Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.5% → 56.3%

Time: 30.0s

Alternatives: 7

Speedup: 151.0×

• could not determine a ground truth

  1. c0 = 7.806303564379024e+255
  2. w = -6.61347221413145e-248
  3. h = 2.396844782087741e-302
  4. D = -8.895383907033312e-271
  5. d = -1.3666285103406704e+167
  6. M = -1.715417853059856e+33

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: 24.5% 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: 56.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := h \cdot \left(w \cdot 0\right)\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\\ t_3 := t_1 \cdot \left(t_2 + \sqrt{t_2 \cdot t_2 - M \cdot M}\right)\\ t_4 := \frac{\frac{c0}{w}}{2}\\ \mathbf{if}\;t_3 \leq -5 \cdot 10^{+53}:\\ \;\;\;\;t_4 \cdot \left(2 \cdot \left(\frac{d}{{D}^{2} \cdot \left(w \cdot h\right)} \cdot \frac{d}{\frac{1}{c0}}\right)\right)\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;t_4 \cdot \left(\mathsf{fma}\left(0.5, \frac{{D}^{2}}{\frac{{d}^{2}}{t_0}}, 0\right) - -0.5 \cdot \left(\frac{{D}^{2}}{c0} \cdot \frac{\left(w \cdot h\right) \cdot \left({M}^{2} + {\left(\frac{0.5 \cdot \left({D}^{2} \cdot t_0\right)}{{d}^{2}}\right)}^{2}\right)}{{d}^{2}}\right)\right)\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;{\left(\sqrt[3]{\left(2 \cdot t_1\right) \cdot {\left(\frac{d}{\frac{D}{\sqrt{\frac{c0}{w}} \cdot \sqrt{\frac{1}{h}}}}\right)}^{2}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* h (* w 0.0)))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (/ (* c0 (* d d)) (* (* D D) (* w h))))
        (t_3 (* t_1 (+ t_2 (sqrt (- (* t_2 t_2) (* M M))))))
        (t_4 (/ (/ c0 w) 2.0)))
   (if (<= t_3 -5e+53)
     (* t_4 (* 2.0 (* (/ d (* (pow D 2.0) (* w h))) (/ d (/ 1.0 c0)))))
     (if (<= t_3 0.0)
       (*
        t_4
        (-
         (fma 0.5 (/ (pow D 2.0) (/ (pow d 2.0) t_0)) 0.0)
         (*
          -0.5
          (*
           (/ (pow D 2.0) c0)
           (/
            (*
             (* w h)
             (+
              (pow M 2.0)
              (pow (/ (* 0.5 (* (pow D 2.0) t_0)) (pow d 2.0)) 2.0)))
            (pow d 2.0))))))
       (if (<= t_3 INFINITY)
         (pow
          (cbrt
           (*
            (* 2.0 t_1)
            (pow (/ d (/ D (* (sqrt (/ c0 w)) (sqrt (/ 1.0 h))))) 2.0)))
          3.0)
         0.0)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = h * (w * 0.0);
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((D * D) * (w * h));
	double t_3 = t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))));
	double t_4 = (c0 / w) / 2.0;
	double tmp;
	if (t_3 <= -5e+53) {
		tmp = t_4 * (2.0 * ((d / (pow(D, 2.0) * (w * h))) * (d / (1.0 / c0))));
	} else if (t_3 <= 0.0) {
		tmp = t_4 * (fma(0.5, (pow(D, 2.0) / (pow(d, 2.0) / t_0)), 0.0) - (-0.5 * ((pow(D, 2.0) / c0) * (((w * h) * (pow(M, 2.0) + pow(((0.5 * (pow(D, 2.0) * t_0)) / pow(d, 2.0)), 2.0))) / pow(d, 2.0)))));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = pow(cbrt(((2.0 * t_1) * pow((d / (D / (sqrt((c0 / w)) * sqrt((1.0 / h))))), 2.0))), 3.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(h * Float64(w * 0.0))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(D * D) * Float64(w * h)))
	t_3 = Float64(t_1 * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M)))))
	t_4 = Float64(Float64(c0 / w) / 2.0)
	tmp = 0.0
	if (t_3 <= -5e+53)
		tmp = Float64(t_4 * Float64(2.0 * Float64(Float64(d / Float64((D ^ 2.0) * Float64(w * h))) * Float64(d / Float64(1.0 / c0)))));
	elseif (t_3 <= 0.0)
		tmp = Float64(t_4 * Float64(fma(0.5, Float64((D ^ 2.0) / Float64((d ^ 2.0) / t_0)), 0.0) - Float64(-0.5 * Float64(Float64((D ^ 2.0) / c0) * Float64(Float64(Float64(w * h) * Float64((M ^ 2.0) + (Float64(Float64(0.5 * Float64((D ^ 2.0) * t_0)) / (d ^ 2.0)) ^ 2.0))) / (d ^ 2.0))))));
	elseif (t_3 <= Inf)
		tmp = cbrt(Float64(Float64(2.0 * t_1) * (Float64(d / Float64(D / Float64(sqrt(Float64(c0 / w)) * sqrt(Float64(1.0 / h))))) ^ 2.0))) ^ 3.0;
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(h * N[(w * 0.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = 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]}, Block[{t$95$4 = N[(N[(c0 / w), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$3, -5e+53], N[(t$95$4 * N[(2.0 * N[(N[(d / N[(N[Power[D, 2.0], $MachinePrecision] * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[(1.0 / c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(t$95$4 * N[(N[(0.5 * N[(N[Power[D, 2.0], $MachinePrecision] / N[(N[Power[d, 2.0], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision] - N[(-0.5 * N[(N[(N[Power[D, 2.0], $MachinePrecision] / c0), $MachinePrecision] * N[(N[(N[(w * h), $MachinePrecision] * N[(N[Power[M, 2.0], $MachinePrecision] + N[Power[N[(N[(0.5 * N[(N[Power[D, 2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Power[N[Power[N[(N[(2.0 * t$95$1), $MachinePrecision] * N[Power[N[(d / N[(D / N[(N[Sqrt[N[(c0 / w), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision], 0.0]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (/.f64 c0 (*.f64 2 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))))) < -5.0000000000000004e53

