Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.9% → 63.4%

Time: 19.0s

Alternatives: 12

Speedup: 156.0×

• could not determine a ground truth

  1. c0 = -8.452621210247073e+74
  2. w = -3.8133471816174465e-123
  3. h = -1.9515259155988317e-47
  4. D = 4.8513322892512456e-290
  5. d = 7.251743481038968e+273
  6. M = 6.660153978116142e+56

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: 24.9% 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: 63.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_1 := \frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\left(d \cdot \left(c0 \cdot \left(2 \cdot d\right)\right)\right) \cdot \left(c0 \cdot 0.5\right)}{D \cdot \left(D \cdot \left(w \cdot \left(w \cdot h\right)\right)\right)}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{2 \cdot \left(c0 \cdot \left(c0 \cdot d\right)\right)}{2 \cdot \left(w \cdot \left(h \cdot D\right)\right)} \cdot \frac{d}{w \cdot D}\\ \mathbf{else}:\\ \;\;\;\;\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d} \cdot \frac{D}{d \cdot 4}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_1 (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))))
   (if (<= t_1 0.0)
     (/ (* (* d (* c0 (* 2.0 d))) (* c0 0.5)) (* D (* D (* w (* w h)))))
     (if (<= t_1 INFINITY)
       (* (/ (* 2.0 (* c0 (* c0 d))) (* 2.0 (* w (* h D)))) (/ d (* w D)))
       (* (/ (* D (* h (* M M))) d) (/ D (* d 4.0)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_1 = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = ((d * (c0 * (2.0 * d))) * (c0 * 0.5)) / (D * (D * (w * (w * h))));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((2.0 * (c0 * (c0 * d))) / (2.0 * (w * (h * D)))) * (d / (w * D));
	} else {
		tmp = ((D * (h * (M * M))) / d) * (D / (d * 4.0));
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_1 = (c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = ((d * (c0 * (2.0 * d))) * (c0 * 0.5)) / (D * (D * (w * (w * h))));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = ((2.0 * (c0 * (c0 * d))) / (2.0 * (w * (h * D)))) * (d / (w * D));
	} else {
		tmp = ((D * (h * (M * M))) / d) * (D / (d * 4.0));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	t_1 = (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))
	tmp = 0
	if t_1 <= 0.0:
		tmp = ((d * (c0 * (2.0 * d))) * (c0 * 0.5)) / (D * (D * (w * (w * h))))
	elif t_1 <= math.inf:
		tmp = ((2.0 * (c0 * (c0 * d))) / (2.0 * (w * (h * D)))) * (d / (w * D))
	else:
		tmp = ((D * (h * (M * M))) / d) * (D / (d * 4.0))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	t_1 = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(Float64(Float64(d * Float64(c0 * Float64(2.0 * d))) * Float64(c0 * 0.5)) / Float64(D * Float64(D * Float64(w * Float64(w * h)))));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(2.0 * Float64(c0 * Float64(c0 * d))) / Float64(2.0 * Float64(w * Float64(h * D)))) * Float64(d / Float64(w * D)));
	else
		tmp = Float64(Float64(Float64(D * Float64(h * Float64(M * M))) / d) * Float64(D / Float64(d * 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	t_1 = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = ((d * (c0 * (2.0 * d))) * (c0 * 0.5)) / (D * (D * (w * (w * h))));
	elseif (t_1 <= Inf)
		tmp = ((2.0 * (c0 * (c0 * d))) / (2.0 * (w * (h * D)))) * (d / (w * D));
	else
		tmp = ((D * (h * (M * M))) / d) * (D / (d * 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, 0.0], N[(N[(N[(d * N[(c0 * N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c0 * 0.5), $MachinePrecision]), $MachinePrecision] / N[(D * N[(D * N[(w * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(2.0 * N[(c0 * N[(c0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[(w * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(D * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(D / N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 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))))) < -0.0

   1. Initial program 82.5%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 75.4

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

   5. Simplified 75.4%

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

   7. Applied egg-rr 78.8%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 83.4

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

   9. Applied egg-rr 83.4%

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

   if -0.0 < (*.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))))) < +inf.0

   1. Initial program 78.2%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 82.4

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

   5. Simplified 82.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

   7. Applied egg-rr 84.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 86.3

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

   9. Applied egg-rr 86.3%

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

   11. Applied egg-rr 92.1%

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

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

   1. Initial program 0.0%

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

   3. 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 19.5%

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

   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-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 40.2

