \[\leadsto \frac{1}{d} \cdot \left(\frac{\color{blue}{\left(D \cdot \left(M \cdot M\right)\right) \cdot h}}{\frac{d}{\frac{1}{4}}} \cdot D\right) \]

       10. *-commutative N/A

           \[\leadsto \frac{1}{d} \cdot \left(\frac{\color{blue}{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}}{\frac{d}{\frac{1}{4}}} \cdot D\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \frac{1}{d} \cdot \left(\frac{\color{blue}{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}}{\frac{d}{\frac{1}{4}}} \cdot D\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \frac{1}{d} \cdot \left(\frac{h \cdot \color{blue}{\left(D \cdot \left(M \cdot M\right)\right)}}{\frac{d}{\frac{1}{4}}} \cdot D\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \frac{1}{d} \cdot \left(\frac{h \cdot \left(D \cdot \color{blue}{\left(M \cdot M\right)}\right)}{\frac{d}{\frac{1}{4}}} \cdot D\right) \]

       14. div-inv N/A

           \[\leadsto \frac{1}{d} \cdot \left(\frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{\color{blue}{d \cdot \frac{1}{\frac{1}{4}}}} \cdot D\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \frac{1}{d} \cdot \left(\frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{\color{blue}{d \cdot \frac{1}{\frac{1}{4}}}} \cdot D\right) \]

       16. metadata-eval 64.2

           \[\leadsto \frac{1}{d} \cdot \left(\frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d \cdot \color{blue}{4}} \cdot D\right) \]

   11. Applied egg-rr 64.2%

       \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(\frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d \cdot 4} \cdot D\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{\left(c0 \cdot d\right) \cdot \left(\left(c0 \cdot d\right) \cdot -2\right)}{\left(2 \cdot w\right) \cdot \left(D \cdot \left(h \cdot \left(w \cdot \left(-D\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(D \cdot \frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d \cdot 4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;c0 \cdot \frac{2 \cdot \left(d \cdot \left(c0 \cdot d\right)\right)}{2 \cdot \left(w \cdot \left(D \cdot \left(\left(w \cdot h\right) \cdot D\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(D \cdot \frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d \cdot 4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
        INFINITY)
     (* c0 (/ (* 2.0 (* d (* c0 d))) (* 2.0 (* w (* D (* (* w h) D))))))
     (* (/ 1.0 d) (* D (/ (* h (* D (* M M))) (* d 4.0)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= ((double) INFINITY)) {
		tmp = c0 * ((2.0 * (d * (c0 * d))) / (2.0 * (w * (D * ((w * h) * D)))));
	} else {
		tmp = (1.0 / d) * (D * ((h * (D * (M * M))) / (d * 4.0)));
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = c0 * ((2.0 * (d * (c0 * d))) / (2.0 * (w * (D * ((w * h) * D)))));
	} else {
		tmp = (1.0 / d) * (D * ((h * (D * (M * M))) / (d * 4.0)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, 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))))) <= math.inf:
		tmp = c0 * ((2.0 * (d * (c0 * d))) / (2.0 * (w * (D * ((w * h) * D)))))
	else:
		tmp = (1.0 / d) * (D * ((h * (D * (M * M))) / (d * 4.0)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M))))) <= Inf)
		tmp = Float64(c0 * Float64(Float64(2.0 * Float64(d * Float64(c0 * d))) / Float64(2.0 * Float64(w * Float64(D * Float64(Float64(w * h) * D))))));
	else
		tmp = Float64(Float64(1.0 / d) * Float64(D * Float64(Float64(h * Float64(D * Float64(M * M))) / Float64(d * 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, 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))))) <= Inf)
		tmp = c0 * ((2.0 * (d * (c0 * d))) / (2.0 * (w * (D * ((w * h) * D)))));
	else
		tmp = (1.0 / d) * (D * ((h * (D * (M * M))) / (d * 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(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 * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(c0 * N[(N[(2.0 * N[(d * N[(c0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[(w * N[(D * N[(N[(w * h), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / d), $MachinePrecision] * N[(D * N[(N[(h * N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

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 67.9%

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

   3. Taylor expanded in c0 around inf

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)} \]
   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. /-lowering-/.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 71.1

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

   5. Simplified 71.1%

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

      1. frac-times N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. associate-*l* N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

      18. *-lowering-*.f64 N/A

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

      19. *-commutative N/A

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

      20. *-lowering-*.f64 N/A

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

      21. *-lowering-*.f64 70.2

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

   7. Applied egg-rr 70.2%

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

   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. Step-by-step derivation

      1. +-commutative N/A

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

      2. frac-2neg N/A

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

      3. distribute-frac-neg N/A

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

      4. unsub-neg N/A

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

      5. distribute-frac-neg2 N/A

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

      6. --lowering--.f64 N/A

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

   4. Applied egg-rr 0.1%

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

   5. Taylor expanded in c0 around -inf

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{{d}^{2}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{{d}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)}{{d}^{2}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)}{{d}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\color{blue}{d \cdot d}} \]

      12. *-lowering-*.f64 41.9

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

   7. Simplified 41.9%

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

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{d \cdot d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}}} \]

      2. inv-pow N/A

         \[\leadsto \color{blue}{{\left(\frac{d \cdot d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\right)}^{-1}} \]

      3. associate-/l* N/A

         \[\leadsto {\color{blue}{\left(d \cdot \frac{d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\right)}}^{-1} \]

      4. unpow-prod-down N/A

         \[\leadsto \color{blue}{{d}^{-1} \cdot {\left(\frac{d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\right)}^{-1}} \]

      5. inv-pow N/A

         \[\leadsto \color{blue}{\frac{1}{d}} \cdot {\left(\frac{d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\right)}^{-1} \]

      6. inv-pow N/A

         \[\leadsto \frac{1}{d} \cdot \color{blue}{\frac{1}{\frac{d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}}} \]

      7. clear-num N/A

         \[\leadsto \frac{1}{d} \cdot \color{blue}{\frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{d} \cdot \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d}} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{d}} \cdot \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d} \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{d} \cdot \frac{\color{blue}{\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{1}{4}}}{d} \]

      11. associate-/l* N/A

          \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \left(\color{blue}{\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)} \cdot \frac{\frac{1}{4}}{d}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \left(\left(\color{blue}{\left(D \cdot D\right)} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \left(\left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot \left(M \cdot M\right)\right)}\right) \cdot \frac{\frac{1}{4}}{d}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \left(\left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right) \cdot \frac{\frac{1}{4}}{d}\right) \]

      17. /-lowering-/.f64 53.9

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

   9. Applied egg-rr 53.9%

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

       1. associate-*l* N/A

          \[\leadsto \frac{1}{d} \cdot \left(\color{blue}{\left(D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)} \cdot \frac{\frac{1}{4}}{d}\right) \]

       2. associate-*l* N/A

          \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(D \cdot \left(\left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right)\right)} \]

       3. *-commutative N/A

          \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(\left(\left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot D\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(\left(\left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot D\right)} \]

       5. clear-num N/A

          \[\leadsto \frac{1}{d} \cdot \left(\left(\left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \color{blue}{\frac{1}{\frac{d}{\frac{1}{4}}}}\right) \cdot D\right) \]

       6. un-div-inv N/A

          \[\leadsto \frac{1}{d} \cdot \left(\color{blue}{\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d}{\frac{1}{4}}}} \cdot D\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \left(\color{blue}{\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d}{\frac{1}{4}}}} \cdot D\right) \]

       8. *-commutative N/A

          \[\leadsto \frac{1}{d} \cdot \left(\frac{D \cdot \color{blue}{\left(\left(M \cdot M\right) \cdot h\right)}}{\frac{d}{\frac{1}{4}}} \cdot D\right) \]

       9. associate-*r* N/A

          \[\leadsto \frac{1}{d} \cdot \left(\frac{\color{blue}{\left(D \cdot \left(M \cdot M\right)\right) \cdot h}}{\frac{d}{\frac{1}{4}}} \cdot D\right) \]

       10. *-commutative N/A

           \[\leadsto \frac{1}{d} \cdot \left(\frac{\color{blue}{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}}{\frac{d}{\frac{1}{4}}} \cdot D\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \frac{1}{d} \cdot \left(\frac{\color{blue}{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}}{\frac{d}{\frac{1}{4}}} \cdot D\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \frac{1}{d} \cdot \left(\frac{h \cdot \color{blue}{\left(D \cdot \left(M \cdot M\right)\right)}}{\frac{d}{\frac{1}{4}}} \cdot D\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \frac{1}{d} \cdot \left(\frac{h \cdot \left(D \cdot \color{blue}{\left(M \cdot M\right)}\right)}{\frac{d}{\frac{1}{4}}} \cdot D\right) \]

