Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.6% → 62.3%

Time: 19.2s

Alternatives: 11

Speedup: 156.0×

• could not determine a ground truth

  1. c0 = 5.51241400903992e+59
  2. w = -9.367203264448707e-139
  3. h = 6.286091229486354e-64
  4. D = 1.15784032007148e-252
  5. d = -2.0597959527007505e+305
  6. M = 4.899740733859431e-271

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 24.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 62.3% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* D D) (* w h)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
        INFINITY)
     (/ (* c0 d) (* D (* (* D (* w h)) (/ w (* c0 d)))))
     (/ (* (* D D) (* 0.25 (/ (* h (* M M)) d))) d))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((D * D) * (w * h));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= ((double) INFINITY)) {
		tmp = (c0 * d) / (D * ((D * (w * h)) * (w / (c0 * d))));
	} else {
		tmp = ((D * D) * (0.25 * ((h * (M * M)) / 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)) / ((D * D) * (w * h));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = (c0 * d) / (D * ((D * (w * h)) * (w / (c0 * d))));
	} else {
		tmp = ((D * D) * (0.25 * ((h * (M * M)) / d))) / d;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((D * D) * (w * h))
	tmp = 0
	if ((c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))) <= math.inf:
		tmp = (c0 * d) / (D * ((D * (w * h)) * (w / (c0 * d))))
	else:
		tmp = ((D * D) * (0.25 * ((h * (M * M)) / d))) / d
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(D * D) * Float64(w * h)))
	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(D * Float64(Float64(D * Float64(w * h)) * Float64(w / Float64(c0 * d)))));
	else
		tmp = Float64(Float64(Float64(D * D) * Float64(0.25 * Float64(Float64(h * Float64(M * M)) / d))) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((D * D) * (w * h));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= Inf)
		tmp = (c0 * d) / (D * ((D * (w * h)) * (w / (c0 * d))));
	else
		tmp = ((D * D) * (0.25 * ((h * (M * M)) / 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[(D * D), $MachinePrecision] * N[(w * h), $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[(D * N[(N[(D * N[(w * h), $MachinePrecision]), $MachinePrecision] * N[(w / N[(c0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(D * D), $MachinePrecision] * N[(0.25 * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $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 73.1%

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

   3. Taylor expanded in c0 around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 49.3

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

   5. Simplified 49.3%

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

      1. unswap-sqr N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-*.f64 75.4

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

   7. Applied egg-rr 75.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. clear-num N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 84.4

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

   9. Applied egg-rr 84.4%

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

   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. Taylor expanded in c0 around -inf

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

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

   5. Simplified 20.5%

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

   6. Taylor expanded in c0 around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 42.8

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

   8. Simplified 42.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. associate-*r/ N/A

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

      12. associate-*l/ N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 52.6

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

   10. Applied egg-rr 52.6%

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

4. Final simplification 62.3%

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

Alternative 2: 62.6% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* D D) (* w h)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
        INFINITY)
     (* (/ (* c0 d) (* h (* w D))) (/ (* c0 d) (* w D)))
     (/ (* (* D D) (* 0.25 (/ (* h (* M M)) d))) d))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((D * D) * (w * h));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= ((double) INFINITY)) {
		tmp = ((c0 * d) / (h * (w * D))) * ((c0 * d) / (w * D));
	} else {
		tmp = ((D * D) * (0.25 * ((h * (M * M)) / 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)) / ((D * D) * (w * h));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = ((c0 * d) / (h * (w * D))) * ((c0 * d) / (w * D));
	} else {
		tmp = ((D * D) * (0.25 * ((h * (M * M)) / d))) / d;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((D * D) * (w * h))
	tmp = 0
	if ((c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))) <= math.inf:
		tmp = ((c0 * d) / (h * (w * D))) * ((c0 * d) / (w * D))
	else:
		tmp = ((D * D) * (0.25 * ((h * (M * M)) / d))) / d
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(D * D) * Float64(w * h)))
	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(h * Float64(w * D))) * Float64(Float64(c0 * d) / Float64(w * D)));
	else
		tmp = Float64(Float64(Float64(D * D) * Float64(0.25 * Float64(Float64(h * Float64(M * M)) / d))) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((D * D) * (w * h));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= Inf)
		tmp = ((c0 * d) / (h * (w * D))) * ((c0 * d) / (w * D));
	else
		tmp = ((D * D) * (0.25 * ((h * (M * M)) / 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[(D * D), $MachinePrecision] * N[(w * h), $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[(h * N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(c0 * d), $MachinePrecision] / N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(D * D), $MachinePrecision] * N[(0.25 * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $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 73.1%