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

   8. Simplified 40.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 40.4

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

   10. Applied egg-rr 40.4%

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

   12. Applied egg-rr 59.3%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 0:\\ \;\;\;\;\frac{\left(d \cdot \left(c0 \cdot \left(2 \cdot d\right)\right)\right) \cdot \left(c0 \cdot 0.5\right)}{D \cdot \left(D \cdot \left(w \cdot \left(w \cdot h\right)\right)\right)}\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{2 \cdot \left(c0 \cdot \left(c0 \cdot d\right)\right)}{2 \cdot \left(w \cdot \left(h \cdot D\right)\right)} \cdot \frac{d}{w \cdot D}\\ \mathbf{else}:\\ \;\;\;\;\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d} \cdot \frac{D}{d \cdot 4}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 65.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(N[(c0 * 2.0), $MachinePrecision] / N[(w * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d * N[(c0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(D * (-N[(w * -2.0), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[(N[(D * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(D / N[(d * 4.0), $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 80.2%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 79.2

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

   5. Simplified 79.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

   7. Applied egg-rr 80.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 80.6

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

   9. Applied egg-rr 80.6%

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

   11. Applied egg-rr 85.0%

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

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

   1. Initial program 0.0%

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

   3. 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 19.5%

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

   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-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 40.2

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

   8. Simplified 40.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 40.4

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

   10. Applied egg-rr 40.4%

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

   12. Applied egg-rr 59.3%

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

4. Final simplification 68.4%

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

Alternative 3: 64.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c0 \cdot \left(d \cdot d\right)\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{t\_0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_1 \cdot \left(t\_2 + \sqrt{t\_2 \cdot t\_2 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_1 \cdot \frac{2 \cdot t\_0}{\left(h \cdot D\right) \cdot \left(w \cdot D\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d} \cdot \frac{D}{d \cdot 4}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* c0 (* d d)))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (/ t_0 (* (* w h) (* D D)))))
   (if (<= (* t_1 (+ t_2 (sqrt (- (* t_2 t_2) (* M M))))) INFINITY)
     (* t_1 (/ (* 2.0 t_0) (* (* h D) (* w D))))
     (* (/ (* D (* h (* M M))) d) (/ D (* d 4.0))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 * (d * d);
	double t_1 = c0 / (2.0 * w);
	double t_2 = t_0 / ((w * h) * (D * D));
	double tmp;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_1 * ((2.0 * t_0) / ((h * D) * (w * D)));
	} else {
		tmp = ((D * (h * (M * M))) / d) * (D / (d * 4.0));
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 * (d * d);
	double t_1 = c0 / (2.0 * w);
	double t_2 = t_0 / ((w * h) * (D * D));
	double tmp;
	if ((t_1 * (t_2 + Math.sqrt(((t_2 * t_2) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_1 * ((2.0 * t_0) / ((h * D) * (w * D)));
	} else {
		tmp = ((D * (h * (M * M))) / d) * (D / (d * 4.0));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 * (d * d)
	t_1 = c0 / (2.0 * w)
	t_2 = t_0 / ((w * h) * (D * D))
	tmp = 0
	if (t_1 * (t_2 + math.sqrt(((t_2 * t_2) - (M * M))))) <= math.inf:
		tmp = t_1 * ((2.0 * t_0) / ((h * D) * (w * D)))
	else:
		tmp = ((D * (h * (M * M))) / d) * (D / (d * 4.0))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 * Float64(d * d))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(t_0 / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_1 * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_1 * Float64(Float64(2.0 * t_0) / Float64(Float64(h * D) * Float64(w * D))));
	else
		tmp = Float64(Float64(Float64(D * Float64(h * Float64(M * M))) / d) * Float64(D / Float64(d * 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 * (d * d);
	t_1 = c0 / (2.0 * w);
	t_2 = t_0 / ((w * h) * (D * D));
	tmp = 0.0;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= Inf)
		tmp = t_1 * ((2.0 * t_0) / ((h * D) * (w * D)));
	else
		tmp = ((D * (h * (M * M))) / d) * (D / (d * 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 * N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$1 * N[(N[(2.0 * t$95$0), $MachinePrecision] / N[(N[(h * D), $MachinePrecision] * N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(D * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(D / N[(d * 4.0), $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 80.2%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 79.2