       14. div-inv N/A

           \[\leadsto \frac{1}{d} \cdot \left(\frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{\color{blue}{d \cdot \frac{1}{\frac{1}{4}}}} \cdot D\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \frac{1}{d} \cdot \left(\frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{\color{blue}{d \cdot \frac{1}{\frac{1}{4}}}} \cdot D\right) \]

       16. metadata-eval 64.2

           \[\leadsto \frac{1}{d} \cdot \left(\frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d \cdot \color{blue}{4}} \cdot D\right) \]

   11. Applied egg-rr 64.2%

       \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(\frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d \cdot 4} \cdot D\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;c0 \cdot \frac{2 \cdot \left(d \cdot \left(c0 \cdot d\right)\right)}{2 \cdot \left(w \cdot \left(D \cdot \left(\left(w \cdot h\right) \cdot D\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(D \cdot \frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d \cdot 4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c0 \cdot \left(d \cdot d\right)\\ t_1 := \frac{t\_0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0 \cdot t\_0}{D \cdot \left(D \cdot \left(w \cdot \left(w \cdot h\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(D \cdot \frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d \cdot 4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* c0 (* d d))) (t_1 (/ t_0 (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))
        INFINITY)
     (/ (* c0 t_0) (* D (* D (* w (* w h)))))
     (* (/ 1.0 d) (* D (/ (* h (* D (* M M))) (* d 4.0)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 * (d * d);
	double t_1 = t_0 / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = (c0 * t_0) / (D * (D * (w * (w * h))));
	} else {
		tmp = (1.0 / d) * (D * ((h * (D * (M * M))) / (d * 4.0)));
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 * (d * d);
	double t_1 = t_0 / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = (c0 * t_0) / (D * (D * (w * (w * h))));
	} else {
		tmp = (1.0 / d) * (D * ((h * (D * (M * M))) / (d * 4.0)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 * (d * d)
	t_1 = t_0 / ((w * h) * (D * D))
	tmp = 0
	if ((c0 / (2.0 * w)) * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))) <= math.inf:
		tmp = (c0 * t_0) / (D * (D * (w * (w * h))))
	else:
		tmp = (1.0 / d) * (D * ((h * (D * (M * M))) / (d * 4.0)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 * Float64(d * d))
	t_1 = Float64(t_0 / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))))) <= Inf)
		tmp = Float64(Float64(c0 * t_0) / Float64(D * Float64(D * Float64(w * Float64(w * h)))));
	else
		tmp = Float64(Float64(1.0 / d) * Float64(D * Float64(Float64(h * Float64(D * Float64(M * M))) / Float64(d * 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 * (d * d);
	t_1 = t_0 / ((w * h) * (D * D));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= Inf)
		tmp = (c0 * t_0) / (D * (D * (w * (w * h))));
	else
		tmp = (1.0 / d) * (D * ((h * (D * (M * M))) / (d * 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / 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$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(c0 * t$95$0), $MachinePrecision] / N[(D * N[(D * N[(w * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / d), $MachinePrecision] * N[(D * N[(N[(h * N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

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 67.9%

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

   3. Taylor expanded in c0 around inf

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)} \]
   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. /-lowering-/.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 71.1

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

   5. Simplified 71.1%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. /-lowering-/.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

   7. Applied egg-rr 76.7%

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 77.8

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

   9. Applied egg-rr 77.8%

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

   10. Taylor expanded in c0 around 0

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

       1. /-lowering-/.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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

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

       9. associate-*l* N/A

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

       10. *-lowering-*.f64 N/A

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

       11. *-lowering-*.f64 N/A

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

       12. unpow2 N/A

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

       13. associate-*r* N/A

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

       14. *-commutative N/A

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

       15. *-lowering-*.f64 N/A

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

       16. *-lowering-*.f64 60.6

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

   12. Simplified 60.6%

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

   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. Step-by-step derivation

      1. +-commutative N/A

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

      2. frac-2neg N/A

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

      3. distribute-frac-neg N/A

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

      4. unsub-neg N/A

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

      5. distribute-frac-neg2 N/A

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

      6. --lowering--.f64 N/A

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

   4. Applied egg-rr 0.1%

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

   5. Taylor expanded in c0 around -inf

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{{d}^{2}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{{d}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)}{{d}^{2}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)}{{d}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\color{blue}{d \cdot d}} \]

      12. *-lowering-*.f64 41.9

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

   7. Simplified 41.9%

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

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{d \cdot d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}}} \]

      2. inv-pow N/A

         \[\leadsto \color{blue}{{\left(\frac{d \cdot d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\right)}^{-1}} \]

      3. associate-/l* N/A

         \[\leadsto {\color{blue}{\left(d \cdot \frac{d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\right)}}^{-1} \]

      4. unpow-prod-down N/A

         \[\leadsto \color{blue}{{d}^{-1} \cdot {\left(\frac{d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\right)}^{-1}} \]

      5. inv-pow N/A

         \[\leadsto \color{blue}{\frac{1}{d}} \cdot {\left(\frac{d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\right)}^{-1} \]

      6. inv-pow N/A

         \[\leadsto \frac{1}{d} \cdot \color{blue}{\frac{1}{\frac{d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}}} \]

      7. clear-num N/A

         \[\leadsto \frac{1}{d} \cdot \color{blue}{\frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{d} \cdot \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d}} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{d}} \cdot \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d} \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{d} \cdot \frac{\color{blue}{\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{1}{4}}}{d} \]

      11. associate-/l* N/A

          \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \left(\color{blue}{\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)} \cdot \frac{\frac{1}{4}}{d}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \left(\left(\color{blue}{\left(D \cdot D\right)} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \left(\left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot \left(M \cdot M\right)\right)}\right) \cdot \frac{\frac{1}{4}}{d}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \left(\left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right) \cdot \frac{\frac{1}{4}}{d}\right) \]

      17. /-lowering-/.f64 53.9

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

   9. Applied egg-rr 53.9%

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

       1. associate-*l* N/A

          \[\leadsto \frac{1}{d} \cdot \left(\color{blue}{\left(D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)} \cdot \frac{\frac{1}{4}}{d}\right) \]

       2. associate-*l* N/A

          \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(D \cdot \left(\left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right)\right)} \]

       3. *-commutative N/A

          \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(\left(\left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot D\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(\left(\left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot D\right)} \]

       5. clear-num N/A

          \[\leadsto \frac{1}{d} \cdot \left(\left(\left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \color{blue}{\frac{1}{\frac{d}{\frac{1}{4}}}}\right) \cdot D\right) \]

       6. un-div-inv N/A

          \[\leadsto \frac{1}{d} \cdot \left(\color{blue}{\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d}{\frac{1}{4}}}} \cdot D\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \left(\color{blue}{\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d}{\frac{1}{4}}}} \cdot D\right) \]

       8. *-commutative N/A

          \[\leadsto \frac{1}{d} \cdot \left(\frac{D \cdot \color{blue}{\left(\left(M \cdot M\right) \cdot h\right)}}{\frac{d}{\frac{1}{4}}} \cdot D\right) \]

       9. associate-*r* N/A

          \[\leadsto \frac{1}{d} \cdot \left(\frac{\color{blue}{\left(D \cdot \left(M \cdot M\right)\right) \cdot h}}{\frac{d}{\frac{1}{4}}} \cdot D\right) \]

       10. *-commutative N/A

           \[\leadsto \frac{1}{d} \cdot \left(\frac{\color{blue}{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}}{\frac{d}{\frac{1}{4}}} \cdot D\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \frac{1}{d} \cdot \left(\frac{\color{blue}{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}}{\frac{d}{\frac{1}{4}}} \cdot D\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \frac{1}{d} \cdot \left(\frac{h \cdot \color{blue}{\left(D \cdot \left(M \cdot M\right)\right)}}{\frac{d}{\frac{1}{4}}} \cdot D\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \frac{1}{d} \cdot \left(\frac{h \cdot \left(D \cdot \color{blue}{\left(M \cdot M\right)}\right)}{\frac{d}{\frac{1}{4}}} \cdot D\right) \]

       14. div-inv N/A

           \[\leadsto \frac{1}{d} \cdot \left(\frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{\color{blue}{d \cdot \frac{1}{\frac{1}{4}}}} \cdot D\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \frac{1}{d} \cdot \left(\frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{\color{blue}{d \cdot \frac{1}{\frac{1}{4}}}} \cdot D\right) \]