   1. Initial program 74.0%

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

   3. Simplified 73.7%

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

      1. sqrt-prod 73.7%

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

      2. fma-udef 73.7%

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

      3. *-commutative 73.7%

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

      4. fma-udef 73.7%

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

      5. associate-/r* 73.7%

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

      6. pow2 73.7%

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

      7. pow2 73.7%

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

   5. Applied egg-rr 73.7%

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

      1. *-commutative 73.7%

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

      2. fma-udef 73.7%

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

      3. unsub-neg 73.7%

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

      4. *-commutative 73.7%

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

      5. *-commutative 73.7%

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

      6. associate-*r* 73.7%

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

   7. Simplified 73.7%

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

   8. Taylor expanded in c0 around inf 74.0%

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

      1. associate-/r* 70.3%

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

      2. *-commutative 70.3%

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

      3. *-commutative 70.3%

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

      4. associate-/r* 74.0%

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

      5. associate-/l* 73.8%

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

      6. *-commutative 73.8%

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

   10. Simplified 73.8%

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

       1. unpow2 73.8%

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

       2. div-inv 73.8%

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

       3. times-frac 85.1%

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

       4. *-commutative 85.1%

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

       5. *-commutative 85.1%

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

   12. Applied egg-rr 85.1%

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

   if -5.0000000000000004e53 < (*.f64 (/.f64 c0 (*.f64 2 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 55.4%

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

   3. Simplified 10.2%

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

   4. Taylor expanded in c0 around -inf 69.9%

      \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) + \left(-0.5 \cdot \frac{{D}^{2} \cdot \left(h \cdot \left(w \cdot \left(-1 \cdot {M}^{2} - {\left(0.5 \cdot \frac{{D}^{2} \cdot \left(h \cdot \left(w \cdot \left(-1 \cdot \frac{M \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{M \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)}{{d}^{2}}\right)}^{2}\right)\right)\right)}{c0 \cdot {d}^{2}} + 0.5 \cdot \frac{{D}^{2} \cdot \left(h \cdot \left(w \cdot \left(-1 \cdot \frac{M \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{M \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)}{{d}^{2}}\right)\right)} \]

   6. Simplified 69.8%

      \[\leadsto \frac{\frac{c0}{w}}{2} \cdot \color{blue}{\left(\mathsf{fma}\left(0.5, \frac{{D}^{2}}{\frac{{d}^{2}}{h \cdot \left(w \cdot 0\right)}}, 0\right) + -0.5 \cdot \left(\frac{{D}^{2}}{c0} \cdot \frac{\left(h \cdot w\right) \cdot \left(\left(-{M}^{2}\right) - {\left(\frac{0.5 \cdot \left({D}^{2} \cdot \left(h \cdot \left(w \cdot 0\right)\right)\right)}{{d}^{2}}\right)}^{2}\right)}{{d}^{2}}\right)\right)} \]

   if 0.0 < (*.f64 (/.f64 c0 (*.f64 2 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 83.5%

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

   3. Simplified 81.0%

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

      1. sqrt-prod 80.9%

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

      2. fma-udef 80.9%

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

      3. *-commutative 80.9%

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

      4. fma-udef 80.9%

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

      5. associate-/r* 80.9%

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

      6. pow2 80.9%

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

      7. pow2 80.9%

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

   5. Applied egg-rr 80.9%

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

      1. *-commutative 80.9%

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

      2. fma-udef 80.9%

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

      3. unsub-neg 80.9%

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

      4. *-commutative 80.9%

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

      5. *-commutative 80.9%

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

      6. associate-*r* 80.9%

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

   7. Simplified 80.9%

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

   8. Taylor expanded in c0 around inf 83.5%

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

      1. associate-/r* 85.8%

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

      2. *-commutative 85.8%

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

      3. *-commutative 85.8%

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

      4. associate-/r* 83.5%

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

      5. associate-/l* 83.5%

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

      6. *-commutative 83.5%

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

   10. Simplified 83.5%

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

       1. add-cube-cbrt 83.4%

          \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\frac{c0}{w}}{2} \cdot \left(2 \cdot \frac{{d}^{2}}{\frac{{D}^{2} \cdot \left(h \cdot w\right)}{c0}}\right)} \cdot \sqrt[3]{\frac{\frac{c0}{w}}{2} \cdot \left(2 \cdot \frac{{d}^{2}}{\frac{{D}^{2} \cdot \left(h \cdot w\right)}{c0}}\right)}\right) \cdot \sqrt[3]{\frac{\frac{c0}{w}}{2} \cdot \left(2 \cdot \frac{{d}^{2}}{\frac{{D}^{2} \cdot \left(h \cdot w\right)}{c0}}\right)}} \]

       2. pow3 83.4%

          \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\frac{c0}{w}}{2} \cdot \left(2 \cdot \frac{{d}^{2}}{\frac{{D}^{2} \cdot \left(h \cdot w\right)}{c0}}\right)}\right)}^{3}} \]

   12. Applied egg-rr 92.4%

       \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(\frac{c0}{w \cdot 2} \cdot 2\right) \cdot {\left(\frac{d}{\frac{D}{\sqrt{\frac{c0}{w \cdot h}}}}\right)}^{2}}\right)}^{3}} \]
   13. Step-by-step derivation

       1. pow1/2 92.4%

          \[\leadsto {\left(\sqrt[3]{\left(\frac{c0}{w \cdot 2} \cdot 2\right) \cdot {\left(\frac{d}{\frac{D}{\color{blue}{{\left(\frac{c0}{w \cdot h}\right)}^{0.5}}}}\right)}^{2}}\right)}^{3} \]