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

   5. Simplified 79.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 82.4

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

   7. Applied egg-rr 82.4%

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

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

   1. Initial program 0.0%

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

   3. 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 19.5%

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

   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-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 40.2

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

   8. Simplified 40.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 40.4

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

   10. Applied egg-rr 40.4%

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

   12. Applied egg-rr 59.3%

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

4. Final simplification 67.4%

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

Alternative 4: 64.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(c0 * t$95$0), $MachinePrecision] / N[(w * D), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision], N[(N[(N[(D * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(D / N[(d * 4.0), $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 80.2%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 79.2

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

   5. Simplified 79.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

   7. Applied egg-rr 80.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 80.6

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

   9. Applied egg-rr 80.6%

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

   10. Taylor expanded in c0 around 0

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

       1. lower-/.f64 N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 81.7

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

   12. Simplified 81.7%

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

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

   1. Initial program 0.0%

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

   3. 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 19.5%

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

   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-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 40.2

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

   8. Simplified 40.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 40.4

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

   10. Applied egg-rr 40.4%

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

   12. Applied egg-rr 59.3%

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

4. Final simplification 67.2%

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

Alternative 5: 62.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(c0 * t$95$0), $MachinePrecision] / N[(D * N[(D * N[(w * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(D * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(D / N[(d * 4.0), $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 80.2%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 79.2

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

   5. Simplified 79.2%

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

   7. Applied egg-rr 78.4%

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

   8. Taylor expanded in c0 around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 78.4

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

   10. Simplified 78.4%

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

   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 19.5%

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

   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-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 40.2

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

   8. Simplified 40.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 40.4

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

   10. Applied egg-rr 40.4%

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

   12. Applied egg-rr 59.3%

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

Alternative 6: 61.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 80.2%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 79.2

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

   5. Simplified 79.2%

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

   7. Applied egg-rr 78.4%

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

   8. Taylor expanded in c0 around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 78.4

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

   10. Simplified 78.4%

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

   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 19.5%

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

   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-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 40.2

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

   8. Simplified 40.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 51.0

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

   10. Applied egg-rr 51.0%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. associate-*r* N/A

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

       5. lift-/.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-/l* N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. lift-*.f64 N/A

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

       15. associate-*l* N/A

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

       16. *-commutative N/A

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

       17. lower-*.f64 N/A

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

       18. *-commutative N/A

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

       19. lower-*.f64 56.0

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

   12. Applied egg-rr 56.0%

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

4. Final simplification 63.9%

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

Alternative 7: 60.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 80.2%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 79.2

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

   5. Simplified 79.2%

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

   7. Applied egg-rr 78.4%

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

   8. Taylor expanded in c0 around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 78.4

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

   10. Simplified 78.4%

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

   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 19.5%

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

   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-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 40.2

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

   8. Simplified 40.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 40.4

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

   10. Applied egg-rr 40.4%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*r* N/A

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

       6. times-frac N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. associate-*l/ N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 N/A

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

       13. lower-/.f64 55.3

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

   12. Applied egg-rr 55.3%

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

4. Final simplification 63.4%

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

Alternative 8: 57.3% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* c0 (* d d))) (t_1 (/ t_0 (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))
        INFINITY)
     (/ (* c0 t_0) (* D (* D (* w (* w h)))))
     (* h (* (* D (/ (* 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);
	double t_1 = t_0 / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = (c0 * t_0) / (D * (D * (w * (w * h))));
	} else {
		tmp = h * ((D * ((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);
	double t_1 = t_0 / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = (c0 * t_0) / (D * (D * (w * (w * h))));
	} else {
		tmp = h * ((D * ((M * M) / (d * d))) * (D * 0.25));
	}
	return tmp;
}

Python

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

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 * Float64(d * d))
	t_1 = Float64(t_0 / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))))) <= Inf)
		tmp = Float64(Float64(c0 * t_0) / Float64(D * Float64(D * Float64(w * Float64(w * h)))));
	else
		tmp = Float64(h * Float64(Float64(D * Float64(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);
	t_1 = t_0 / ((w * h) * (D * D));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= Inf)
		tmp = (c0 * t_0) / (D * (D * (w * (w * h))));
	else
		tmp = h * ((D * ((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[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(c0 * t$95$0), $MachinePrecision] / N[(D * N[(D * N[(w * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(h * N[(N[(D * N[(N[(M * M), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D * 0.25), $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 80.2%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 79.2