       16. metadata-eval 64.2

           \[\leadsto \frac{1}{d} \cdot \left(\frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d \cdot \color{blue}{4}} \cdot D\right) \]

   11. Applied egg-rr 64.2%

       \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(\frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d \cdot 4} \cdot D\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{D \cdot \left(D \cdot \left(w \cdot \left(w \cdot h\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(D \cdot \frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d \cdot 4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c0 \cdot \left(d \cdot d\right)\\ t_1 := \frac{t\_0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0 \cdot t\_0}{D \cdot \left(D \cdot \left(w \cdot \left(w \cdot h\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(D \cdot \left(D \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot 4}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* c0 (* d d))) (t_1 (/ t_0 (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))
        INFINITY)
     (/ (* c0 t_0) (* D (* D (* w (* w h)))))
     (* (/ 1.0 d) (* D (* D (/ (* h (* M M)) (* d 4.0))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 * (d * d);
	double t_1 = t_0 / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = (c0 * t_0) / (D * (D * (w * (w * h))));
	} else {
		tmp = (1.0 / d) * (D * (D * ((h * (M * M)) / (d * 4.0))));
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 * (d * d);
	double t_1 = t_0 / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = (c0 * t_0) / (D * (D * (w * (w * h))));
	} else {
		tmp = (1.0 / d) * (D * (D * ((h * (M * M)) / (d * 4.0))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 * (d * d)
	t_1 = t_0 / ((w * h) * (D * D))
	tmp = 0
	if ((c0 / (2.0 * w)) * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))) <= math.inf:
		tmp = (c0 * t_0) / (D * (D * (w * (w * h))))
	else:
		tmp = (1.0 / d) * (D * (D * ((h * (M * M)) / (d * 4.0))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 * Float64(d * d))
	t_1 = Float64(t_0 / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))))) <= Inf)
		tmp = Float64(Float64(c0 * t_0) / Float64(D * Float64(D * Float64(w * Float64(w * h)))));
	else
		tmp = Float64(Float64(1.0 / d) * Float64(D * Float64(D * Float64(Float64(h * Float64(M * M)) / Float64(d * 4.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 * (d * d);
	t_1 = t_0 / ((w * h) * (D * D));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= Inf)
		tmp = (c0 * t_0) / (D * (D * (w * (w * h))));
	else
		tmp = (1.0 / d) * (D * (D * ((h * (M * M)) / (d * 4.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / 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$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(c0 * t$95$0), $MachinePrecision] / N[(D * N[(D * N[(w * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / d), $MachinePrecision] * N[(D * N[(D * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

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 67.9%

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

   3. Taylor expanded in c0 around inf

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)} \]
   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. /-lowering-/.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 71.1

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

   5. Simplified 71.1%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. /-lowering-/.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

   7. Applied egg-rr 76.7%

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 77.8

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

   9. Applied egg-rr 77.8%

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

   10. Taylor expanded in c0 around 0

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

       1. /-lowering-/.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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

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

       9. associate-*l* N/A

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

       10. *-lowering-*.f64 N/A

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

       11. *-lowering-*.f64 N/A

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

       12. unpow2 N/A

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

       13. associate-*r* N/A

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

       14. *-commutative N/A

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

       15. *-lowering-*.f64 N/A

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

       16. *-lowering-*.f64 60.6

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

   12. Simplified 60.6%

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

   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. Step-by-step derivation

      1. +-commutative N/A

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

      2. frac-2neg N/A

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

      3. distribute-frac-neg N/A

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

      4. unsub-neg N/A

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

      5. distribute-frac-neg2 N/A

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

      6. --lowering--.f64 N/A

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

   4. Applied egg-rr 0.1%

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

   5. Taylor expanded in c0 around -inf

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{{d}^{2}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{{d}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)}{{d}^{2}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)}{{d}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\color{blue}{d \cdot d}} \]

      12. *-lowering-*.f64 41.9

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

   7. Simplified 41.9%

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

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{d \cdot d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}}} \]

      2. inv-pow N/A

         \[\leadsto \color{blue}{{\left(\frac{d \cdot d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\right)}^{-1}} \]

      3. associate-/l* N/A

         \[\leadsto {\color{blue}{\left(d \cdot \frac{d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\right)}}^{-1} \]

      4. unpow-prod-down N/A

         \[\leadsto \color{blue}{{d}^{-1} \cdot {\left(\frac{d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\right)}^{-1}} \]

      5. inv-pow N/A

         \[\leadsto \color{blue}{\frac{1}{d}} \cdot {\left(\frac{d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\right)}^{-1} \]

      6. inv-pow N/A

         \[\leadsto \frac{1}{d} \cdot \color{blue}{\frac{1}{\frac{d}{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}}} \]

      7. clear-num N/A

         \[\leadsto \frac{1}{d} \cdot \color{blue}{\frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{d} \cdot \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d}} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{d}} \cdot \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d} \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{d} \cdot \frac{\color{blue}{\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{1}{4}}}{d} \]

      11. associate-/l* N/A

          \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \left(\color{blue}{\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)} \cdot \frac{\frac{1}{4}}{d}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \left(\left(\color{blue}{\left(D \cdot D\right)} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \left(\left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot \left(M \cdot M\right)\right)}\right) \cdot \frac{\frac{1}{4}}{d}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \left(\left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right) \cdot \frac{\frac{1}{4}}{d}\right) \]

      17. /-lowering-/.f64 53.9

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

   9. Applied egg-rr 53.9%

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

       1. associate-*l* N/A

          \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(\left(D \cdot D\right) \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \frac{\frac{1}{4}}{d}\right)\right)} \]

       2. associate-*l* N/A

          \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(D \cdot \left(D \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \frac{\frac{1}{4}}{d}\right)\right)\right)} \]

       3. *-commutative N/A

          \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(\left(D \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \frac{\frac{1}{4}}{d}\right)\right) \cdot D\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(\left(D \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \frac{\frac{1}{4}}{d}\right)\right) \cdot D\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \left(\color{blue}{\left(D \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \frac{\frac{1}{4}}{d}\right)\right)} \cdot D\right) \]

       6. clear-num N/A

          \[\leadsto \frac{1}{d} \cdot \left(\left(D \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \color{blue}{\frac{1}{\frac{d}{\frac{1}{4}}}}\right)\right) \cdot D\right) \]

       7. un-div-inv N/A

          \[\leadsto \frac{1}{d} \cdot \left(\left(D \cdot \color{blue}{\frac{h \cdot \left(M \cdot M\right)}{\frac{d}{\frac{1}{4}}}}\right) \cdot D\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \left(\left(D \cdot \color{blue}{\frac{h \cdot \left(M \cdot M\right)}{\frac{d}{\frac{1}{4}}}}\right) \cdot D\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{d} \cdot \left(\left(D \cdot \frac{\color{blue}{h \cdot \left(M \cdot M\right)}}{\frac{d}{\frac{1}{4}}}\right) \cdot D\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \frac{1}{d} \cdot \left(\left(D \cdot \frac{h \cdot \color{blue}{\left(M \cdot M\right)}}{\frac{d}{\frac{1}{4}}}\right) \cdot D\right) \]

       11. div-inv N/A

           \[\leadsto \frac{1}{d} \cdot \left(\left(D \cdot \frac{h \cdot \left(M \cdot M\right)}{\color{blue}{d \cdot \frac{1}{\frac{1}{4}}}}\right) \cdot D\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \frac{1}{d} \cdot \left(\left(D \cdot \frac{h \cdot \left(M \cdot M\right)}{\color{blue}{d \cdot \frac{1}{\frac{1}{4}}}}\right) \cdot D\right) \]

       13. metadata-eval 63.1

           \[\leadsto \frac{1}{d} \cdot \left(\left(D \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot \color{blue}{4}}\right) \cdot D\right) \]

   11. Applied egg-rr 63.1%

       \[\leadsto \frac{1}{d} \cdot \color{blue}{\left(\left(D \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot 4}\right) \cdot D\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{D \cdot \left(D \cdot \left(w \cdot \left(w \cdot h\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(D \cdot \left(D \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot 4}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c0 \cdot \left(d \cdot d\right)\\ t_1 := \frac{t\_0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0 \cdot t\_0}{D \cdot \left(D \cdot \left(w \cdot \left(w \cdot h\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot \left(\frac{h \cdot M}{d} \cdot \frac{M}{d}\right)\\ \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 * (d * d);
	t_1 = t_0 / ((w * h) * (D * D));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= Inf)
		tmp = (c0 * t_0) / (D * (D * (w * (w * h))));
	else
		tmp = ((D * D) * 0.25) * (((h * M) / d) * (M / d));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / 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$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(c0 * t$95$0), $MachinePrecision] / N[(D * N[(D * N[(w * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(D * D), $MachinePrecision] * 0.25), $MachinePrecision] * N[(N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