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

   3. Taylor expanded in c0 around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 49.3

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

   5. Simplified 49.3%

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

      1. unswap-sqr N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-*.f64 75.4

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

   7. Applied egg-rr 75.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. clear-num N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 84.4

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

   9. Applied egg-rr 84.4%

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

   11. Applied egg-rr 82.6%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around -inf

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

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

   5. Simplified 20.5%

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

   6. Taylor expanded in c0 around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 42.8

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

   8. Simplified 42.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. associate-*r/ N/A

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

      12. associate-*l/ N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 52.6

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

   10. Applied egg-rr 52.6%

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

4. Final simplification 61.8%

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

Alternative 3: 61.4% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* D D) (* w h)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
        INFINITY)
     (* (/ (* c0 d) D) (/ (* c0 d) (* w (* D (* w h)))))
     (/ (* (* D D) (* 0.25 (/ (* h (* M M)) d))) d))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((D * D) * (w * h));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= ((double) INFINITY)) {
		tmp = ((c0 * d) / D) * ((c0 * d) / (w * (D * (w * h))));
	} else {
		tmp = ((D * D) * (0.25 * ((h * (M * M)) / 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)) / ((D * D) * (w * h));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = ((c0 * d) / D) * ((c0 * d) / (w * (D * (w * h))));
	} else {
		tmp = ((D * D) * (0.25 * ((h * (M * M)) / d))) / d;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((D * D) * (w * h))
	tmp = 0
	if ((c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))) <= math.inf:
		tmp = ((c0 * d) / D) * ((c0 * d) / (w * (D * (w * h))))
	else:
		tmp = ((D * D) * (0.25 * ((h * (M * M)) / d))) / d
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(D * D) * Float64(w * h)))
	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) / D) * Float64(Float64(c0 * d) / Float64(w * Float64(D * Float64(w * h)))));
	else
		tmp = Float64(Float64(Float64(D * D) * Float64(0.25 * Float64(Float64(h * Float64(M * M)) / d))) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((D * D) * (w * h));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= Inf)
		tmp = ((c0 * d) / D) * ((c0 * d) / (w * (D * (w * h))));
	else
		tmp = ((D * D) * (0.25 * ((h * (M * M)) / 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[(D * D), $MachinePrecision] * N[(w * h), $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] / D), $MachinePrecision] * N[(N[(c0 * d), $MachinePrecision] / N[(w * N[(D * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(D * D), $MachinePrecision] * N[(0.25 * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $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 73.1%