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

   5. Simplified 79.2%

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

   7. Applied egg-rr 78.4%

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

   8. Taylor expanded in c0 around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 78.4

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

   10. Simplified 78.4%

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

   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 19.5%

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

   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-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 40.2

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

   8. Simplified 40.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 40.4

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

   10. Applied egg-rr 40.4%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-/l* N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*r* N/A

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

       9. associate-*l* N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. lower-*.f64 N/A

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

       13. lower-/.f64 N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 47.5

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

   12. Applied egg-rr 47.5%

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

Alternative 9: 55.0% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
        INFINITY)
     (* (* c0 c0) (/ (* d d) (* (* D D) (* w (* w h)))))
     (* h (* (* D (/ (* 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)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= ((double) INFINITY)) {
		tmp = (c0 * c0) * ((d * d) / ((D * D) * (w * (w * h))));
	} else {
		tmp = h * ((D * ((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)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = (c0 * c0) * ((d * d) / ((D * D) * (w * (w * h))));
	} else {
		tmp = h * ((D * ((M * M) / (d * d))) * (D * 0.25));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if ((c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))) <= math.inf:
		tmp = (c0 * c0) * ((d * d) / ((D * D) * (w * (w * h))))
	else:
		tmp = h * ((D * ((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(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M))))) <= Inf)
		tmp = Float64(Float64(c0 * c0) * Float64(Float64(d * d) / Float64(Float64(D * D) * Float64(w * Float64(w * h)))));
	else
		tmp = Float64(h * Float64(Float64(D * Float64(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)) / ((w * h) * (D * D));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= Inf)
		tmp = (c0 * c0) * ((d * d) / ((D * D) * (w * (w * h))));
	else
		tmp = h * ((D * ((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[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(c0 * c0), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(w * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(h * N[(N[(D * N[(N[(M * M), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D * 0.25), $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 80.2%

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

   3. Taylor expanded in c0 around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   5. Simplified 64.0%

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

      1. unswap-sqr N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 64.0

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

   7. Applied egg-rr 64.0%

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

   8. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 69.5

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

   10. Simplified 69.5%

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

   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 19.5%

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

   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-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 40.2

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

   8. Simplified 40.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 40.4

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

   10. Applied egg-rr 40.4%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-/l* N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*r* N/A

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

       9. associate-*l* N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. lower-*.f64 N/A

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

       13. lower-/.f64 N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 47.5

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

   12. Applied egg-rr 47.5%

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

Alternative 10: 54.2% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
        INFINITY)
     (* (* c0 c0) (/ (* d d) (* (* D D) (* h (* w w)))))
     (* h (* (* D (/ (* 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)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= ((double) INFINITY)) {
		tmp = (c0 * c0) * ((d * d) / ((D * D) * (h * (w * w))));
	} else {
		tmp = h * ((D * ((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)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = (c0 * c0) * ((d * d) / ((D * D) * (h * (w * w))));
	} else {
		tmp = h * ((D * ((M * M) / (d * d))) * (D * 0.25));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if ((c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))) <= math.inf:
		tmp = (c0 * c0) * ((d * d) / ((D * D) * (h * (w * w))))
	else:
		tmp = h * ((D * ((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(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M))))) <= Inf)
		tmp = Float64(Float64(c0 * c0) * Float64(Float64(d * d) / Float64(Float64(D * D) * Float64(h * Float64(w * w)))));
	else
		tmp = Float64(h * Float64(Float64(D * Float64(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)) / ((w * h) * (D * D));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= Inf)
		tmp = (c0 * c0) * ((d * d) / ((D * D) * (h * (w * w))));
	else
		tmp = h * ((D * ((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[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(c0 * c0), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(h * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(h * N[(N[(D * N[(N[(M * M), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D * 0.25), $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 80.2%

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

   3. Taylor expanded in c0 around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   5. Simplified 64.0%

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

   6. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 67.3

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

   8. Simplified 67.3%

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

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

   1. Initial program 0.0%

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

   3. 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 19.5%

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

   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-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 40.2

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

   8. Simplified 40.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 40.4

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

   10. Applied egg-rr 40.4%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

          \[\leadsto h \cdot \frac{\color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right)} \cdot \left(M \cdot M\right)}{d \cdot d} \]

       3. lift-*.f64 N/A

          \[\leadsto h \cdot \frac{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\left(M \cdot M\right)}}{d \cdot d} \]

       4. lift-*.f64 N/A

          \[\leadsto h \cdot \frac{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot M\right)}{\color{blue}{d \cdot d}} \]