       2. associate-/r* 92.4%

          \[\leadsto {\left(\sqrt[3]{\left(\frac{c0}{w \cdot 2} \cdot 2\right) \cdot {\left(\frac{d}{\frac{D}{{\color{blue}{\left(\frac{\frac{c0}{w}}{h}\right)}}^{0.5}}}\right)}^{2}}\right)}^{3} \]

       3. div-inv 92.4%

          \[\leadsto {\left(\sqrt[3]{\left(\frac{c0}{w \cdot 2} \cdot 2\right) \cdot {\left(\frac{d}{\frac{D}{{\color{blue}{\left(\frac{c0}{w} \cdot \frac{1}{h}\right)}}^{0.5}}}\right)}^{2}}\right)}^{3} \]

       4. unpow-prod-down 92.4%

          \[\leadsto {\left(\sqrt[3]{\left(\frac{c0}{w \cdot 2} \cdot 2\right) \cdot {\left(\frac{d}{\frac{D}{\color{blue}{{\left(\frac{c0}{w}\right)}^{0.5} \cdot {\left(\frac{1}{h}\right)}^{0.5}}}}\right)}^{2}}\right)}^{3} \]

       5. pow1/2 92.4%

          \[\leadsto {\left(\sqrt[3]{\left(\frac{c0}{w \cdot 2} \cdot 2\right) \cdot {\left(\frac{d}{\frac{D}{\color{blue}{\sqrt{\frac{c0}{w}}} \cdot {\left(\frac{1}{h}\right)}^{0.5}}}\right)}^{2}}\right)}^{3} \]

   14. Applied egg-rr 92.4%

       \[\leadsto {\left(\sqrt[3]{\left(\frac{c0}{w \cdot 2} \cdot 2\right) \cdot {\left(\frac{d}{\frac{D}{\color{blue}{\sqrt{\frac{c0}{w}} \cdot {\left(\frac{1}{h}\right)}^{0.5}}}}\right)}^{2}}\right)}^{3} \]
   15. Step-by-step derivation

       1. unpow1/2 92.4%

          \[\leadsto {\left(\sqrt[3]{\left(\frac{c0}{w \cdot 2} \cdot 2\right) \cdot {\left(\frac{d}{\frac{D}{\sqrt{\frac{c0}{w}} \cdot \color{blue}{\sqrt{\frac{1}{h}}}}}\right)}^{2}}\right)}^{3} \]

   16. Simplified 92.4%

       \[\leadsto {\left(\sqrt[3]{\left(\frac{c0}{w \cdot 2} \cdot 2\right) \cdot {\left(\frac{d}{\frac{D}{\color{blue}{\sqrt{\frac{c0}{w}} \cdot \sqrt{\frac{1}{h}}}}}\right)}^{2}}\right)}^{3} \]

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 2 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%

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

   3. Simplified 0.1%

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

   4. Taylor expanded in c0 around -inf 1.3%

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

      1. mul-1-neg 1.3%

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

      2. distribute-lft-in 1.3%

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

   6. Simplified 37.4%

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

   7. Taylor expanded in c0 around 0 44.1%

      \[\leadsto \color{blue}{0} \]
3. Recombined 4 regimes into one program.

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq -5 \cdot 10^{+53}:\\ \;\;\;\;\frac{\frac{c0}{w}}{2} \cdot \left(2 \cdot \left(\frac{d}{{D}^{2} \cdot \left(w \cdot h\right)} \cdot \frac{d}{\frac{1}{c0}}\right)\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq 0:\\ \;\;\;\;\frac{\frac{c0}{w}}{2} \cdot \left(\mathsf{fma}\left(0.5, \frac{{D}^{2}}{\frac{{d}^{2}}{h \cdot \left(w \cdot 0\right)}}, 0\right) - -0.5 \cdot \left(\frac{{D}^{2}}{c0} \cdot \frac{\left(w \cdot h\right) \cdot \left({M}^{2} + {\left(\frac{0.5 \cdot \left({D}^{2} \cdot \left(h \cdot \left(w \cdot 0\right)\right)\right)}{{d}^{2}}\right)}^{2}\right)}{{d}^{2}}\right)\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;{\left(\sqrt[3]{\left(2 \cdot \frac{c0}{2 \cdot w}\right) \cdot {\left(\frac{d}{\frac{D}{\sqrt{\frac{c0}{w}} \cdot \sqrt{\frac{1}{h}}}}\right)}^{2}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 2: 56.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\\ t_2 := t_0 \cdot \left(t_1 + \sqrt{t_1 \cdot t_1 - M \cdot M}\right)\\ \mathbf{if}\;t_2 \leq -5 \cdot 10^{+53}:\\ \;\;\;\;\frac{\frac{c0}{w}}{2} \cdot \left(2 \cdot \left(\frac{d}{{D}^{2} \cdot \left(w \cdot h\right)} \cdot \frac{d}{\frac{1}{c0}}\right)\right)\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;t_0 \cdot \mathsf{fma}\left(0.5, \frac{\frac{{D}^{2}}{c0} \cdot \left(\left(w \cdot h\right) \cdot {M}^{2}\right)}{{d}^{2}}, 0\right)\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;{\left(\sqrt[3]{\left(2 \cdot t_0\right) \cdot {\left(\frac{d}{\frac{D}{\sqrt{\frac{c0}{w}} \cdot \sqrt{\frac{1}{h}}}}\right)}^{2}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w)))
        (t_1 (/ (* c0 (* d d)) (* (* D D) (* w h))))
        (t_2 (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))))
   (if (<= t_2 -5e+53)
     (*
      (/ (/ c0 w) 2.0)
      (* 2.0 (* (/ d (* (pow D 2.0) (* w h))) (/ d (/ 1.0 c0)))))
     (if (<= t_2 0.0)
       (*
        t_0
        (fma
         0.5
         (/ (* (/ (pow D 2.0) c0) (* (* w h) (pow M 2.0))) (pow d 2.0))
         0.0))
       (if (<= t_2 INFINITY)
         (pow
          (cbrt
           (*
            (* 2.0 t_0)
            (pow (/ d (/ D (* (sqrt (/ c0 w)) (sqrt (/ 1.0 h))))) 2.0)))
          3.0)
         0.0)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((D * D) * (w * h));
	double t_2 = t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= -5e+53) {
		tmp = ((c0 / w) / 2.0) * (2.0 * ((d / (pow(D, 2.0) * (w * h))) * (d / (1.0 / c0))));
	} else if (t_2 <= 0.0) {
		tmp = t_0 * fma(0.5, (((pow(D, 2.0) / c0) * ((w * h) * pow(M, 2.0))) / pow(d, 2.0)), 0.0);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = pow(cbrt(((2.0 * t_0) * pow((d / (D / (sqrt((c0 / w)) * sqrt((1.0 / h))))), 2.0))), 3.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(D * D) * Float64(w * h)))
	t_2 = Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M)))))
	tmp = 0.0
	if (t_2 <= -5e+53)
		tmp = Float64(Float64(Float64(c0 / w) / 2.0) * Float64(2.0 * Float64(Float64(d / Float64((D ^ 2.0) * Float64(w * h))) * Float64(d / Float64(1.0 / c0)))));
	elseif (t_2 <= 0.0)
		tmp = Float64(t_0 * fma(0.5, Float64(Float64(Float64((D ^ 2.0) / c0) * Float64(Float64(w * h) * (M ^ 2.0))) / (d ^ 2.0)), 0.0));
	elseif (t_2 <= Inf)
		tmp = cbrt(Float64(Float64(2.0 * t_0) * (Float64(d / Float64(D / Float64(sqrt(Float64(c0 / w)) * sqrt(Float64(1.0 / h))))) ^ 2.0))) ^ 3.0;
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+53], N[(N[(N[(c0 / w), $MachinePrecision] / 2.0), $MachinePrecision] * N[(2.0 * N[(N[(d / N[(N[Power[D, 2.0], $MachinePrecision] * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[(1.0 / c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(t$95$0 * N[(0.5 * N[(N[(N[(N[Power[D, 2.0], $MachinePrecision] / c0), $MachinePrecision] * N[(N[(w * h), $MachinePrecision] * N[Power[M, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Power[N[Power[N[(N[(2.0 * t$95$0), $MachinePrecision] * N[Power[N[(d / N[(D / N[(N[Sqrt[N[(c0 / w), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision], 0.0]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (/.f64 c0 (*.f64 2 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))))) < -5.0000000000000004e53