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

   3. Taylor expanded in c0 around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 49.3

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

   5. Simplified 49.3%

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

      1. unswap-sqr N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-*.f64 75.4

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

   7. Applied egg-rr 75.4%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around -inf

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

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

   5. Simplified 20.5%

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

   6. Taylor expanded in c0 around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 42.8

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

   8. Simplified 42.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. associate-*r/ N/A

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

      12. associate-*l/ N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 52.6

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

   10. Applied egg-rr 52.6%

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

4. Final simplification 59.6%

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

Alternative 4: 59.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c0 \cdot \left(d \cdot d\right)\\ t_1 := \frac{t\_0}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{D} \cdot \frac{t\_0}{w \cdot \left(D \cdot \left(w \cdot h\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(D \cdot D\right) \cdot \left(0.25 \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* c0 (* d d))) (t_1 (/ t_0 (* (* D D) (* w h)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))
        INFINITY)
     (* (/ c0 D) (/ t_0 (* w (* D (* w h)))))
     (/ (* (* D D) (* 0.25 (/ (* h (* M M)) d))) d))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 * (d * d);
	double t_1 = t_0 / ((D * D) * (w * h));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = (c0 / D) * (t_0 / (w * (D * (w * h))));
	} else {
		tmp = ((D * D) * (0.25 * ((h * (M * M)) / d))) / d;
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 * (d * d);
	double t_1 = t_0 / ((D * D) * (w * h));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = (c0 / D) * (t_0 / (w * (D * (w * h))));
	} else {
		tmp = ((D * D) * (0.25 * ((h * (M * M)) / d))) / d;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 * (d * d)
	t_1 = t_0 / ((D * D) * (w * h))
	tmp = 0
	if ((c0 / (2.0 * w)) * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))) <= math.inf:
		tmp = (c0 / D) * (t_0 / (w * (D * (w * h))))
	else:
		tmp = ((D * D) * (0.25 * ((h * (M * M)) / d))) / d
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 * Float64(d * d))
	t_1 = Float64(t_0 / Float64(Float64(D * D) * Float64(w * h)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))))) <= Inf)
		tmp = Float64(Float64(c0 / D) * Float64(t_0 / Float64(w * Float64(D * Float64(w * h)))));
	else
		tmp = Float64(Float64(Float64(D * D) * Float64(0.25 * Float64(Float64(h * Float64(M * M)) / d))) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 * (d * d);
	t_1 = t_0 / ((D * D) * (w * h));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= Inf)
		tmp = (c0 / D) * (t_0 / (w * (D * (w * h))));
	else
		tmp = ((D * D) * (0.25 * ((h * (M * M)) / d))) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(N[(D * D), $MachinePrecision] * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(c0 / D), $MachinePrecision] * N[(t$95$0 / N[(w * N[(D * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(D * D), $MachinePrecision] * N[(0.25 * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $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 73.1%

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

   3. Taylor expanded in c0 around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 49.3

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

   5. Simplified 49.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-*.f64 75.3

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

   7. Applied egg-rr 75.3%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around -inf

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

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

   5. Simplified 20.5%

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

   6. Taylor expanded in c0 around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 42.8

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

   8. Simplified 42.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. associate-*r/ N/A

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

      12. associate-*l/ N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 52.6

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

   10. Applied egg-rr 52.6%

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

4. Final simplification 59.5%

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

Alternative 5: 59.9% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* D D) (* w h)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
        INFINITY)
     (/ (* (* c0 d) (* c0 d)) (* (* w D) (* h (* w D))))
     (/ (* (* D D) (* 0.25 (/ (* h (* M M)) d))) d))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((D * D) * (w * h));
	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) * (h * (w * D)));
	} else {
		tmp = ((D * D) * (0.25 * ((h * (M * M)) / 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)) / ((D * D) * (w * h));
	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) * (h * (w * D)));
	} else {
		tmp = ((D * D) * (0.25 * ((h * (M * M)) / d))) / d;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((D * D) * (w * h))
	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) * (h * (w * D)))
	else:
		tmp = ((D * D) * (0.25 * ((h * (M * M)) / d))) / d
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(D * D) * Float64(w * h)))
	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(c0 * d)) / Float64(Float64(w * D) * Float64(h * Float64(w * D))));
	else
		tmp = Float64(Float64(Float64(D * D) * Float64(0.25 * Float64(Float64(h * Float64(M * M)) / d))) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((D * D) * (w * h));
	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) * (h * (w * D)));
	else
		tmp = ((D * D) * (0.25 * ((h * (M * M)) / 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[(D * D), $MachinePrecision] * N[(w * h), $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[(c0 * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * D), $MachinePrecision] * N[(h * N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(D * D), $MachinePrecision] * N[(0.25 * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $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 73.1%

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

   3. Taylor expanded in c0 around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 49.3

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

   5. Simplified 49.3%

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

      1. unswap-sqr N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-*.f64 75.4

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

   7. Applied egg-rr 75.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift-/.f64 N/A

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

      12. frac-2neg N/A

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

      13. frac-times N/A

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

   9. Applied egg-rr 73.1%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-neg.f64 N/A

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

       4. lift-neg.f64 N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lift-*.f64 N/A