       5. associate-/l* N/A

          \[\leadsto h \cdot \color{blue}{\left(\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot M}{d \cdot d}\right)} \]

       6. lift-*.f64 N/A

          \[\leadsto h \cdot \left(\color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right)} \cdot \frac{M \cdot M}{d \cdot d}\right) \]

       7. lift-*.f64 N/A

          \[\leadsto h \cdot \left(\left(\frac{1}{4} \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot \frac{M \cdot M}{d \cdot d}\right) \]

       8. associate-*r* N/A

          \[\leadsto h \cdot \left(\color{blue}{\left(\left(\frac{1}{4} \cdot D\right) \cdot D\right)} \cdot \frac{M \cdot M}{d \cdot d}\right) \]

       9. associate-*l* N/A

          \[\leadsto h \cdot \color{blue}{\left(\left(\frac{1}{4} \cdot D\right) \cdot \left(D \cdot \frac{M \cdot M}{d \cdot d}\right)\right)} \]

       10. *-commutative N/A

           \[\leadsto h \cdot \color{blue}{\left(\left(D \cdot \frac{M \cdot M}{d \cdot d}\right) \cdot \left(\frac{1}{4} \cdot D\right)\right)} \]

       11. lower-*.f64 N/A

           \[\leadsto h \cdot \color{blue}{\left(\left(D \cdot \frac{M \cdot M}{d \cdot d}\right) \cdot \left(\frac{1}{4} \cdot D\right)\right)} \]

       12. lower-*.f64 N/A

           \[\leadsto h \cdot \left(\color{blue}{\left(D \cdot \frac{M \cdot M}{d \cdot d}\right)} \cdot \left(\frac{1}{4} \cdot D\right)\right) \]

       13. lower-/.f64 N/A

           \[\leadsto h \cdot \left(\left(D \cdot \color{blue}{\frac{M \cdot M}{d \cdot d}}\right) \cdot \left(\frac{1}{4} \cdot D\right)\right) \]

       14. *-commutative N/A

           \[\leadsto h \cdot \left(\left(D \cdot \frac{M \cdot M}{d \cdot d}\right) \cdot \color{blue}{\left(D \cdot \frac{1}{4}\right)}\right) \]

       15. lower-*.f64 47.5

           \[\leadsto h \cdot \left(\left(D \cdot \frac{M \cdot M}{d \cdot d}\right) \cdot \color{blue}{\left(D \cdot 0.25\right)}\right) \]

   12. Applied egg-rr 47.5%

       \[\leadsto h \cdot \color{blue}{\left(\left(D \cdot \frac{M \cdot M}{d \cdot d}\right) \cdot \left(D \cdot 0.25\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 42.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \cdot M \leq 0:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;h \cdot \left(\left(D \cdot \frac{M \cdot M}{d \cdot d}\right) \cdot \left(D \cdot 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= (* M M) 0.0) 0.0 (* h (* (* D (/ (* M M) (* d d))) (* D 0.25)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((M * M) <= 0.0) {
		tmp = 0.0;
	} else {
		tmp = h * ((D * ((M * M) / (d * d))) * (D * 0.25));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((m * m) <= 0.0d0) then
        tmp = 0.0d0
    else
        tmp = h * ((d * ((m * m) / (d_1 * d_1))) * (d * 0.25d0))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((M * M) <= 0.0) {
		tmp = 0.0;
	} else {
		tmp = h * ((D * ((M * M) / (d * d))) * (D * 0.25));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (M * M) <= 0.0:
		tmp = 0.0
	else:
		tmp = h * ((D * ((M * M) / (d * d))) * (D * 0.25))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (Float64(M * M) <= 0.0)
		tmp = 0.0;
	else
		tmp = Float64(h * Float64(Float64(D * Float64(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)
	tmp = 0.0;
	if ((M * M) <= 0.0)
		tmp = 0.0;
	else
		tmp = h * ((D * ((M * M) / (d * d))) * (D * 0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[N[(M * M), $MachinePrecision], 0.0], 0.0, N[(h * N[(N[(D * N[(N[(M * M), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M M) < 0.0

   1. Initial program 35.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{{c0}^{2} \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w}} \]
   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 43.7

         \[\leadsto \color{blue}{0} \]

   5. Simplified 43.7%

      \[\leadsto \color{blue}{0} \]

   if 0.0 < (*.f64 M M)