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 67.9%

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

   3. Taylor expanded in c0 around inf

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)} \]
   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. /-lowering-/.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 71.1

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

   5. Simplified 71.1%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. /-lowering-/.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

   7. Applied egg-rr 76.7%

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 77.8

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

   9. Applied egg-rr 77.8%

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

   10. Taylor expanded in c0 around 0

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

       1. /-lowering-/.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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

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

       9. associate-*l* N/A

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

       10. *-lowering-*.f64 N/A

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

       11. *-lowering-*.f64 N/A

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

       12. unpow2 N/A

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

       13. associate-*r* N/A

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

       14. *-commutative N/A

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

       15. *-lowering-*.f64 N/A

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

       16. *-lowering-*.f64 60.6

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

   12. Simplified 60.6%

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

   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. Step-by-step derivation

      1. +-commutative N/A

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

      2. frac-2neg N/A

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

      3. distribute-frac-neg N/A

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

      4. unsub-neg N/A

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

      5. distribute-frac-neg2 N/A

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

      6. --lowering--.f64 N/A

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

   4. Applied egg-rr 0.1%

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

   5. Taylor expanded in c0 around -inf

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{{d}^{2}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{{d}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)}{{d}^{2}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)}{{d}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\color{blue}{d \cdot d}} \]

      12. *-lowering-*.f64 41.9

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

   7. Simplified 41.9%

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

      1. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}}{d \cdot d} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right)} \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\frac{h \cdot \left(M \cdot M\right)}{d \cdot d}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{\color{blue}{h \cdot \left(M \cdot M\right)}}{d \cdot d} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{h \cdot \color{blue}{\left(M \cdot M\right)}}{d \cdot d} \]

      9. *-lowering-*.f64 41.3

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

   9. Applied egg-rr 41.3%

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

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{\color{blue}{\left(h \cdot M\right) \cdot M}}{d \cdot d} \]

       2. times-frac N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\left(\frac{h \cdot M}{d} \cdot \frac{M}{d}\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\left(\frac{h \cdot M}{d} \cdot \frac{M}{d}\right)} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \left(\color{blue}{\frac{h \cdot M}{d}} \cdot \frac{M}{d}\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \left(\frac{\color{blue}{h \cdot M}}{d} \cdot \frac{M}{d}\right) \]

       6. /-lowering-/.f64 57.3

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

   11. Applied egg-rr 57.3%

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

4. Final simplification 58.4%

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

Alternative 10: 56.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c0 \cdot \left(d \cdot d\right)\\ t_1 := \frac{t\_0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0 \cdot t\_0}{D \cdot \left(D \cdot \left(w \cdot \left(w \cdot h\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{D \cdot \left(\left(h \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot 0.25\right)}{d \cdot d}\\ \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 * (d * d);
	t_1 = t_0 / ((w * h) * (D * D));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= Inf)
		tmp = (c0 * t_0) / (D * (D * (w * (w * h))));
	else
		tmp = (D * ((h * (D * (M * M))) * 0.25)) / (d * d);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / 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$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(c0 * t$95$0), $MachinePrecision] / N[(D * N[(D * N[(w * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(D * N[(N[(h * N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]]]]

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 67.9%

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

   3. Taylor expanded in c0 around inf

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)} \]
   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. /-lowering-/.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 71.1

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

   5. Simplified 71.1%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. /-lowering-/.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

   7. Applied egg-rr 76.7%

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 77.8

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

   9. Applied egg-rr 77.8%

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

   10. Taylor expanded in c0 around 0

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

       1. /-lowering-/.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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

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

       9. associate-*l* N/A

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

       10. *-lowering-*.f64 N/A

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

       11. *-lowering-*.f64 N/A

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

       12. unpow2 N/A

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

       13. associate-*r* N/A

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

       14. *-commutative N/A

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

       15. *-lowering-*.f64 N/A

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

       16. *-lowering-*.f64 60.6

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

   12. Simplified 60.6%

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

   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. Step-by-step derivation

      1. +-commutative N/A

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

      2. frac-2neg N/A

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

      3. distribute-frac-neg N/A

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

      4. unsub-neg N/A

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

      5. distribute-frac-neg2 N/A

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

      6. --lowering--.f64 N/A

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

   4. Applied egg-rr 0.1%

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

   5. Taylor expanded in c0 around -inf

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{{d}^{2}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{{d}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)}{{d}^{2}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)}{{d}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\color{blue}{d \cdot d}} \]

      12. *-lowering-*.f64 41.9

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

   7. Simplified 41.9%

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

      1. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}}{d \cdot d} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right)} \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\frac{h \cdot \left(M \cdot M\right)}{d \cdot d}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{\color{blue}{h \cdot \left(M \cdot M\right)}}{d \cdot d} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{h \cdot \color{blue}{\left(M \cdot M\right)}}{d \cdot d} \]

      9. *-lowering-*.f64 41.3

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

   9. Applied egg-rr 41.3%

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

   10. Taylor expanded in D around 0

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

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

       3. *-commutative N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2}} \]

       4. unpow2 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2}} \]

       5. associate-*r* N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2}} \]

       6. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{1}{4} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2}} \]

       7. *-commutative N/A

          \[\leadsto \frac{\color{blue}{D \cdot \left(\frac{1}{4} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2}} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \frac{\color{blue}{D \cdot \left(\frac{1}{4} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2}} \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \frac{D \cdot \color{blue}{\left(\frac{1}{4} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2}} \]

       10. *-commutative N/A

           \[\leadsto \frac{D \cdot \left(\frac{1}{4} \cdot \left(\color{blue}{\left(h \cdot {M}^{2}\right)} \cdot D\right)\right)}{{d}^{2}} \]

       11. associate-*l* N/A

           \[\leadsto \frac{D \cdot \left(\frac{1}{4} \cdot \color{blue}{\left(h \cdot \left({M}^{2} \cdot D\right)\right)}\right)}{{d}^{2}} \]

       12. *-commutative N/A

           \[\leadsto \frac{D \cdot \left(\frac{1}{4} \cdot \left(h \cdot \color{blue}{\left(D \cdot {M}^{2}\right)}\right)\right)}{{d}^{2}} \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \frac{D \cdot \left(\frac{1}{4} \cdot \color{blue}{\left(h \cdot \left(D \cdot {M}^{2}\right)\right)}\right)}{{d}^{2}} \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \frac{D \cdot \left(\frac{1}{4} \cdot \left(h \cdot \color{blue}{\left(D \cdot {M}^{2}\right)}\right)\right)}{{d}^{2}} \]

       15. unpow2 N/A

           \[\leadsto \frac{D \cdot \left(\frac{1}{4} \cdot \left(h \cdot \left(D \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2}} \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \frac{D \cdot \left(\frac{1}{4} \cdot \left(h \cdot \left(D \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2}} \]

       17. unpow2 N/A

           \[\leadsto \frac{D \cdot \left(\frac{1}{4} \cdot \left(h \cdot \left(D \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot d}} \]

       18. *-lowering-*.f64 48.7

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

   12. Simplified 48.7%

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

4. Final simplification 52.7%

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

Alternative 11: 53.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\left(d \cdot d\right) \cdot \frac{c0 \cdot c0}{D \cdot \left(h \cdot \left(D \cdot \left(w \cdot w\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{D \cdot \left(\left(h \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot 0.25\right)}{d \cdot d}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d 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)))))
        INFINITY)
     (* (* d d) (/ (* c0 c0) (* D (* h (* D (* w w))))))
     (/ (* D (* (* h (* D (* M M))) 0.25)) (* d d)))))

C

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

Java

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

Python

def code(c0, w, h, D, d, 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))))) <= math.inf:
		tmp = (d * d) * ((c0 * c0) / (D * (h * (D * (w * w)))))
	else:
		tmp = (D * ((h * (D * (M * M))) * 0.25)) / (d * d)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= Inf)
		tmp = (d * d) * ((c0 * c0) / (D * (h * (D * (w * w)))));
	else
		tmp = (D * ((h * (D * (M * M))) * 0.25)) / (d * d);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, 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 * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(d * d), $MachinePrecision] * N[(N[(c0 * c0), $MachinePrecision] / N[(D * N[(h * N[(D * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(D * N[(N[(h * N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]]]