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

       10. associate-*l* N/A

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

       11. remove-double-neg N/A

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

       12. lift-neg.f64 N/A

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

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

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

       14. distribute-rgt-neg-out N/A

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

       15. lift-neg.f64 N/A

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

       16. lower-*.f64 N/A

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

       17. lift-neg.f64 N/A

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

       18. distribute-rgt-neg-out N/A

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

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

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

       20. lift-neg.f64 N/A

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

       21. remove-double-neg N/A

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

       22. *-commutative N/A

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

       23. lower-*.f64 N/A

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

       24. lift-neg.f64 N/A

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

   11. Applied egg-rr 74.4%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around -inf

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

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

   5. Simplified 20.5%

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

   6. Taylor expanded in c0 around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 42.8

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

   8. Simplified 42.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. associate-*r/ N/A

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

      12. associate-*l/ N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 52.6

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

   10. Applied egg-rr 52.6%

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

4. Final simplification 59.3%

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

Alternative 6: 61.9% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* D D) (* w h)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
        INFINITY)
     (/ (* (* c0 d) (* c0 d)) (* (* w D) (* h (* w D))))
     (* (* (* D D) 0.25) (* (/ (* h M) d) (/ M d))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((D * D) * (w * h));
	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) * (h * (w * D)));
	} else {
		tmp = ((D * D) * 0.25) * (((h * M) / d) * (M / d));
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((D * D) * (w * h));
	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) * (h * (w * D)));
	} else {
		tmp = ((D * D) * 0.25) * (((h * M) / d) * (M / d));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((D * D) * (w * h))
	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) * (h * (w * D)))
	else:
		tmp = ((D * D) * 0.25) * (((h * M) / d) * (M / d))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(D * D) * Float64(w * h)))
	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(c0 * d)) / Float64(Float64(w * D) * Float64(h * Float64(w * D))));
	else
		tmp = Float64(Float64(Float64(D * D) * 0.25) * Float64(Float64(Float64(h * M) / d) * Float64(M / d)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((D * D) * (w * h));
	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) * (h * (w * D)));
	else
		tmp = ((D * D) * 0.25) * (((h * M) / d) * (M / d));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(w * h), $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[(c0 * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * D), $MachinePrecision] * N[(h * N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(D * D), $MachinePrecision] * 0.25), $MachinePrecision] * N[(N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 73.1%

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

   3. Taylor expanded in c0 around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 49.3

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

   5. Simplified 49.3%

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

      1. unswap-sqr N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-*.f64 75.4

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

   7. Applied egg-rr 75.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift-/.f64 N/A

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

      12. frac-2neg N/A

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

      13. frac-times N/A

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

   9. Applied egg-rr 73.1%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-neg.f64 N/A

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

       4. lift-neg.f64 N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lift-*.f64 N/A