   1. Initial program 24.8%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
   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 12.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c0 \cdot c0, \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}, 0\right)} \]

   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-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{4} \cdot {D}^{2}\right) \cdot \left({M}^{2} \cdot h\right)}}{{d}^{2}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{4} \cdot {D}^{2}\right) \cdot \left({M}^{2} \cdot h\right)}}{{d}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{4} \cdot {D}^{2}\right)} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \]

      6. unpow2 N/A

         \[\leadsto \frac{\left(\frac{1}{4} \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{4} \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \]

      8. *-commutative N/A

         \[\leadsto \frac{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{{d}^{2}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{{d}^{2}} \]

      10. unpow2 N/A

          \[\leadsto \frac{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)}{{d}^{2}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)}{{d}^{2}} \]

      12. unpow2 N/A

          \[\leadsto \frac{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\color{blue}{d \cdot d}} \]

      13. lower-*.f64 31.3

          \[\leadsto \frac{\left(0.25 \cdot \left(D \cdot D\right)\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\color{blue}{d \cdot d}} \]

   8. Simplified 31.3%

      \[\leadsto \color{blue}{\frac{\left(0.25 \cdot \left(D \cdot D\right)\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{4} \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right)} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)}{d \cdot d} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\left(h \cdot \left(M \cdot M\right)\right)}}{d \cdot d} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(h \cdot \left(M \cdot M\right)\right) \cdot \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right)}}{d \cdot d} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(h \cdot \left(M \cdot M\right)\right) \cdot \color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right)}}{d \cdot d} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(h \cdot \left(M \cdot M\right)\right) \cdot \color{blue}{\left(\left(D \cdot D\right) \cdot \frac{1}{4}\right)}}{d \cdot d} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(h \cdot \left(M \cdot M\right)\right) \cdot \color{blue}{\left(\left(D \cdot D\right) \cdot \frac{1}{4}\right)}}{d \cdot d} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(h \cdot \left(M \cdot M\right)\right)} \cdot \left(\left(D \cdot D\right) \cdot \frac{1}{4}\right)}{d \cdot d} \]

      10. associate-*l* N/A

          \[\leadsto \frac{\color{blue}{h \cdot \left(\left(M \cdot M\right) \cdot \left(\left(D \cdot D\right) \cdot \frac{1}{4}\right)\right)}}{d \cdot d} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{h \cdot \left(\left(M \cdot M\right) \cdot \left(\left(D \cdot D\right) \cdot \frac{1}{4}\right)\right)}{\color{blue}{d \cdot d}} \]

      12. associate-/l* N/A

          \[\leadsto \color{blue}{h \cdot \frac{\left(M \cdot M\right) \cdot \left(\left(D \cdot D\right) \cdot \frac{1}{4}\right)}{d \cdot d}} \]

      13. lower-*.f64 N/A

          \[\leadsto \color{blue}{h \cdot \frac{\left(M \cdot M\right) \cdot \left(\left(D \cdot D\right) \cdot \frac{1}{4}\right)}{d \cdot d}} \]

      14. lower-/.f64 N/A

          \[\leadsto h \cdot \color{blue}{\frac{\left(M \cdot M\right) \cdot \left(\left(D \cdot D\right) \cdot \frac{1}{4}\right)}{d \cdot d}} \]

      15. lift-*.f64 N/A

          \[\leadsto h \cdot \frac{\left(M \cdot M\right) \cdot \color{blue}{\left(\left(D \cdot D\right) \cdot \frac{1}{4}\right)}}{d \cdot d} \]

      16. *-commutative N/A

          \[\leadsto h \cdot \frac{\left(M \cdot M\right) \cdot \color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right)}}{d \cdot d} \]

      17. lift-*.f64 N/A

          \[\leadsto h \cdot \frac{\left(M \cdot M\right) \cdot \color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right)}}{d \cdot d} \]

      18. *-commutative N/A

          \[\leadsto h \cdot \frac{\color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot M\right)}}{d \cdot d} \]

      19. lower-*.f64 31.5

          \[\leadsto h \cdot \frac{\color{blue}{\left(0.25 \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot M\right)}}{d \cdot d} \]

   10. Applied egg-rr 31.5%

       \[\leadsto \color{blue}{h \cdot \frac{\left(0.25 \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot M\right)}{d \cdot d}} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto h \cdot \frac{\left(\frac{1}{4} \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot \left(M \cdot M\right)}{d \cdot d} \]