   1. Initial program 74.0%

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

   3. Simplified 73.7%

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

      1. sqrt-prod 73.7%

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

      2. fma-udef 73.7%

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

      3. *-commutative 73.7%

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

      4. fma-udef 73.7%

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

      5. associate-/r* 73.7%

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

      6. pow2 73.7%

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

      7. pow2 73.7%

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

   5. Applied egg-rr 73.7%

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

      1. *-commutative 73.7%

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

      2. fma-udef 73.7%

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

      3. unsub-neg 73.7%

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

      4. *-commutative 73.7%

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

      5. *-commutative 73.7%

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

      6. associate-*r* 73.7%

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

   7. Simplified 73.7%

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

   8. Taylor expanded in c0 around inf 74.0%

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

      1. associate-/r* 70.3%

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

      2. *-commutative 70.3%

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

      3. *-commutative 70.3%

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

      4. associate-/r* 74.0%

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

      5. associate-/l* 73.8%

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

      6. *-commutative 73.8%

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

   10. Simplified 73.8%

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

       1. unpow2 73.8%

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

       2. div-inv 73.8%

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

       3. times-frac 85.1%

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

       4. *-commutative 85.1%

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

       5. *-commutative 85.1%

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

   12. Applied egg-rr 85.1%

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

   if -5.0000000000000004e53 < (*.f64 (/.f64 c0 (*.f64 2 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 55.4%

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

   3. Simplified 41.9%

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

   4. Taylor expanded in c0 around -inf 63.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) + 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right)} \]

   6. Simplified 63.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{\frac{{D}^{2}}{c0} \cdot \left(\left(h \cdot w\right) \cdot {M}^{2}\right)}{{d}^{2}}, 0\right)} \]

   if 0.0 < (*.f64 (/.f64 c0 (*.f64 2 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 83.5%

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

   3. Simplified 81.0%

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

      1. sqrt-prod 80.9%

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

      2. fma-udef 80.9%

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

      3. *-commutative 80.9%

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

      4. fma-udef 80.9%

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

      5. associate-/r* 80.9%

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

      6. pow2 80.9%

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

      7. pow2 80.9%

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

   5. Applied egg-rr 80.9%

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

      1. *-commutative 80.9%

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

      2. fma-udef 80.9%

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

      3. unsub-neg 80.9%

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

      4. *-commutative 80.9%

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

      5. *-commutative 80.9%

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

      6. associate-*r* 80.9%

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

   7. Simplified 80.9%

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

   8. Taylor expanded in c0 around inf 83.5%

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

      1. associate-/r* 85.8%

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

      2. *-commutative 85.8%

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

      3. *-commutative 85.8%

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

      4. associate-/r* 83.5%

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

      5. associate-/l* 83.5%

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

      6. *-commutative 83.5%

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

   10. Simplified 83.5%

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

       1. add-cube-cbrt 83.4%

          \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\frac{c0}{w}}{2} \cdot \left(2 \cdot \frac{{d}^{2}}{\frac{{D}^{2} \cdot \left(h \cdot w\right)}{c0}}\right)} \cdot \sqrt[3]{\frac{\frac{c0}{w}}{2} \cdot \left(2 \cdot \frac{{d}^{2}}{\frac{{D}^{2} \cdot \left(h \cdot w\right)}{c0}}\right)}\right) \cdot \sqrt[3]{\frac{\frac{c0}{w}}{2} \cdot \left(2 \cdot \frac{{d}^{2}}{\frac{{D}^{2} \cdot \left(h \cdot w\right)}{c0}}\right)}} \]

