Henrywood and Agarwal, Equation (13)

Percentage Accurate: 25.0% → 59.6%

Time: 18.3s

Alternatives: 7

Speedup: 156.0×

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: 25.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 59.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0 \cdot d}{w \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot d}{w \cdot h}\\ \mathbf{else}:\\ \;\;\;\;\frac{D \cdot \left(D \cdot \left(h \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right)\right)}{d \cdot d}\\ \end{array} \end{array} \]

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)
     (* (/ (* c0 d) (* w (* D D))) (/ (* c0 d) (* w h)))
     (/ (* D (* D (* h (* (* 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 = ((c0 * d) / (w * (D * D))) * ((c0 * d) / (w * h));
	} else {
		tmp = (D * (D * (h * ((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 = ((c0 * d) / (w * (D * D))) * ((c0 * d) / (w * h));
	} else {
		tmp = (D * (D * (h * ((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 = ((c0 * d) / (w * (D * D))) * ((c0 * d) / (w * h))
	else:
		tmp = (D * (D * (h * ((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(Float64(c0 * d) / Float64(w * Float64(D * D))) * Float64(Float64(c0 * d) / Float64(w * h)));
	else
		tmp = Float64(Float64(D * Float64(D * Float64(h * Float64(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 = ((c0 * d) / (w * (D * D))) * ((c0 * d) / (w * h));
	else
		tmp = (D * (D * (h * ((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[(N[(c0 * d), $MachinePrecision] / N[(w * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(c0 * d), $MachinePrecision] / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(D * N[(D * N[(h * N[(N[(M * M), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $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 76.8%

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

   3. Taylor expanded in c0 around inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-/l* N/A

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

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

         \[\leadsto \left(c0 \cdot c0\right) \cdot \color{blue}{\mathsf{fma}\left(d, \frac{d}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}, \frac{-1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]

   5. Simplified 56.9%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \mathsf{fma}\left(d, \frac{d}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}, \frac{-0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}\right)} \]

   6. Taylor expanded in d around inf

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 58.5

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

   8. Simplified 58.5%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

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

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

      5. *-commutative N/A

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

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

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

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

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

      8. *-lowering-*.f64 63.1

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

   10. Applied egg-rr 63.1%

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

       1. associate-*r/ N/A

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

       2. unswap-sqr N/A

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

       3. *-commutative N/A

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

       4. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

       12. *-lowering-*.f64 81.8

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

   12. Applied egg-rr 81.8%

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

   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. frac-times N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

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

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

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

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

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

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

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

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

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

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

      19. *-lowering-*.f64 0.7

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

   4. Applied egg-rr 0.7%

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

   5. Taylor expanded in c0 around -inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

   7. Simplified 16.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c0 \cdot c0, \left(\left(D \cdot D\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}\right) \cdot 0.25, 0\right)} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. 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. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

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

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

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

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

      15. unpow2 N/A

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f64 44.2

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

   10. Simplified 44.2%

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

4. Final simplification 56.7%

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

Alternative 2: 59.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\left(c0 \cdot d\right) \cdot \frac{c0 \cdot d}{w \cdot \left(D \cdot \left(\left(w \cdot h\right) \cdot D\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{D \cdot \left(D \cdot \left(h \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right)\right)}{d \cdot d}\\ \end{array} \end{array} \]

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)
     (* (* c0 d) (/ (* c0 d) (* w (* D (* (* w h) D)))))
     (/ (* D (* D (* h (* (* 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 = (c0 * d) * ((c0 * d) / (w * (D * ((w * h) * D))));
	} else {
		tmp = (D * (D * (h * ((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 = (c0 * d) * ((c0 * d) / (w * (D * ((w * h) * D))));
	} else {
		tmp = (D * (D * (h * ((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 = (c0 * d) * ((c0 * d) / (w * (D * ((w * h) * D))))
	else:
		tmp = (D * (D * (h * ((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(c0 * d) * Float64(Float64(c0 * d) / Float64(w * Float64(D * Float64(Float64(w * h) * D)))));
	else
		tmp = Float64(Float64(D * Float64(D * Float64(h * Float64(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 = (c0 * d) * ((c0 * d) / (w * (D * ((w * h) * D))));
	else
		tmp = (D * (D * (h * ((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[(c0 * d), $MachinePrecision] * N[(N[(c0 * d), $MachinePrecision] / N[(w * N[(D * N[(N[(w * h), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(D * N[(D * N[(h * N[(N[(M * M), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $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 76.8%