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 67.9%

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

   3. Taylor expanded in c0 around inf

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)} \]
   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. /-lowering-/.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 71.1

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

   5. Simplified 71.1%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. /-lowering-/.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

   7. Applied egg-rr 76.7%

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

   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. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 54.8

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

   10. Simplified 54.8%

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

   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. Step-by-step derivation

      1. +-commutative N/A

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

      2. frac-2neg N/A

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

      3. distribute-frac-neg N/A

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

      4. unsub-neg N/A

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

      5. distribute-frac-neg2 N/A

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

      6. --lowering--.f64 N/A

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

   4. Applied egg-rr 0.1%

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

   5. Taylor expanded in c0 around -inf

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{{d}^{2}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{{d}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)}{{d}^{2}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)}{{d}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\color{blue}{d \cdot d}} \]

      12. *-lowering-*.f64 41.9

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

   7. Simplified 41.9%

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

      1. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}}{d \cdot d} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right)} \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\frac{h \cdot \left(M \cdot M\right)}{d \cdot d}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{\color{blue}{h \cdot \left(M \cdot M\right)}}{d \cdot d} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{h \cdot \color{blue}{\left(M \cdot M\right)}}{d \cdot d} \]

      9. *-lowering-*.f64 41.3

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

   9. Applied egg-rr 41.3%

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

   10. Taylor expanded in D around 0

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

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

       3. *-commutative N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2}} \]

       4. unpow2 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2}} \]

       5. associate-*r* N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2}} \]

       6. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{1}{4} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2}} \]

       7. *-commutative N/A

          \[\leadsto \frac{\color{blue}{D \cdot \left(\frac{1}{4} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2}} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \frac{\color{blue}{D \cdot \left(\frac{1}{4} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2}} \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \frac{D \cdot \color{blue}{\left(\frac{1}{4} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2}} \]

       10. *-commutative N/A

           \[\leadsto \frac{D \cdot \left(\frac{1}{4} \cdot \left(\color{blue}{\left(h \cdot {M}^{2}\right)} \cdot D\right)\right)}{{d}^{2}} \]

       11. associate-*l* N/A

           \[\leadsto \frac{D \cdot \left(\frac{1}{4} \cdot \color{blue}{\left(h \cdot \left({M}^{2} \cdot D\right)\right)}\right)}{{d}^{2}} \]

       12. *-commutative N/A

           \[\leadsto \frac{D \cdot \left(\frac{1}{4} \cdot \left(h \cdot \color{blue}{\left(D \cdot {M}^{2}\right)}\right)\right)}{{d}^{2}} \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \frac{D \cdot \left(\frac{1}{4} \cdot \color{blue}{\left(h \cdot \left(D \cdot {M}^{2}\right)\right)}\right)}{{d}^{2}} \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \frac{D \cdot \left(\frac{1}{4} \cdot \left(h \cdot \color{blue}{\left(D \cdot {M}^{2}\right)}\right)\right)}{{d}^{2}} \]

       15. unpow2 N/A

           \[\leadsto \frac{D \cdot \left(\frac{1}{4} \cdot \left(h \cdot \left(D \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2}} \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \frac{D \cdot \left(\frac{1}{4} \cdot \left(h \cdot \left(D \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2}} \]

       17. unpow2 N/A

           \[\leadsto \frac{D \cdot \left(\frac{1}{4} \cdot \left(h \cdot \left(D \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot d}} \]

       18. *-lowering-*.f64 48.7

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

   12. Simplified 48.7%

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

4. Final simplification 50.8%

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

Alternative 12: 40.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \cdot M \leq 10^{-133}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \cdot M \leq 2 \cdot 10^{+272}:\\ \;\;\;\;D \cdot \left(\frac{h \cdot \left(M \cdot M\right)}{d \cdot d} \cdot \left(D \cdot 0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot \left(\left(h \cdot M\right) \cdot \frac{M}{d \cdot d}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= (* M M) 1e-133)
   0.0
   (if (<= (* M M) 2e+272)
     (* D (* (/ (* h (* M M)) (* d d)) (* D 0.25)))
     (* (* (* D D) 0.25) (* (* h M) (/ M (* d d)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((M * M) <= 1e-133) {
		tmp = 0.0;
	} else if ((M * M) <= 2e+272) {
		tmp = D * (((h * (M * M)) / (d * d)) * (D * 0.25));
	} else {
		tmp = ((D * D) * 0.25) * ((h * M) * (M / (d * d)));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((m * m) <= 1d-133) then
        tmp = 0.0d0
    else if ((m * m) <= 2d+272) then
        tmp = d * (((h * (m * m)) / (d_1 * d_1)) * (d * 0.25d0))
    else
        tmp = ((d * d) * 0.25d0) * ((h * m) * (m / (d_1 * d_1)))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((M * M) <= 1e-133) {
		tmp = 0.0;
	} else if ((M * M) <= 2e+272) {
		tmp = D * (((h * (M * M)) / (d * d)) * (D * 0.25));
	} else {
		tmp = ((D * D) * 0.25) * ((h * M) * (M / (d * d)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (M * M) <= 1e-133:
		tmp = 0.0
	elif (M * M) <= 2e+272:
		tmp = D * (((h * (M * M)) / (d * d)) * (D * 0.25))
	else:
		tmp = ((D * D) * 0.25) * ((h * M) * (M / (d * d)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (Float64(M * M) <= 1e-133)
		tmp = 0.0;
	elseif (Float64(M * M) <= 2e+272)
		tmp = Float64(D * Float64(Float64(Float64(h * Float64(M * M)) / Float64(d * d)) * Float64(D * 0.25)));
	else
		tmp = Float64(Float64(Float64(D * D) * 0.25) * Float64(Float64(h * M) * Float64(M / Float64(d * d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((M * M) <= 1e-133)
		tmp = 0.0;
	elseif ((M * M) <= 2e+272)
		tmp = D * (((h * (M * M)) / (d * d)) * (D * 0.25));
	else
		tmp = ((D * D) * 0.25) * ((h * M) * (M / (d * d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[N[(M * M), $MachinePrecision], 1e-133], 0.0, If[LessEqual[N[(M * M), $MachinePrecision], 2e+272], N[(D * N[(N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * N[(D * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(D * D), $MachinePrecision] * 0.25), $MachinePrecision] * N[(N[(h * M), $MachinePrecision] * N[(M / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 M M) < 1.0000000000000001e-133

   1. Initial program 30.2%

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

   3. Taylor expanded in c0 around -inf

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

      1. associate-/l* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. div0 N/A

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

      6. mul0-rgt N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{0} \]

      7. metadata-eval 46.1

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

   5. Simplified 46.1%

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

   if 1.0000000000000001e-133 < (*.f64 M M) < 2.0000000000000001e272

   1. Initial program 25.7%

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

      1. +-commutative N/A

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

      2. frac-2neg N/A

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

      3. distribute-frac-neg N/A

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

      4. unsub-neg N/A

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

      5. distribute-frac-neg2 N/A

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

      6. --lowering--.f64 N/A

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

   4. Applied egg-rr 19.0%

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

   5. Taylor expanded in c0 around -inf

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{{d}^{2}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{{d}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)}{{d}^{2}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)}{{d}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\color{blue}{d \cdot d}} \]

      12. *-lowering-*.f64 27.5

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

   7. Simplified 27.5%

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

      1. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}}{d \cdot d} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right)} \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\frac{h \cdot \left(M \cdot M\right)}{d \cdot d}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{\color{blue}{h \cdot \left(M \cdot M\right)}}{d \cdot d} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{h \cdot \color{blue}{\left(M \cdot M\right)}}{d \cdot d} \]

      9. *-lowering-*.f64 28.7

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

   9. Applied egg-rr 28.7%

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

       1. *-commutative N/A

          \[\leadsto \color{blue}{\frac{h \cdot \left(M \cdot M\right)}{d \cdot d} \cdot \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right)} \]