       2. pow3 83.4%

          \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\frac{c0}{w}}{2} \cdot \left(2 \cdot \frac{{d}^{2}}{\frac{{D}^{2} \cdot \left(h \cdot w\right)}{c0}}\right)}\right)}^{3}} \]

   12. Applied egg-rr 92.4%

       \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(\frac{c0}{w \cdot 2} \cdot 2\right) \cdot {\left(\frac{d}{\frac{D}{\sqrt{\frac{c0}{w \cdot h}}}}\right)}^{2}}\right)}^{3}} \]
   13. Step-by-step derivation

       1. pow1/2 92.4%

          \[\leadsto {\left(\sqrt[3]{\left(\frac{c0}{w \cdot 2} \cdot 2\right) \cdot {\left(\frac{d}{\frac{D}{\color{blue}{{\left(\frac{c0}{w \cdot h}\right)}^{0.5}}}}\right)}^{2}}\right)}^{3} \]

       2. associate-/r* 92.4%

          \[\leadsto {\left(\sqrt[3]{\left(\frac{c0}{w \cdot 2} \cdot 2\right) \cdot {\left(\frac{d}{\frac{D}{{\color{blue}{\left(\frac{\frac{c0}{w}}{h}\right)}}^{0.5}}}\right)}^{2}}\right)}^{3} \]

       3. div-inv 92.4%

          \[\leadsto {\left(\sqrt[3]{\left(\frac{c0}{w \cdot 2} \cdot 2\right) \cdot {\left(\frac{d}{\frac{D}{{\color{blue}{\left(\frac{c0}{w} \cdot \frac{1}{h}\right)}}^{0.5}}}\right)}^{2}}\right)}^{3} \]

       4. unpow-prod-down 92.4%

          \[\leadsto {\left(\sqrt[3]{\left(\frac{c0}{w \cdot 2} \cdot 2\right) \cdot {\left(\frac{d}{\frac{D}{\color{blue}{{\left(\frac{c0}{w}\right)}^{0.5} \cdot {\left(\frac{1}{h}\right)}^{0.5}}}}\right)}^{2}}\right)}^{3} \]

       5. pow1/2 92.4%

          \[\leadsto {\left(\sqrt[3]{\left(\frac{c0}{w \cdot 2} \cdot 2\right) \cdot {\left(\frac{d}{\frac{D}{\color{blue}{\sqrt{\frac{c0}{w}}} \cdot {\left(\frac{1}{h}\right)}^{0.5}}}\right)}^{2}}\right)}^{3} \]

   14. Applied egg-rr 92.4%

       \[\leadsto {\left(\sqrt[3]{\left(\frac{c0}{w \cdot 2} \cdot 2\right) \cdot {\left(\frac{d}{\frac{D}{\color{blue}{\sqrt{\frac{c0}{w}} \cdot {\left(\frac{1}{h}\right)}^{0.5}}}}\right)}^{2}}\right)}^{3} \]
   15. Step-by-step derivation

       1. unpow1/2 92.4%

          \[\leadsto {\left(\sqrt[3]{\left(\frac{c0}{w \cdot 2} \cdot 2\right) \cdot {\left(\frac{d}{\frac{D}{\sqrt{\frac{c0}{w}} \cdot \color{blue}{\sqrt{\frac{1}{h}}}}}\right)}^{2}}\right)}^{3} \]

   16. Simplified 92.4%

       \[\leadsto {\left(\sqrt[3]{\left(\frac{c0}{w \cdot 2} \cdot 2\right) \cdot {\left(\frac{d}{\frac{D}{\color{blue}{\sqrt{\frac{c0}{w}} \cdot \sqrt{\frac{1}{h}}}}}\right)}^{2}}\right)}^{3} \]

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 2 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%

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

   3. Simplified 0.1%

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

   4. Taylor expanded in c0 around -inf 1.3%

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

      1. mul-1-neg 1.3%

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

      2. distribute-lft-in 1.3%

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

   6. Simplified 37.4%

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

   7. Taylor expanded in c0 around 0 44.1%

      \[\leadsto \color{blue}{0} \]
3. Recombined 4 regimes into one program.