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

       10. associate-*l* N/A

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

       11. remove-double-neg N/A

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

       12. lift-neg.f64 N/A

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

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

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

       14. distribute-rgt-neg-out N/A

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

       15. lift-neg.f64 N/A

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

       16. lower-*.f64 N/A

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

       17. lift-neg.f64 N/A

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

       18. distribute-rgt-neg-out N/A

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

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

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

       20. lift-neg.f64 N/A

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

       21. remove-double-neg N/A

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

       22. *-commutative N/A

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

       23. lower-*.f64 N/A

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

       24. lift-neg.f64 N/A

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

   11. Applied egg-rr 74.4%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around -inf

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

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

   5. Simplified 20.5%

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

   6. Taylor expanded in c0 around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 42.8

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

   8. Simplified 42.8%

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 52.5

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

   10. Applied egg-rr 52.5%

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

4. Final simplification 59.2%

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

Alternative 7: 59.4% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* D D) (* w h)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
        INFINITY)
     (/ (* (* c0 d) (* c0 d)) (* (* w D) (* h (* w D))))
     (* (/ (* D (* h (* M M))) (* d d)) (* D 0.25)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((D * D) * (w * h));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= ((double) INFINITY)) {
		tmp = ((c0 * d) * (c0 * d)) / ((w * D) * (h * (w * D)));
	} else {
		tmp = ((D * (h * (M * M))) / (d * d)) * (D * 0.25);
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((D * D) * (w * h));
	double 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) * (h * (w * D)));
	} else {
		tmp = ((D * (h * (M * M))) / (d * d)) * (D * 0.25);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((D * D) * (w * h))
	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) * (h * (w * D)))
	else:
		tmp = ((D * (h * (M * M))) / (d * d)) * (D * 0.25)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(D * D) * Float64(w * h)))
	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(c0 * d)) / Float64(Float64(w * D) * Float64(h * Float64(w * D))));
	else
		tmp = Float64(Float64(Float64(D * Float64(h * Float64(M * M))) / Float64(d * d)) * Float64(D * 0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((D * D) * (w * h));
	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) * (h * (w * D)));
	else
		tmp = ((D * (h * (M * M))) / (d * d)) * (D * 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, 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[(c0 * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * D), $MachinePrecision] * N[(h * N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(D * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * N[(D * 0.25), $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 73.1%

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

   3. Taylor expanded in c0 around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 49.3

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

   5. Simplified 49.3%

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

      1. unswap-sqr N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-*.f64 75.4

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

   7. Applied egg-rr 75.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift-/.f64 N/A

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

      12. frac-2neg N/A

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

      13. frac-times N/A

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

   9. Applied egg-rr 73.1%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-neg.f64 N/A

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

       4. lift-neg.f64 N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lift-*.f64 N/A

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

       10. associate-*l* N/A

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

       11. remove-double-neg N/A

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

       12. lift-neg.f64 N/A

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

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

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

       14. distribute-rgt-neg-out N/A

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

       15. lift-neg.f64 N/A

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

       16. lower-*.f64 N/A

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

       17. lift-neg.f64 N/A

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

       18. distribute-rgt-neg-out N/A

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

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

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

       20. lift-neg.f64 N/A

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

       21. remove-double-neg N/A

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

       22. *-commutative N/A

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

       23. lower-*.f64 N/A

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

       24. lift-neg.f64 N/A

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

   11. Applied egg-rr 74.4%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around -inf

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

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

   5. Simplified 20.5%

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

   6. Taylor expanded in c0 around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 42.8

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

   8. Simplified 42.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. times-frac N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

   10. Applied egg-rr 51.1%

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

4. Final simplification 58.2%

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

Alternative 8: 59.2% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* D D) (* w h)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
        INFINITY)
     (* c0 (* (* c0 d) (/ d (* (* w h) (* D (* w D))))))
     (* (/ (* D (* h (* M M))) (* d d)) (* D 0.25)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((D * D) * (w * h));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= ((double) INFINITY)) {
		tmp = c0 * ((c0 * d) * (d / ((w * h) * (D * (w * D)))));
	} else {
		tmp = ((D * (h * (M * M))) / (d * d)) * (D * 0.25);
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((D * D) * (w * h));
	double 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) * (D * (w * D)))));
	} else {
		tmp = ((D * (h * (M * M))) / (d * d)) * (D * 0.25);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((D * D) * (w * h))
	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) * (D * (w * D)))))
	else:
		tmp = ((D * (h * (M * M))) / (d * d)) * (D * 0.25)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(D * D) * Float64(w * h)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M))))) <= Inf)
		tmp = Float64(c0 * Float64(Float64(c0 * d) * Float64(d / Float64(Float64(w * h) * Float64(D * Float64(w * D))))));
	else
		tmp = Float64(Float64(Float64(D * Float64(h * Float64(M * M))) / Float64(d * d)) * Float64(D * 0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((D * D) * (w * h));
	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) * (D * (w * D)))));
	else
		tmp = ((D * (h * (M * M))) / (d * d)) * (D * 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(c0 * N[(N[(c0 * d), $MachinePrecision] * N[(d / N[(N[(w * h), $MachinePrecision] * N[(D * N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(D * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * N[(D * 0.25), $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 73.1%