       2. lift-*.f64 N/A

          \[\leadsto h \cdot \frac{\color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right)} \cdot \left(M \cdot M\right)}{d \cdot d} \]

       3. lift-*.f64 N/A

          \[\leadsto h \cdot \frac{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\left(M \cdot M\right)}}{d \cdot d} \]

       4. lift-*.f64 N/A

          \[\leadsto h \cdot \frac{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot M\right)}{\color{blue}{d \cdot d}} \]

       5. associate-/l* N/A

          \[\leadsto h \cdot \color{blue}{\left(\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot M}{d \cdot d}\right)} \]

       6. lift-*.f64 N/A

          \[\leadsto h \cdot \left(\color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right)} \cdot \frac{M \cdot M}{d \cdot d}\right) \]

       7. lift-*.f64 N/A

          \[\leadsto h \cdot \left(\left(\frac{1}{4} \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot \frac{M \cdot M}{d \cdot d}\right) \]

       8. associate-*r* N/A

          \[\leadsto h \cdot \left(\color{blue}{\left(\left(\frac{1}{4} \cdot D\right) \cdot D\right)} \cdot \frac{M \cdot M}{d \cdot d}\right) \]

       9. associate-*l* N/A

          \[\leadsto h \cdot \color{blue}{\left(\left(\frac{1}{4} \cdot D\right) \cdot \left(D \cdot \frac{M \cdot M}{d \cdot d}\right)\right)} \]

       10. *-commutative N/A

           \[\leadsto h \cdot \color{blue}{\left(\left(D \cdot \frac{M \cdot M}{d \cdot d}\right) \cdot \left(\frac{1}{4} \cdot D\right)\right)} \]

       11. lower-*.f64 N/A

           \[\leadsto h \cdot \color{blue}{\left(\left(D \cdot \frac{M \cdot M}{d \cdot d}\right) \cdot \left(\frac{1}{4} \cdot D\right)\right)} \]

       12. lower-*.f64 N/A

           \[\leadsto h \cdot \left(\color{blue}{\left(D \cdot \frac{M \cdot M}{d \cdot d}\right)} \cdot \left(\frac{1}{4} \cdot D\right)\right) \]

       13. lower-/.f64 N/A

           \[\leadsto h \cdot \left(\left(D \cdot \color{blue}{\frac{M \cdot M}{d \cdot d}}\right) \cdot \left(\frac{1}{4} \cdot D\right)\right) \]

       14. *-commutative N/A

           \[\leadsto h \cdot \left(\left(D \cdot \frac{M \cdot M}{d \cdot d}\right) \cdot \color{blue}{\left(D \cdot \frac{1}{4}\right)}\right) \]

       15. lower-*.f64 39.7

           \[\leadsto h \cdot \left(\left(D \cdot \frac{M \cdot M}{d \cdot d}\right) \cdot \color{blue}{\left(D \cdot 0.25\right)}\right) \]

   12. Applied egg-rr 39.7%

       \[\leadsto h \cdot \color{blue}{\left(\left(D \cdot \frac{M \cdot M}{d \cdot d}\right) \cdot \left(D \cdot 0.25\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 33.9% 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 28.2%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
2. Add Preprocessing

3. Taylor expanded in c0 around -inf

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{{c0}^{2} \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w}} \]
4. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \frac{-1}{2} \cdot \color{blue}{\left({c0}^{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)} \]

   2. distribute-lft1-in N/A

      \[\leadsto \frac{-1}{2} \cdot \left({c0}^{2} \cdot \frac{\color{blue}{\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}}{w}\right) \]

   3. metadata-eval N/A

      \[\leadsto \frac{-1}{2} \cdot \left({c0}^{2} \cdot \frac{\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right) \]

   4. mul0-lft N/A

      \[\leadsto \frac{-1}{2} \cdot \left({c0}^{2} \cdot \frac{\color{blue}{0}}{w}\right) \]

   5. div0 N/A

      \[\leadsto \frac{-1}{2} \cdot \left({c0}^{2} \cdot \color{blue}{0}\right) \]

   6. mul0-rgt N/A

      \[\leadsto \frac{-1}{2} \cdot \color{blue}{0} \]

   7. metadata-eval 31.2

      \[\leadsto \color{blue}{0} \]

5. Simplified 31.2%

   \[\leadsto \color{blue}{0} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024211 
(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))))))