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq -5 \cdot 10^{+53}:\\ \;\;\;\;\frac{\frac{c0}{w}}{2} \cdot \left(2 \cdot \left(\frac{d}{{D}^{2} \cdot \left(w \cdot h\right)} \cdot \frac{d}{\frac{1}{c0}}\right)\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq 0:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(0.5, \frac{\frac{{D}^{2}}{c0} \cdot \left(\left(w \cdot h\right) \cdot {M}^{2}\right)}{{d}^{2}}, 0\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;{\left(\sqrt[3]{\left(2 \cdot \frac{c0}{2 \cdot w}\right) \cdot {\left(\frac{d}{\frac{D}{\sqrt{\frac{c0}{w}} \cdot \sqrt{\frac{1}{h}}}}\right)}^{2}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 3: 56.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\\ t_2 := t_0 \cdot \left(t_1 + \sqrt{t_1 \cdot t_1 - M \cdot M}\right)\\ \mathbf{if}\;t_2 \leq -5 \cdot 10^{+53}:\\ \;\;\;\;\frac{\frac{c0}{w}}{2} \cdot \left(2 \cdot \left(\frac{d}{{D}^{2} \cdot \left(w \cdot h\right)} \cdot \frac{d}{\frac{1}{c0}}\right)\right)\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;t_0 \cdot \mathsf{fma}\left(0.5, \frac{\frac{{D}^{2}}{c0} \cdot \left(\left(w \cdot h\right) \cdot {M}^{2}\right)}{{d}^{2}}, 0\right)\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;{\left(\left(\frac{d}{D} \cdot \sqrt{\frac{\frac{c0}{w}}{h}}\right) \cdot \sqrt{2 \cdot t_0}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w)))
        (t_1 (/ (* c0 (* d d)) (* (* D D) (* w h))))
        (t_2 (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))))
   (if (<= t_2 -5e+53)
     (*
      (/ (/ c0 w) 2.0)
      (* 2.0 (* (/ d (* (pow D 2.0) (* w h))) (/ d (/ 1.0 c0)))))
     (if (<= t_2 0.0)
       (*
        t_0
        (fma
         0.5
         (/ (* (/ (pow D 2.0) c0) (* (* w h) (pow M 2.0))) (pow d 2.0))
         0.0))
       (if (<= t_2 INFINITY)
         (pow (* (* (/ d D) (sqrt (/ (/ c0 w) h))) (sqrt (* 2.0 t_0))) 2.0)
         0.0)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((D * D) * (w * h));
	double t_2 = t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= -5e+53) {
		tmp = ((c0 / w) / 2.0) * (2.0 * ((d / (pow(D, 2.0) * (w * h))) * (d / (1.0 / c0))));
	} else if (t_2 <= 0.0) {
		tmp = t_0 * fma(0.5, (((pow(D, 2.0) / c0) * ((w * h) * pow(M, 2.0))) / pow(d, 2.0)), 0.0);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = pow((((d / D) * sqrt(((c0 / w) / h))) * sqrt((2.0 * t_0))), 2.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(D * D) * Float64(w * h)))
	t_2 = Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M)))))
	tmp = 0.0
	if (t_2 <= -5e+53)
		tmp = Float64(Float64(Float64(c0 / w) / 2.0) * Float64(2.0 * Float64(Float64(d / Float64((D ^ 2.0) * Float64(w * h))) * Float64(d / Float64(1.0 / c0)))));
	elseif (t_2 <= 0.0)
		tmp = Float64(t_0 * fma(0.5, Float64(Float64(Float64((D ^ 2.0) / c0) * Float64(Float64(w * h) * (M ^ 2.0))) / (d ^ 2.0)), 0.0));
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(Float64(d / D) * sqrt(Float64(Float64(c0 / w) / h))) * sqrt(Float64(2.0 * t_0))) ^ 2.0;
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+53], N[(N[(N[(c0 / w), $MachinePrecision] / 2.0), $MachinePrecision] * N[(2.0 * N[(N[(d / N[(N[Power[D, 2.0], $MachinePrecision] * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[(1.0 / c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(t$95$0 * N[(0.5 * N[(N[(N[(N[Power[D, 2.0], $MachinePrecision] / c0), $MachinePrecision] * N[(N[(w * h), $MachinePrecision] * N[Power[M, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Power[N[(N[(N[(d / D), $MachinePrecision] * N[Sqrt[N[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 0.0]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (/.f64 c0 (*.f64 2 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))))) < -5.0000000000000004e53

   1. Initial program 74.0%

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

   3. Simplified 73.7%

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

      1. sqrt-prod 73.7%

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

      2. fma-udef 73.7%

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

      3. *-commutative 73.7%

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

      4. fma-udef 73.7%

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

      5. associate-/r* 73.7%

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

      6. pow2 73.7%

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

      7. pow2 73.7%

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

   5. Applied egg-rr 73.7%

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

      1. *-commutative 73.7%

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

      2. fma-udef 73.7%

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

      3. unsub-neg 73.7%

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

      4. *-commutative 73.7%

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

      5. *-commutative 73.7%

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

      6. associate-*r* 73.7%

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

   7. Simplified 73.7%

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

   8. Taylor expanded in c0 around inf 74.0%

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

      1. associate-/r* 70.3%

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

      2. *-commutative 70.3%

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

      3. *-commutative 70.3%

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

      4. associate-/r* 74.0%

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

      5. associate-/l* 73.8%

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

      6. *-commutative 73.8%

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

   10. Simplified 73.8%

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

       1. unpow2 73.8%

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

       2. div-inv 73.8%

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

       3. times-frac 85.1%

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

       4. *-commutative 85.1%

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

       5. *-commutative 85.1%

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

   12. Applied egg-rr 85.1%

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

   if -5.0000000000000004e53 < (*.f64 (/.f64 c0 (*.f64 2 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 55.4%

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

   3. Simplified 41.9%

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

   4. Taylor expanded in c0 around -inf 63.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) + 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right)} \]

   6. Simplified 63.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{\frac{{D}^{2}}{c0} \cdot \left(\left(h \cdot w\right) \cdot {M}^{2}\right)}{{d}^{2}}, 0\right)} \]

   if 0.0 < (*.f64 (/.f64 c0 (*.f64 2 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 83.5%

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

   3. Simplified 81.0%

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

      1. sqrt-prod 80.9%

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

      2. fma-udef 80.9%

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

      3. *-commutative 80.9%

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

      4. fma-udef 80.9%

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

      5. associate-/r* 80.9%

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

      6. pow2 80.9%

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

      7. pow2 80.9%

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

   5. Applied egg-rr 80.9%

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

      1. *-commutative 80.9%

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

      2. fma-udef 80.9%

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

      3. unsub-neg 80.9%

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

      4. *-commutative 80.9%

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

      5. *-commutative 80.9%

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

      6. associate-*r* 80.9%

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

   7. Simplified 80.9%

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

   8. Taylor expanded in c0 around inf 83.5%

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

      1. associate-/r* 85.8%

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

      2. *-commutative 85.8%

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

      3. *-commutative 85.8%

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

      4. associate-/r* 83.5%

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

      5. associate-/l* 83.5%

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

      6. *-commutative 83.5%

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

   10. Simplified 83.5%

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

       1. add-sqr-sqrt 83.4%

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

       2. pow2 83.4%

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

   12. Applied egg-rr 92.5%

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

       1. *-commutative 92.5%

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

       2. associate-/r/ 92.4%

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

       3. associate-/r* 92.4%

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

       4. *-commutative 92.4%

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

       5. *-commutative 92.4%

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

   14. Simplified 92.4%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 2 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%

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

   3. Simplified 0.1%

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

   4. Taylor expanded in c0 around -inf 1.3%

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

      1. mul-1-neg 1.3%

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

      2. distribute-lft-in 1.3%

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

   6. Simplified 37.4%

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

   7. Taylor expanded in c0 around 0 44.1%

      \[\leadsto \color{blue}{0} \]
3. Recombined 4 regimes into one program.