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

   3. Taylor expanded in c0 around inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-/l* N/A

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

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

         \[\leadsto \left(c0 \cdot c0\right) \cdot \color{blue}{\mathsf{fma}\left(d, \frac{d}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}, \frac{-1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]

   5. Simplified 56.9%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \mathsf{fma}\left(d, \frac{d}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}, \frac{-0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}\right)} \]

   6. Taylor expanded in d around inf

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 58.5

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

   8. Simplified 58.5%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

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

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

      5. *-commutative N/A

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

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

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

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

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

      8. *-lowering-*.f64 63.1

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

   10. Applied egg-rr 63.1%

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

       1. associate-*r/ N/A

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

       2. unswap-sqr N/A

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

       3. associate-/l* N/A

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

       4. *-commutative N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. associate-*r* N/A

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

       10. *-commutative N/A

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

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

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

   12. Applied egg-rr 80.3%

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

   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. frac-times N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

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

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

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

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

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

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

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

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

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

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

      19. *-lowering-*.f64 0.7

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

   4. Applied egg-rr 0.7%

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

   5. Taylor expanded in c0 around -inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

   7. Simplified 16.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c0 \cdot c0, \left(\left(D \cdot D\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}\right) \cdot 0.25, 0\right)} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. 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. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

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

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

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

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

      15. unpow2 N/A

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f64 44.2

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

   10. Simplified 44.2%

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

4. Final simplification 56.2%

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

Alternative 3: 57.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;c0 \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(w \cdot \left(h \cdot D\right)\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{D \cdot \left(D \cdot \left(h \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right)\right)}{d \cdot d}\\ \end{array} \end{array} \]

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)
     (* c0 (* c0 (* d (/ d (* D (* w (* w (* h D))))))))
     (/ (* D (* D (* h (* (* 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 = c0 * (c0 * (d * (d / (D * (w * (w * (h * D)))))));
	} else {
		tmp = (D * (D * (h * ((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 = c0 * (c0 * (d * (d / (D * (w * (w * (h * D)))))));
	} else {
		tmp = (D * (D * (h * ((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 = c0 * (c0 * (d * (d / (D * (w * (w * (h * D)))))))
	else:
		tmp = (D * (D * (h * ((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(c0 * Float64(c0 * Float64(d * Float64(d / Float64(D * Float64(w * Float64(w * Float64(h * D))))))));
	else
		tmp = Float64(Float64(D * Float64(D * Float64(h * Float64(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 = c0 * (c0 * (d * (d / (D * (w * (w * (h * D)))))));
	else
		tmp = (D * (D * (h * ((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[(c0 * N[(c0 * N[(d * N[(d / N[(D * N[(w * N[(w * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(D * N[(D * N[(h * N[(N[(M * M), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $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 76.8%

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

   3. Taylor expanded in c0 around inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-/l* N/A

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

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

         \[\leadsto \left(c0 \cdot c0\right) \cdot \color{blue}{\mathsf{fma}\left(d, \frac{d}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}, \frac{-1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]

   5. Simplified 56.9%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \mathsf{fma}\left(d, \frac{d}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}, \frac{-0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}\right)} \]

   6. Taylor expanded in d around inf

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 58.5

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

   8. Simplified 58.5%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

   10. Applied egg-rr 78.2%

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

   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. frac-times N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