       2. associate-*r* N/A

          \[\leadsto \frac{h \cdot \left(M \cdot M\right)}{d \cdot d} \cdot \color{blue}{\left(\left(\frac{1}{4} \cdot D\right) \cdot D\right)} \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{h \cdot \left(M \cdot M\right)}{d \cdot d} \cdot \left(\frac{1}{4} \cdot D\right)\right) \cdot D} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{h \cdot \left(M \cdot M\right)}{d \cdot d} \cdot \left(\frac{1}{4} \cdot D\right)\right) \cdot D} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{h \cdot \left(M \cdot M\right)}{d \cdot d} \cdot \left(\frac{1}{4} \cdot D\right)\right)} \cdot D \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{h \cdot \left(M \cdot M\right)}{d \cdot d}} \cdot \left(\frac{1}{4} \cdot D\right)\right) \cdot D \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \left(\frac{\color{blue}{h \cdot \left(M \cdot M\right)}}{d \cdot d} \cdot \left(\frac{1}{4} \cdot D\right)\right) \cdot D \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \left(\frac{h \cdot \color{blue}{\left(M \cdot M\right)}}{d \cdot d} \cdot \left(\frac{1}{4} \cdot D\right)\right) \cdot D \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \left(\frac{h \cdot \left(M \cdot M\right)}{\color{blue}{d \cdot d}} \cdot \left(\frac{1}{4} \cdot D\right)\right) \cdot D \]

       10. *-commutative N/A

           \[\leadsto \left(\frac{h \cdot \left(M \cdot M\right)}{d \cdot d} \cdot \color{blue}{\left(D \cdot \frac{1}{4}\right)}\right) \cdot D \]

       11. *-lowering-*.f64 33.4

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

   11. Applied egg-rr 33.4%

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

   if 2.0000000000000001e272 < (*.f64 M M)

   1. Initial program 3.6%

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

      1. +-commutative N/A

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

      2. frac-2neg N/A

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

      3. distribute-frac-neg N/A

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

      4. unsub-neg N/A

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

      5. distribute-frac-neg2 N/A

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

      6. --lowering--.f64 N/A

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

   4. Applied egg-rr 1.8%

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

   5. Taylor expanded in c0 around -inf

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{{d}^{2}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{{d}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)}{{d}^{2}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)}{{d}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\color{blue}{d \cdot d}} \]

      12. *-lowering-*.f64 20.9

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

   7. Simplified 20.9%

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

      1. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}}{d \cdot d} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right)} \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\frac{h \cdot \left(M \cdot M\right)}{d \cdot d}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{\color{blue}{h \cdot \left(M \cdot M\right)}}{d \cdot d} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{h \cdot \color{blue}{\left(M \cdot M\right)}}{d \cdot d} \]

      9. *-lowering-*.f64 20.6

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

   9. Applied egg-rr 20.6%

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

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{\color{blue}{\left(h \cdot M\right) \cdot M}}{d \cdot d} \]

       2. associate-/l* N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\left(\left(h \cdot M\right) \cdot \frac{M}{d \cdot d}\right)} \]

       3. *-commutative N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\left(\frac{M}{d \cdot d} \cdot \left(h \cdot M\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\left(\frac{M}{d \cdot d} \cdot \left(h \cdot M\right)\right)} \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \left(\color{blue}{\frac{M}{d \cdot d}} \cdot \left(h \cdot M\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \left(\frac{M}{\color{blue}{d \cdot d}} \cdot \left(h \cdot M\right)\right) \]

       7. *-lowering-*.f64 38.9

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

   11. Applied egg-rr 38.9%

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

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot M \leq 10^{-133}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \cdot M \leq 2 \cdot 10^{+272}:\\ \;\;\;\;D \cdot \left(\frac{h \cdot \left(M \cdot M\right)}{d \cdot d} \cdot \left(D \cdot 0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot \left(\left(h \cdot M\right) \cdot \frac{M}{d \cdot d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 37.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 3 \cdot 10^{-229}:\\ \;\;\;\;0\\ \mathbf{elif}\;D \leq 5.6 \cdot 10^{-131}:\\ \;\;\;\;\frac{D \cdot \left(\left(h \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot 0.25\right)}{d \cdot d}\\ \mathbf{elif}\;D \leq 3.4 \cdot 10^{+56}:\\ \;\;\;\;\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot \left(\left(h \cdot M\right) \cdot \frac{M}{d \cdot d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 \cdot \left(D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot d}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= D 3e-229)
   0.0
   (if (<= D 5.6e-131)
     (/ (* D (* (* h (* D (* M M))) 0.25)) (* d d))
     (if (<= D 3.4e+56)
       (* (* (* D D) 0.25) (* (* h M) (/ M (* d d))))
       (/ (* 0.25 (* D (* D (* h (* M M))))) (* d d))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (D <= 3e-229) {
		tmp = 0.0;
	} else if (D <= 5.6e-131) {
		tmp = (D * ((h * (D * (M * M))) * 0.25)) / (d * d);
	} else if (D <= 3.4e+56) {
		tmp = ((D * D) * 0.25) * ((h * M) * (M / (d * d)));
	} else {
		tmp = (0.25 * (D * (D * (h * (M * M))))) / (d * d);
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (d <= 3d-229) then
        tmp = 0.0d0
    else if (d <= 5.6d-131) then
        tmp = (d * ((h * (d * (m * m))) * 0.25d0)) / (d_1 * d_1)
    else if (d <= 3.4d+56) then
        tmp = ((d * d) * 0.25d0) * ((h * m) * (m / (d_1 * d_1)))
    else
        tmp = (0.25d0 * (d * (d * (h * (m * m))))) / (d_1 * d_1)
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (D <= 3e-229) {
		tmp = 0.0;
	} else if (D <= 5.6e-131) {
		tmp = (D * ((h * (D * (M * M))) * 0.25)) / (d * d);
	} else if (D <= 3.4e+56) {
		tmp = ((D * D) * 0.25) * ((h * M) * (M / (d * d)));
	} else {
		tmp = (0.25 * (D * (D * (h * (M * M))))) / (d * d);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if D <= 3e-229:
		tmp = 0.0
	elif D <= 5.6e-131:
		tmp = (D * ((h * (D * (M * M))) * 0.25)) / (d * d)
	elif D <= 3.4e+56:
		tmp = ((D * D) * 0.25) * ((h * M) * (M / (d * d)))
	else:
		tmp = (0.25 * (D * (D * (h * (M * M))))) / (d * d)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (D <= 3e-229)
		tmp = 0.0;
	elseif (D <= 5.6e-131)
		tmp = Float64(Float64(D * Float64(Float64(h * Float64(D * Float64(M * M))) * 0.25)) / Float64(d * d));
	elseif (D <= 3.4e+56)
		tmp = Float64(Float64(Float64(D * D) * 0.25) * Float64(Float64(h * M) * Float64(M / Float64(d * d))));
	else
		tmp = Float64(Float64(0.25 * Float64(D * Float64(D * Float64(h * Float64(M * M))))) / Float64(d * d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (D <= 3e-229)
		tmp = 0.0;
	elseif (D <= 5.6e-131)
		tmp = (D * ((h * (D * (M * M))) * 0.25)) / (d * d);
	elseif (D <= 3.4e+56)
		tmp = ((D * D) * 0.25) * ((h * M) * (M / (d * d)));
	else
		tmp = (0.25 * (D * (D * (h * (M * M))))) / (d * d);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[D, 3e-229], 0.0, If[LessEqual[D, 5.6e-131], N[(N[(D * N[(N[(h * N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[D, 3.4e+56], N[(N[(N[(D * D), $MachinePrecision] * 0.25), $MachinePrecision] * N[(N[(h * M), $MachinePrecision] * N[(M / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 * N[(D * N[(D * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if D < 3.00000000000000002e-229

   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

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

      1. associate-/l* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. div0 N/A

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

      6. mul0-rgt N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{0} \]

      7. metadata-eval 34.9

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

   5. Simplified 34.9%

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

   if 3.00000000000000002e-229 < D < 5.5999999999999999e-131

   1. Initial program 35.8%

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

      1. +-commutative N/A

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

      2. frac-2neg N/A

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

      3. distribute-frac-neg N/A

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

      4. unsub-neg N/A

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

      5. distribute-frac-neg2 N/A

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

      6. --lowering--.f64 N/A

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

   4. Applied egg-rr 25.2%

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

   5. Taylor expanded in c0 around -inf

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{{d}^{2}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{{d}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)}{{d}^{2}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)}{{d}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\color{blue}{d \cdot d}} \]

      12. *-lowering-*.f64 29.9

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

   7. Simplified 29.9%

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

      1. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}}{d \cdot d} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right)} \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\frac{h \cdot \left(M \cdot M\right)}{d \cdot d}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{\color{blue}{h \cdot \left(M \cdot M\right)}}{d \cdot d} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{h \cdot \color{blue}{\left(M \cdot M\right)}}{d \cdot d} \]

      9. *-lowering-*.f64 29.7

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

   9. Applied egg-rr 29.7%

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

   10. Taylor expanded in D around 0

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

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

       3. *-commutative N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2}} \]

       4. unpow2 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2}} \]

       5. associate-*r* N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2}} \]