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq -5 \cdot 10^{+53}:\\ \;\;\;\;\frac{\frac{c0}{w}}{2} \cdot \left(2 \cdot \left(\frac{d}{{D}^{2} \cdot \left(w \cdot h\right)} \cdot \frac{d}{\frac{1}{c0}}\right)\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq 0:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(0.5, \frac{\frac{{D}^{2}}{c0} \cdot \left(\left(w \cdot h\right) \cdot {M}^{2}\right)}{{d}^{2}}, 0\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;{\left(\left(\frac{d}{D} \cdot \sqrt{\frac{\frac{c0}{w}}{h}}\right) \cdot \sqrt{2 \cdot \frac{c0}{2 \cdot w}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 4: 55.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t_0 + \sqrt{t_0 \cdot t_0 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{\frac{c0}{w}}{2} \cdot \left(2 \cdot \left(\frac{d}{{D}^{2} \cdot \left(w \cdot h\right)} \cdot \frac{d}{\frac{1}{c0}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* D D) (* w h)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
        INFINITY)
     (*
      (/ (/ c0 w) 2.0)
      (* 2.0 (* (/ d (* (pow D 2.0) (* w h))) (/ d (/ 1.0 c0)))))
     0.0)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((D * D) * (w * h));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= ((double) INFINITY)) {
		tmp = ((c0 / w) / 2.0) * (2.0 * ((d / (pow(D, 2.0) * (w * h))) * (d / (1.0 / c0))));
	} else {
		tmp = 0.0;
	}
	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)) / ((D * D) * (w * h));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = ((c0 / w) / 2.0) * (2.0 * ((d / (Math.pow(D, 2.0) * (w * h))) * (d / (1.0 / c0))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((D * D) * (w * h))
	tmp = 0
	if ((c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))) <= math.inf:
		tmp = ((c0 / w) / 2.0) * (2.0 * ((d / (math.pow(D, 2.0) * (w * h))) * (d / (1.0 / c0))))
	else:
		tmp = 0.0
	return tmp

Julia

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

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((D * D) * (w * h));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= Inf)
		tmp = ((c0 / w) / 2.0) * (2.0 * ((d / ((D ^ 2.0) * (w * h))) * (d / (1.0 / c0))));
	else
		tmp = 0.0;
	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[(D * D), $MachinePrecision] * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(c0 / w), $MachinePrecision] / 2.0), $MachinePrecision] * N[(2.0 * N[(N[(d / N[(N[Power[D, 2.0], $MachinePrecision] * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[(1.0 / c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 2 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 75.3%

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

   3. Simplified 65.7%

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

      1. sqrt-prod 63.9%

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

      2. fma-udef 63.9%

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

      3. *-commutative 63.9%

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

      4. fma-udef 63.9%

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

      5. associate-/r* 63.9%

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

      6. pow2 63.9%

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

      7. pow2 63.9%

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

   5. Applied egg-rr 63.9%

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

      1. *-commutative 63.9%

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

      2. fma-udef 63.9%

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

      3. unsub-neg 63.9%

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

      4. *-commutative 63.9%

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

      5. *-commutative 63.9%

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

      6. associate-*r* 65.1%

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

   7. Simplified 65.1%

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

   8. Taylor expanded in c0 around inf 70.4%

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

      1. associate-/r* 67.9%

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

      2. *-commutative 67.9%

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

      3. *-commutative 67.9%

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

      4. associate-/r* 70.4%

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

      5. associate-/l* 70.3%

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

      6. *-commutative 70.3%

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

   10. Simplified 70.3%

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

       1. unpow2 70.3%

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

       2. div-inv 70.3%

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

       3. times-frac 76.2%

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

       4. *-commutative 76.2%

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

       5. *-commutative 76.2%

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

   12. Applied egg-rr 76.2%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 2 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%