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

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

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

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

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

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

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

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

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

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

      19. *-lowering-*.f64 0.7

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

   4. Applied egg-rr 0.7%

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

   5. Taylor expanded in c0 around -inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

   7. Simplified 16.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c0 \cdot c0, \left(\left(D \cdot D\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}\right) \cdot 0.25, 0\right)} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. 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. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

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

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

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

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

      15. unpow2 N/A

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f64 44.2

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

   10. Simplified 44.2%

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

4. Final simplification 55.5%

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

Alternative 4: 54.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left(w \cdot h\right) \cdot \left(w \cdot \left(D \cdot D\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{D \cdot \left(D \cdot \left(h \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right)\right)}{d \cdot d}\\ \end{array} \end{array} \]

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)
     (* (* c0 c0) (/ (* d d) (* (* w h) (* w (* D D)))))
     (/ (* D (* D (* h (* (* 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 = (c0 * c0) * ((d * d) / ((w * h) * (w * (D * D))));
	} else {
		tmp = (D * (D * (h * ((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 = (c0 * c0) * ((d * d) / ((w * h) * (w * (D * D))));
	} else {
		tmp = (D * (D * (h * ((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 = (c0 * c0) * ((d * d) / ((w * h) * (w * (D * D))))
	else:
		tmp = (D * (D * (h * ((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(c0 * c0) * Float64(Float64(d * d) / Float64(Float64(w * h) * Float64(w * Float64(D * D)))));
	else
		tmp = Float64(Float64(D * Float64(D * Float64(h * Float64(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 = (c0 * c0) * ((d * d) / ((w * h) * (w * (D * D))));
	else
		tmp = (D * (D * (h * ((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[(c0 * c0), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(w * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(D * N[(D * N[(h * N[(N[(M * M), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $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 76.8%

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

   3. Taylor expanded in c0 around inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-/l* N/A

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

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

         \[\leadsto \left(c0 \cdot c0\right) \cdot \color{blue}{\mathsf{fma}\left(d, \frac{d}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}, \frac{-1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]

   5. Simplified 56.9%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \mathsf{fma}\left(d, \frac{d}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}, \frac{-0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}\right)} \]

   6. Taylor expanded in d around inf

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 58.5

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

   8. Simplified 58.5%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

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

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

      5. *-commutative N/A

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

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

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

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

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

      8. *-lowering-*.f64 63.1

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

   10. Applied egg-rr 63.1%

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

   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. frac-times N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

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

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

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

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

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

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

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

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

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

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

      19. *-lowering-*.f64 0.7

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

   4. Applied egg-rr 0.7%

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

   5. Taylor expanded in c0 around -inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

   7. Simplified 16.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c0 \cdot c0, \left(\left(D \cdot D\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}\right) \cdot 0.25, 0\right)} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. 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. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

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

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

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

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

      15. unpow2 N/A

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f64 44.2

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

   10. Simplified 44.2%

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

4. Final simplification 50.5%

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

Alternative 5: 52.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{D \cdot \left(D \cdot \left(h \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right)\right)}{d \cdot d}\\ \end{array} \end{array} \]

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)
     (* (* c0 c0) (/ (* d d) (* (* D D) (* h (* w w)))))
     (/ (* D (* D (* h (* (* 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 = (c0 * c0) * ((d * d) / ((D * D) * (h * (w * w))));
	} else {
		tmp = (D * (D * (h * ((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 = (c0 * c0) * ((d * d) / ((D * D) * (h * (w * w))));
	} else {
		tmp = (D * (D * (h * ((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 = (c0 * c0) * ((d * d) / ((D * D) * (h * (w * w))))
	else:
		tmp = (D * (D * (h * ((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(c0 * c0) * Float64(Float64(d * d) / Float64(Float64(D * D) * Float64(h * Float64(w * w)))));
	else
		tmp = Float64(Float64(D * Float64(D * Float64(h * Float64(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 = (c0 * c0) * ((d * d) / ((D * D) * (h * (w * w))));
	else
		tmp = (D * (D * (h * ((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[(c0 * c0), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(h * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(D * N[(D * N[(h * N[(N[(M * M), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $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 76.8%