       6. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{1}{4} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2}} \]

       7. *-commutative N/A

          \[\leadsto \frac{\color{blue}{D \cdot \left(\frac{1}{4} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2}} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \frac{\color{blue}{D \cdot \left(\frac{1}{4} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2}} \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \frac{D \cdot \color{blue}{\left(\frac{1}{4} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2}} \]

       10. *-commutative N/A

           \[\leadsto \frac{D \cdot \left(\frac{1}{4} \cdot \left(\color{blue}{\left(h \cdot {M}^{2}\right)} \cdot D\right)\right)}{{d}^{2}} \]

       11. associate-*l* N/A

           \[\leadsto \frac{D \cdot \left(\frac{1}{4} \cdot \color{blue}{\left(h \cdot \left({M}^{2} \cdot D\right)\right)}\right)}{{d}^{2}} \]

       12. *-commutative N/A

           \[\leadsto \frac{D \cdot \left(\frac{1}{4} \cdot \left(h \cdot \color{blue}{\left(D \cdot {M}^{2}\right)}\right)\right)}{{d}^{2}} \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \frac{D \cdot \left(\frac{1}{4} \cdot \color{blue}{\left(h \cdot \left(D \cdot {M}^{2}\right)\right)}\right)}{{d}^{2}} \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \frac{D \cdot \left(\frac{1}{4} \cdot \left(h \cdot \color{blue}{\left(D \cdot {M}^{2}\right)}\right)\right)}{{d}^{2}} \]

       15. unpow2 N/A

           \[\leadsto \frac{D \cdot \left(\frac{1}{4} \cdot \left(h \cdot \left(D \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2}} \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \frac{D \cdot \left(\frac{1}{4} \cdot \left(h \cdot \left(D \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2}} \]

       17. unpow2 N/A

           \[\leadsto \frac{D \cdot \left(\frac{1}{4} \cdot \left(h \cdot \left(D \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot d}} \]

       18. *-lowering-*.f64 37.0

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

   12. Simplified 37.0%

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

   if 5.5999999999999999e-131 < D < 3.40000000000000001e56

   1. Initial program 36.5%

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

      1. +-commutative N/A

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

      2. frac-2neg N/A

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

      3. distribute-frac-neg N/A

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

      4. unsub-neg N/A

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

      5. distribute-frac-neg2 N/A

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

      6. --lowering--.f64 N/A

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

   4. Applied egg-rr 28.4%

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

   5. Taylor expanded in c0 around -inf

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{{d}^{2}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{{d}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)}{{d}^{2}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)}{{d}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\color{blue}{d \cdot d}} \]

      12. *-lowering-*.f64 31.7

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

   7. Simplified 31.7%

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

      1. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}}{d \cdot d} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right)} \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\frac{h \cdot \left(M \cdot M\right)}{d \cdot d}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{\color{blue}{h \cdot \left(M \cdot M\right)}}{d \cdot d} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{h \cdot \color{blue}{\left(M \cdot M\right)}}{d \cdot d} \]

      9. *-lowering-*.f64 35.6

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

   9. Applied egg-rr 35.6%

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

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{\color{blue}{\left(h \cdot M\right) \cdot M}}{d \cdot d} \]

       2. associate-/l* N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\left(\left(h \cdot M\right) \cdot \frac{M}{d \cdot d}\right)} \]

       3. *-commutative N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\left(\frac{M}{d \cdot d} \cdot \left(h \cdot M\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\left(\frac{M}{d \cdot d} \cdot \left(h \cdot M\right)\right)} \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \left(\color{blue}{\frac{M}{d \cdot d}} \cdot \left(h \cdot M\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \left(\frac{M}{\color{blue}{d \cdot d}} \cdot \left(h \cdot M\right)\right) \]

       7. *-lowering-*.f64 45.4

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

   11. Applied egg-rr 45.4%

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

   if 3.40000000000000001e56 < D

   1. Initial program 7.6%

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

      1. +-commutative N/A

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

      2. frac-2neg N/A

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

      3. distribute-frac-neg N/A

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

      4. unsub-neg N/A

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

      5. distribute-frac-neg2 N/A

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

      6. --lowering--.f64 N/A

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

   4. Applied egg-rr 3.8%

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

   5. Taylor expanded in c0 around -inf

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{{d}^{2}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{{d}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)}{{d}^{2}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)}{{d}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\color{blue}{d \cdot d}} \]

      12. *-lowering-*.f64 16.9

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

   7. Simplified 16.9%

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

   8. Taylor expanded in D around 0

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

      1. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{d \cdot d} \]

      2. associate-*l* N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{\left(D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{d \cdot d} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{\left(D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{d \cdot d} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(D \cdot \color{blue}{\left(D \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{d \cdot d} \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(D \cdot \left(D \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{d \cdot d} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(D \cdot \left(D \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{d \cdot d} \]

      7. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(D \cdot \left(D \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{d \cdot d} \]

      8. *-lowering-*.f64 27.4

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

   10. Simplified 27.4%

       \[\leadsto \frac{0.25 \cdot \color{blue}{\left(D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}}{d \cdot d} \]
3. Recombined 4 regimes into one program.

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 3 \cdot 10^{-229}:\\ \;\;\;\;0\\ \mathbf{elif}\;D \leq 5.6 \cdot 10^{-131}:\\ \;\;\;\;\frac{D \cdot \left(\left(h \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot 0.25\right)}{d \cdot d}\\ \mathbf{elif}\;D \leq 3.4 \cdot 10^{+56}:\\ \;\;\;\;\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot \left(\left(h \cdot M\right) \cdot \frac{M}{d \cdot d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 \cdot \left(D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot d}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 36.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.25 \cdot \left(D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot d}\\ \mathbf{if}\;D \leq 1.9 \cdot 10^{-228}:\\ \;\;\;\;0\\ \mathbf{elif}\;D \leq 5 \cdot 10^{-131}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;D \leq 4.2 \cdot 10^{+56}:\\ \;\;\;\;\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot \left(\left(h \cdot M\right) \cdot \frac{M}{d \cdot d}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* 0.25 (* D (* D (* h (* M M))))) (* d d))))
   (if (<= D 1.9e-228)
     0.0
     (if (<= D 5e-131)
       t_0
       (if (<= D 4.2e+56)
         (* (* (* D D) 0.25) (* (* h M) (/ M (* d d))))
         t_0)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (0.25 * (D * (D * (h * (M * M))))) / (d * d);
	double tmp;
	if (D <= 1.9e-228) {
		tmp = 0.0;
	} else if (D <= 5e-131) {
		tmp = t_0;
	} else if (D <= 4.2e+56) {
		tmp = ((D * D) * 0.25) * ((h * M) * (M / (d * d)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (0.25d0 * (d * (d * (h * (m * m))))) / (d_1 * d_1)
    if (d <= 1.9d-228) then
        tmp = 0.0d0
    else if (d <= 5d-131) then
        tmp = t_0
    else if (d <= 4.2d+56) then
        tmp = ((d * d) * 0.25d0) * ((h * m) * (m / (d_1 * d_1)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (0.25 * (D * (D * (h * (M * M))))) / (d * d);
	double tmp;
	if (D <= 1.9e-228) {
		tmp = 0.0;
	} else if (D <= 5e-131) {
		tmp = t_0;
	} else if (D <= 4.2e+56) {
		tmp = ((D * D) * 0.25) * ((h * M) * (M / (d * d)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (0.25 * (D * (D * (h * (M * M))))) / (d * d)
	tmp = 0
	if D <= 1.9e-228:
		tmp = 0.0
	elif D <= 5e-131:
		tmp = t_0
	elif D <= 4.2e+56:
		tmp = ((D * D) * 0.25) * ((h * M) * (M / (d * d)))
	else:
		tmp = t_0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(0.25 * Float64(D * Float64(D * Float64(h * Float64(M * M))))) / Float64(d * d))
	tmp = 0.0
	if (D <= 1.9e-228)
		tmp = 0.0;
	elseif (D <= 5e-131)
		tmp = t_0;
	elseif (D <= 4.2e+56)
		tmp = Float64(Float64(Float64(D * D) * 0.25) * Float64(Float64(h * M) * Float64(M / Float64(d * d))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (0.25 * (D * (D * (h * (M * M))))) / (d * d);
	tmp = 0.0;
	if (D <= 1.9e-228)
		tmp = 0.0;
	elseif (D <= 5e-131)
		tmp = t_0;
	elseif (D <= 4.2e+56)
		tmp = ((D * D) * 0.25) * ((h * M) * (M / (d * d)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(0.25 * N[(D * N[(D * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[D, 1.9e-228], 0.0, If[LessEqual[D, 5e-131], t$95$0, If[LessEqual[D, 4.2e+56], N[(N[(N[(D * D), $MachinePrecision] * 0.25), $MachinePrecision] * N[(N[(h * M), $MachinePrecision] * N[(M / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if D < 1.8999999999999999e-228