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

   3. Taylor expanded in c0 around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 49.3

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

   5. Simplified 49.3%

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

      1. unswap-sqr N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-*.f64 75.4

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

   7. Applied egg-rr 75.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. frac-times N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

   9. Applied egg-rr 72.3%

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

   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. Taylor expanded in c0 around -inf

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

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

   5. Simplified 20.5%

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

   6. Taylor expanded in c0 around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 42.8

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

   8. Simplified 42.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. times-frac N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

   10. Applied egg-rr 51.1%

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

4. Final simplification 57.5%

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

Alternative 9: 41.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \cdot d \leq 10^{+292}:\\ \;\;\;\;\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d} \cdot \left(D \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= (* d d) 1e+292) (* (/ (* D (* h (* M M))) (* d d)) (* D 0.25)) 0.0))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((d * d) <= 1e+292) {
		tmp = ((D * (h * (M * M))) / (d * d)) * (D * 0.25);
	} 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 * d_1) <= 1d+292) then
        tmp = ((d * (h * (m * m))) / (d_1 * d_1)) * (d * 0.25d0)
    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 * d) <= 1e+292) {
		tmp = ((D * (h * (M * M))) / (d * d)) * (D * 0.25);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (d * d) <= 1e+292:
		tmp = ((D * (h * (M * M))) / (d * d)) * (D * 0.25)
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (Float64(d * d) <= 1e+292)
		tmp = Float64(Float64(Float64(D * Float64(h * Float64(M * M))) / Float64(d * d)) * Float64(D * 0.25));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((d * d) <= 1e+292)
		tmp = ((D * (h * (M * M))) / (d * d)) * (D * 0.25);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[N[(d * d), $MachinePrecision], 1e+292], N[(N[(N[(D * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * N[(D * 0.25), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 d d) < 1e292

   1. Initial program 22.4%

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

   3. Taylor expanded in c0 around -inf

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

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

   5. Simplified 16.4%

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

   6. Taylor expanded in c0 around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 38.0

          \[\leadsto \left(\left(D \cdot D\right) \cdot 0.25\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{\color{blue}{d \cdot d}} \]

   8. Simplified 38.0%

      \[\leadsto \color{blue}{\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(D \cdot D\right)} \cdot \frac{1}{4}\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(D \cdot D\right) \cdot \frac{1}{4}\right)} \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(D \cdot D\right) \cdot \frac{1}{4}\right) \cdot \frac{h \cdot \color{blue}{\left(M \cdot M\right)}}{d \cdot d} \]

      4. times-frac N/A

         \[\leadsto \left(\left(D \cdot D\right) \cdot \frac{1}{4}\right) \cdot \color{blue}{\left(\frac{h}{d} \cdot \frac{M \cdot M}{d}\right)} \]

      5. times-frac N/A

         \[\leadsto \left(\left(D \cdot D\right) \cdot \frac{1}{4}\right) \cdot \color{blue}{\frac{h \cdot \left(M \cdot M\right)}{d \cdot d}} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(D \cdot D\right) \cdot \frac{1}{4}\right) \cdot \frac{\color{blue}{h \cdot \left(M \cdot M\right)}}{d \cdot d} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(D \cdot D\right) \cdot \frac{1}{4}\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{\color{blue}{d \cdot d}} \]

      8. lift-/.f64 N/A

         \[\leadsto \left(\left(D \cdot D\right) \cdot \frac{1}{4}\right) \cdot \color{blue}{\frac{h \cdot \left(M \cdot M\right)}{d \cdot d}} \]

      9. *-commutative N/A

         \[\leadsto \color{blue}{\frac{h \cdot \left(M \cdot M\right)}{d \cdot d} \cdot \left(\left(D \cdot D\right) \cdot \frac{1}{4}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{h \cdot \left(M \cdot M\right)}{d \cdot d} \cdot \color{blue}{\left(\left(D \cdot D\right) \cdot \frac{1}{4}\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{h \cdot \left(M \cdot M\right)}{d \cdot d} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \frac{1}{4}\right) \]