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

   3. Taylor expanded in c0 around inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-/l* N/A

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

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

         \[\leadsto \left(c0 \cdot c0\right) \cdot \color{blue}{\mathsf{fma}\left(d, \frac{d}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}, \frac{-1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]

   5. Simplified 56.9%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \mathsf{fma}\left(d, \frac{d}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}, \frac{-0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}\right)} \]

   6. Taylor expanded in d around inf

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 58.5

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

   8. Simplified 58.5%

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

   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. frac-times N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

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

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

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

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

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

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

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

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

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

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

      19. *-lowering-*.f64 0.7

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

   4. Applied egg-rr 0.7%

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

   5. Taylor expanded in c0 around -inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

   7. Simplified 16.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c0 \cdot c0, \left(\left(D \cdot D\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}\right) \cdot 0.25, 0\right)} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. 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. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

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

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

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

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

      15. unpow2 N/A

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f64 44.2

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

   10. Simplified 44.2%

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

4. Final simplification 48.9%

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

Alternative 6: 38.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;0\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{D \cdot \left(D \cdot \left(h \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right)\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= d 1.6e-162)
   0.0
   (if (<= d 1.35e+154) (/ (* D (* D (* h (* (* M M) 0.25)))) (* d d)) 0.0)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (d <= 1.6e-162) {
		tmp = 0.0;
	} else if (d <= 1.35e+154) {
		tmp = (D * (D * (h * ((M * M) * 0.25)))) / (d * d);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (d_1 <= 1.6d-162) then
        tmp = 0.0d0
    else if (d_1 <= 1.35d+154) then
        tmp = (d * (d * (h * ((m * m) * 0.25d0)))) / (d_1 * d_1)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (d <= 1.6e-162) {
		tmp = 0.0;
	} else if (d <= 1.35e+154) {
		tmp = (D * (D * (h * ((M * M) * 0.25)))) / (d * d);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if d <= 1.6e-162:
		tmp = 0.0
	elif d <= 1.35e+154:
		tmp = (D * (D * (h * ((M * M) * 0.25)))) / (d * d)
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (d <= 1.6e-162)
		tmp = 0.0;
	elseif (d <= 1.35e+154)
		tmp = Float64(Float64(D * Float64(D * Float64(h * Float64(Float64(M * M) * 0.25)))) / Float64(d * d));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (d <= 1.6e-162)
		tmp = 0.0;
	elseif (d <= 1.35e+154)
		tmp = (D * (D * (h * ((M * M) * 0.25)))) / (d * d);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[d, 1.6e-162], 0.0, If[LessEqual[d, 1.35e+154], N[(N[(D * N[(D * N[(h * N[(N[(M * M), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if d < 1.59999999999999988e-162 or 1.35000000000000003e154 < d

   1. Initial program 24.7%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. 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 31.4

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

   5. Simplified 31.4%

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

   if 1.59999999999999988e-162 < d < 1.35000000000000003e154

   1. Initial program 28.6%

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

      1. frac-times N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

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

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

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

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

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

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

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

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

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

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

      19. *-lowering-*.f64 26.7

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

   4. Applied egg-rr 26.7%

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

   5. Taylor expanded in c0 around -inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

   7. Simplified 18.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c0 \cdot c0, \left(\left(D \cdot D\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}\right) \cdot 0.25, 0\right)} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. 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. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

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

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

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

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

      15. unpow2 N/A

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f64 51.1

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

   10. Simplified 51.1%

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

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;0\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{D \cdot \left(D \cdot \left(h \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right)\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 34.3% 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 25.5%

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

3. Taylor expanded in c0 around -inf

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{{c0}^{2} \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w}} \]
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 32.6

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

5. Simplified 32.6%

   \[\leadsto \color{blue}{0} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024201 
(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))))))