   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

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

      1. associate-/l* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. div0 N/A

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

      6. mul0-rgt N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{0} \]

      7. metadata-eval 34.9

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

   5. Simplified 34.9%

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

   if 1.8999999999999999e-228 < D < 5.0000000000000004e-131 or 4.20000000000000034e56 < D

   1. Initial program 21.0%

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

      1. +-commutative N/A

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

      2. frac-2neg N/A

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

      3. distribute-frac-neg N/A

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

      4. unsub-neg N/A

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

      5. distribute-frac-neg2 N/A

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

      6. --lowering--.f64 N/A

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

   4. Applied egg-rr 14.0%

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

   5. Taylor expanded in c0 around -inf

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{{d}^{2}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{{d}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)}{{d}^{2}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)}{{d}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\color{blue}{d \cdot d}} \]

      12. *-lowering-*.f64 23.1

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

   7. Simplified 23.1%

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

   8. Taylor expanded in D around 0

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

      1. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{d \cdot d} \]

      2. associate-*l* N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{\left(D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{d \cdot d} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{\left(D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{d \cdot d} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(D \cdot \color{blue}{\left(D \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{d \cdot d} \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(D \cdot \left(D \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{d \cdot d} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(D \cdot \left(D \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{d \cdot d} \]

      7. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(D \cdot \left(D \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{d \cdot d} \]

      8. *-lowering-*.f64 31.9

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

   10. Simplified 31.9%

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

   if 5.0000000000000004e-131 < D < 4.20000000000000034e56

   1. Initial program 36.5%

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

      1. +-commutative N/A

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

      2. frac-2neg N/A

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

      3. distribute-frac-neg N/A

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

      4. unsub-neg N/A

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

      5. distribute-frac-neg2 N/A

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

      6. --lowering--.f64 N/A

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

   4. Applied egg-rr 28.4%

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

   5. Taylor expanded in c0 around -inf

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{{d}^{2}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{{d}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)}{{d}^{2}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)}{{d}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\color{blue}{d \cdot d}} \]

      12. *-lowering-*.f64 31.7

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

   7. Simplified 31.7%

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

      1. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}}{d \cdot d} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right)} \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\frac{h \cdot \left(M \cdot M\right)}{d \cdot d}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{\color{blue}{h \cdot \left(M \cdot M\right)}}{d \cdot d} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{h \cdot \color{blue}{\left(M \cdot M\right)}}{d \cdot d} \]

      9. *-lowering-*.f64 35.6

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

   9. Applied egg-rr 35.6%

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

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{\color{blue}{\left(h \cdot M\right) \cdot M}}{d \cdot d} \]

       2. associate-/l* N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\left(\left(h \cdot M\right) \cdot \frac{M}{d \cdot d}\right)} \]

       3. *-commutative N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\left(\frac{M}{d \cdot d} \cdot \left(h \cdot M\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\left(\frac{M}{d \cdot d} \cdot \left(h \cdot M\right)\right)} \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \left(\color{blue}{\frac{M}{d \cdot d}} \cdot \left(h \cdot M\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \left(\frac{M}{\color{blue}{d \cdot d}} \cdot \left(h \cdot M\right)\right) \]

       7. *-lowering-*.f64 45.4

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

   11. Applied egg-rr 45.4%

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

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 1.9 \cdot 10^{-228}:\\ \;\;\;\;0\\ \mathbf{elif}\;D \leq 5 \cdot 10^{-131}:\\ \;\;\;\;\frac{0.25 \cdot \left(D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot d}\\ \mathbf{elif}\;D \leq 4.2 \cdot 10^{+56}:\\ \;\;\;\;\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot \left(\left(h \cdot M\right) \cdot \frac{M}{d \cdot d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 \cdot \left(D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot d}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 38.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \cdot M \leq 10^{-133}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot \left(\left(h \cdot M\right) \cdot \frac{M}{d \cdot d}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= (* M M) 1e-133) 0.0 (* (* (* D D) 0.25) (* (* h M) (/ M (* d d))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((M * M) <= 1e-133) {
		tmp = 0.0;
	} else {
		tmp = ((D * D) * 0.25) * ((h * M) * (M / (d * d)));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((m * m) <= 1d-133) then
        tmp = 0.0d0
    else
        tmp = ((d * d) * 0.25d0) * ((h * m) * (m / (d_1 * d_1)))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((M * M) <= 1e-133) {
		tmp = 0.0;
	} else {
		tmp = ((D * D) * 0.25) * ((h * M) * (M / (d * d)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (M * M) <= 1e-133:
		tmp = 0.0
	else:
		tmp = ((D * D) * 0.25) * ((h * M) * (M / (d * d)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (Float64(M * M) <= 1e-133)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(D * D) * 0.25) * Float64(Float64(h * M) * Float64(M / Float64(d * d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((M * M) <= 1e-133)
		tmp = 0.0;
	else
		tmp = ((D * D) * 0.25) * ((h * M) * (M / (d * d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[N[(M * M), $MachinePrecision], 1e-133], 0.0, N[(N[(N[(D * D), $MachinePrecision] * 0.25), $MachinePrecision] * N[(N[(h * M), $MachinePrecision] * N[(M / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M M) < 1.0000000000000001e-133

   1. Initial program 30.2%

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

   3. Taylor expanded in c0 around -inf

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

      1. associate-/l* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. div0 N/A

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

      6. mul0-rgt N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{0} \]

      7. metadata-eval 46.1

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

   5. Simplified 46.1%

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

   if 1.0000000000000001e-133 < (*.f64 M M)

   1. Initial program 16.7%

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

      1. +-commutative N/A

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

      2. frac-2neg N/A

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

      3. distribute-frac-neg N/A

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

      4. unsub-neg N/A

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

      5. distribute-frac-neg2 N/A

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

      6. --lowering--.f64 N/A

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

   4. Applied egg-rr 12.0%

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

   5. Taylor expanded in c0 around -inf

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{{d}^{2}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{{d}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)}{{d}^{2}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)}{{d}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\color{blue}{d \cdot d}} \]

      12. *-lowering-*.f64 24.8

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

   7. Simplified 24.8%

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

      1. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}}{d \cdot d} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right)} \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\frac{h \cdot \left(M \cdot M\right)}{d \cdot d}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{\color{blue}{h \cdot \left(M \cdot M\right)}}{d \cdot d} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{h \cdot \color{blue}{\left(M \cdot M\right)}}{d \cdot d} \]

      9. *-lowering-*.f64 25.4

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

   9. Applied egg-rr 25.4%

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

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{\color{blue}{\left(h \cdot M\right) \cdot M}}{d \cdot d} \]

       2. associate-/l* N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\left(\left(h \cdot M\right) \cdot \frac{M}{d \cdot d}\right)} \]

       3. *-commutative N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\left(\frac{M}{d \cdot d} \cdot \left(h \cdot M\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\left(\frac{M}{d \cdot d} \cdot \left(h \cdot M\right)\right)} \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \left(\color{blue}{\frac{M}{d \cdot d}} \cdot \left(h \cdot M\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \left(\frac{M}{\color{blue}{d \cdot d}} \cdot \left(h \cdot M\right)\right) \]

       7. *-lowering-*.f64 32.2

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

   11. Applied egg-rr 32.2%

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

4. Final simplification 38.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot M \leq 10^{-133}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot \left(\left(h \cdot M\right) \cdot \frac{M}{d \cdot d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 33.2% accurate, 156.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

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

C

double code(double c0, double w, double h, double D, double d, double M) {
	return 0.0;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = 0.0d0
end function

Java

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

Python

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

Julia

function code(c0, w, h, D, d, M)
	return 0.0
end

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := 0.0

Derivation

1. Initial program 23.1%

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

3. Taylor expanded in c0 around -inf

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

   1. associate-/l* N/A

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

   2. distribute-lft1-in N/A

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

   3. metadata-eval N/A

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

   4. mul0-lft N/A

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

   5. div0 N/A

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

   6. mul0-rgt N/A

      \[\leadsto \frac{-1}{2} \cdot \color{blue}{0} \]

   7. metadata-eval 31.1

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

5. Simplified 31.1%

   \[\leadsto \color{blue}{0} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024198 
(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))))))