      12. associate-*l* N/A

          \[\leadsto \frac{h \cdot \left(M \cdot M\right)}{d \cdot d} \cdot \color{blue}{\left(D \cdot \left(D \cdot \frac{1}{4}\right)\right)} \]

      13. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{h \cdot \left(M \cdot M\right)}{d \cdot d} \cdot D\right) \cdot \left(D \cdot \frac{1}{4}\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{h \cdot \left(M \cdot M\right)}{d \cdot d} \cdot D\right) \cdot \left(D \cdot \frac{1}{4}\right)} \]

   10. Applied egg-rr 47.8%

       \[\leadsto \color{blue}{\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d} \cdot \left(D \cdot 0.25\right)} \]

   if 1e292 < (*.f64 d d)

   1. Initial program 22.1%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{{c0}^{2} \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\left({c0}^{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)} \]

      2. distribute-lft1-in N/A

         \[\leadsto \frac{-1}{2} \cdot \left({c0}^{2} \cdot \frac{\color{blue}{\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}}{w}\right) \]

      3. metadata-eval N/A

         \[\leadsto \frac{-1}{2} \cdot \left({c0}^{2} \cdot \frac{\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right) \]

      4. mul0-lft N/A

         \[\leadsto \frac{-1}{2} \cdot \left({c0}^{2} \cdot \frac{\color{blue}{0}}{w}\right) \]

      5. div0 N/A

         \[\leadsto \frac{-1}{2} \cdot \left({c0}^{2} \cdot \color{blue}{0}\right) \]

      6. mul0-rgt N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{0} \]

      7. metadata-eval 36.4

         \[\leadsto \color{blue}{0} \]

   5. Simplified 36.4%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 41.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \cdot d \leq 10^{+292}:\\ \;\;\;\;D \cdot \left(\left(D \cdot 0.25\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= (* d d) 1e+292) (* D (* (* D 0.25) (/ (* h (* M M)) (* d d)))) 0.0))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((d * d) <= 1e+292) {
		tmp = D * ((D * 0.25) * ((h * (M * M)) / (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 * d_1) <= 1d+292) then
        tmp = d * ((d * 0.25d0) * ((h * (m * m)) / (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 * d) <= 1e+292) {
		tmp = D * ((D * 0.25) * ((h * (M * M)) / (d * d)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (d * d) <= 1e+292:
		tmp = D * ((D * 0.25) * ((h * (M * M)) / (d * d)))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (Float64(d * d) <= 1e+292)
		tmp = Float64(D * Float64(Float64(D * 0.25) * Float64(Float64(h * Float64(M * M)) / 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 * d) <= 1e+292)
		tmp = D * ((D * 0.25) * ((h * (M * M)) / (d * d)));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[N[(d * d), $MachinePrecision], 1e+292], N[(D * N[(N[(D * 0.25), $MachinePrecision] * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 d d) < 1e292

   1. Initial program 22.4%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
   4. 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

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right) + {c0}^{2} \cdot \color{blue}{\frac{\frac{-1}{2} \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w}} \]

   5. Simplified 16.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c0 \cdot c0, \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}, 0\right)} \]

   6. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{1}{4} \cdot \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot {D}^{2}\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot {D}^{2}\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left({D}^{2} \cdot \frac{1}{4}\right)} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({D}^{2} \cdot \frac{1}{4}\right)} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}} \]

      6. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(D \cdot D\right)} \cdot \frac{1}{4}\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2}} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(D \cdot D\right)} \cdot \frac{1}{4}\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\left(D \cdot D\right) \cdot \frac{1}{4}\right) \cdot \color{blue}{\frac{{M}^{2} \cdot h}{{d}^{2}}} \]

      9. *-commutative N/A

         \[\leadsto \left(\left(D \cdot D\right) \cdot \frac{1}{4}\right) \cdot \frac{\color{blue}{h \cdot {M}^{2}}}{{d}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(D \cdot D\right) \cdot \frac{1}{4}\right) \cdot \frac{\color{blue}{h \cdot {M}^{2}}}{{d}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \left(\left(D \cdot D\right) \cdot \frac{1}{4}\right) \cdot \frac{h \cdot \color{blue}{\left(M \cdot M\right)}}{{d}^{2}} \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\left(D \cdot D\right) \cdot \frac{1}{4}\right) \cdot \frac{h \cdot \color{blue}{\left(M \cdot M\right)}}{{d}^{2}} \]

      13. unpow2 N/A

          \[\leadsto \left(\left(D \cdot D\right) \cdot \frac{1}{4}\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{\color{blue}{d \cdot d}} \]

      14. lower-*.f64 38.0

          \[\leadsto \left(\left(D \cdot D\right) \cdot 0.25\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{\color{blue}{d \cdot d}} \]

   8. Simplified 38.0%

      \[\leadsto \color{blue}{\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}} \]
   9. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \color{blue}{\left(D \cdot \left(D \cdot \frac{1}{4}\right)\right)} \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(D \cdot \left(D \cdot \frac{1}{4}\right)\right) \cdot \frac{h \cdot \color{blue}{\left(M \cdot M\right)}}{d \cdot d} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(D \cdot \left(D \cdot \frac{1}{4}\right)\right) \cdot \frac{\color{blue}{h \cdot \left(M \cdot M\right)}}{d \cdot d} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(D \cdot \left(D \cdot \frac{1}{4}\right)\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{\color{blue}{d \cdot d}} \]

      5. lift-/.f64 N/A

         \[\leadsto \left(D \cdot \left(D \cdot \frac{1}{4}\right)\right) \cdot \color{blue}{\frac{h \cdot \left(M \cdot M\right)}{d \cdot d}} \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{D \cdot \left(\left(D \cdot \frac{1}{4}\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{D \cdot \left(\left(D \cdot \frac{1}{4}\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto D \cdot \color{blue}{\left(\left(D \cdot \frac{1}{4}\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}\right)} \]

      9. lower-*.f64 46.5

         \[\leadsto D \cdot \left(\color{blue}{\left(D \cdot 0.25\right)} \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}\right) \]

   10. Applied egg-rr 46.5%

       \[\leadsto \color{blue}{D \cdot \left(\left(D \cdot 0.25\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}\right)} \]

   if 1e292 < (*.f64 d d)

   1. Initial program 22.1%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{{c0}^{2} \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\left({c0}^{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)} \]

      2. distribute-lft1-in N/A

         \[\leadsto \frac{-1}{2} \cdot \left({c0}^{2} \cdot \frac{\color{blue}{\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}}{w}\right) \]

      3. metadata-eval N/A

         \[\leadsto \frac{-1}{2} \cdot \left({c0}^{2} \cdot \frac{\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right) \]

      4. mul0-lft N/A

         \[\leadsto \frac{-1}{2} \cdot \left({c0}^{2} \cdot \frac{\color{blue}{0}}{w}\right) \]

      5. div0 N/A

         \[\leadsto \frac{-1}{2} \cdot \left({c0}^{2} \cdot \color{blue}{0}\right) \]

      6. mul0-rgt N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{0} \]

      7. metadata-eval 36.4

         \[\leadsto \color{blue}{0} \]

   5. Simplified 36.4%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 34.2% accurate, 156.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (c0 w h D d M) :precision binary64 0.0)

C

double code(double c0, double w, double h, double D, double d, double M) {
	return 0.0;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = 0.0d0
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return 0.0;
}

Python

def code(c0, w, h, D, d, M):
	return 0.0

Julia

function code(c0, w, h, D, d, M)
	return 0.0
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = 0.0;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := 0.0

Derivation

1. Initial program 22.3%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
2. 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 32.4

      \[\leadsto \color{blue}{0} \]

5. Simplified 32.4%

   \[\leadsto \color{blue}{0} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024210 
(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))))))