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

   3. Simplified 0.1%

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

   4. Taylor expanded in c0 around -inf 1.3%

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

      1. mul-1-neg 1.3%

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

      2. distribute-lft-in 1.3%

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

   6. Simplified 37.4%

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

   7. Taylor expanded in c0 around 0 44.1%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{\frac{c0}{w}}{2} \cdot \left(2 \cdot \left(\frac{d}{{D}^{2} \cdot \left(w \cdot h\right)} \cdot \frac{d}{\frac{1}{c0}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5: 42.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.043:\\ \;\;\;\;0\\ \mathbf{elif}\;w \leq 1.7 \cdot 10^{-238}:\\ \;\;\;\;\frac{\frac{c0}{w}}{2} \cdot \left(2 \cdot \left(c0 \cdot \left(\frac{1}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;w \leq 7.5 \cdot 10^{-202}:\\ \;\;\;\;0\\ \mathbf{elif}\;w \leq 10^{+99}:\\ \;\;\;\;c0 \cdot \frac{c0 \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w \cdot h}}{w}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= w -0.043)
   0.0
   (if (<= w 1.7e-238)
     (* (/ (/ c0 w) 2.0) (* 2.0 (* c0 (* (/ 1.0 (* w h)) (pow (/ d D) 2.0)))))
     (if (<= w 7.5e-202)
       0.0
       (if (<= w 1e+99)
         (* c0 (/ (* c0 (/ (* (/ d D) (/ d D)) (* w h))) w))
         0.0)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (w <= -0.043) {
		tmp = 0.0;
	} else if (w <= 1.7e-238) {
		tmp = ((c0 / w) / 2.0) * (2.0 * (c0 * ((1.0 / (w * h)) * pow((d / D), 2.0))));
	} else if (w <= 7.5e-202) {
		tmp = 0.0;
	} else if (w <= 1e+99) {
		tmp = c0 * ((c0 * (((d / D) * (d / D)) / (w * h))) / w);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (w <= (-0.043d0)) then
        tmp = 0.0d0
    else if (w <= 1.7d-238) then
        tmp = ((c0 / w) / 2.0d0) * (2.0d0 * (c0 * ((1.0d0 / (w * h)) * ((d_1 / d) ** 2.0d0))))
    else if (w <= 7.5d-202) then
        tmp = 0.0d0
    else if (w <= 1d+99) then
        tmp = c0 * ((c0 * (((d_1 / d) * (d_1 / d)) / (w * h))) / w)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (w <= -0.043) {
		tmp = 0.0;
	} else if (w <= 1.7e-238) {
		tmp = ((c0 / w) / 2.0) * (2.0 * (c0 * ((1.0 / (w * h)) * Math.pow((d / D), 2.0))));
	} else if (w <= 7.5e-202) {
		tmp = 0.0;
	} else if (w <= 1e+99) {
		tmp = c0 * ((c0 * (((d / D) * (d / D)) / (w * h))) / w);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if w <= -0.043:
		tmp = 0.0
	elif w <= 1.7e-238:
		tmp = ((c0 / w) / 2.0) * (2.0 * (c0 * ((1.0 / (w * h)) * math.pow((d / D), 2.0))))
	elif w <= 7.5e-202:
		tmp = 0.0
	elif w <= 1e+99:
		tmp = c0 * ((c0 * (((d / D) * (d / D)) / (w * h))) / w)
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (w <= -0.043)
		tmp = 0.0;
	elseif (w <= 1.7e-238)
		tmp = Float64(Float64(Float64(c0 / w) / 2.0) * Float64(2.0 * Float64(c0 * Float64(Float64(1.0 / Float64(w * h)) * (Float64(d / D) ^ 2.0)))));
	elseif (w <= 7.5e-202)
		tmp = 0.0;
	elseif (w <= 1e+99)
		tmp = Float64(c0 * Float64(Float64(c0 * Float64(Float64(Float64(d / D) * Float64(d / D)) / Float64(w * h))) / w));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (w <= -0.043)
		tmp = 0.0;
	elseif (w <= 1.7e-238)
		tmp = ((c0 / w) / 2.0) * (2.0 * (c0 * ((1.0 / (w * h)) * ((d / D) ^ 2.0))));
	elseif (w <= 7.5e-202)
		tmp = 0.0;
	elseif (w <= 1e+99)
		tmp = c0 * ((c0 * (((d / D) * (d / D)) / (w * h))) / w);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[w, -0.043], 0.0, If[LessEqual[w, 1.7e-238], N[(N[(N[(c0 / w), $MachinePrecision] / 2.0), $MachinePrecision] * N[(2.0 * N[(c0 * N[(N[(1.0 / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 7.5e-202], 0.0, If[LessEqual[w, 1e+99], N[(c0 * N[(N[(c0 * N[(N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 3 regimes

2. if w < -0.042999999999999997 or 1.69999999999999992e-238 < w < 7.50000000000000005e-202 or 9.9999999999999997e98 < w

   1. Initial program 13.6%

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

   3. Simplified 11.5%

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

   4. Taylor expanded in c0 around -inf 11.7%

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

      1. mul-1-neg 11.7%

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

      2. distribute-lft-in 11.7%

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

   6. Simplified 52.1%

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

   7. Taylor expanded in c0 around 0 55.2%

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

   if -0.042999999999999997 < w < 1.69999999999999992e-238

   1. Initial program 32.6%

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

   3. Simplified 43.0%

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

      1. sqrt-prod 42.6%

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

      2. fma-udef 42.6%

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

      3. *-commutative 42.6%

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

      4. fma-udef 42.6%

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

      5. associate-/r* 42.6%

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

      6. pow2 42.6%

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

      7. pow2 42.6%

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

   5. Applied egg-rr 42.6%

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

      1. *-commutative 42.6%

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

      2. fma-udef 42.6%

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

      3. unsub-neg 42.6%

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

      4. *-commutative 42.6%

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

      5. *-commutative 42.6%

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

      6. associate-*r* 43.6%

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

   7. Simplified 43.6%

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

   8. Taylor expanded in c0 around inf 42.3%

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

      1. associate-/r* 42.4%

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

      2. *-commutative 42.4%

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

      3. *-commutative 42.4%

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

      4. associate-/r* 42.3%

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

      5. associate-/l* 42.4%

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

      6. *-commutative 42.4%

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

   10. Simplified 42.4%

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

       1. associate-/r/ 42.4%

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

       2. *-un-lft-identity 42.4%

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

       3. *-commutative 42.4%

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

       4. *-commutative 42.4%

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

       5. times-frac 42.4%

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

       6. unpow2 42.4%

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

       7. unpow2 42.4%

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

       8. frac-times 53.8%

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

       9. pow2 53.8%

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

   12. Applied egg-rr 53.8%

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

   if 7.50000000000000005e-202 < w < 9.9999999999999997e98

   1. Initial program 27.4%

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

   3. Simplified 37.9%

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

      1. sqrt-prod 36.0%

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

      2. fma-udef 36.0%

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

      3. *-commutative 36.0%

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

      4. fma-udef 36.0%

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

      5. associate-/r* 36.0%

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

      6. pow2 36.0%

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

      7. pow2 36.0%

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

   5. Applied egg-rr 36.0%

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

      1. *-commutative 36.0%

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

      2. fma-udef 36.0%

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

      3. unsub-neg 36.0%

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

      4. *-commutative 36.0%

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

      5. *-commutative 36.0%

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

      6. associate-*r* 37.7%

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

   7. Simplified 37.7%

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

   8. Taylor expanded in c0 around inf 36.7%

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

      1. associate-/r* 38.4%

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

      2. *-commutative 38.4%

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

      3. *-commutative 38.4%

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

      4. associate-/r* 36.7%

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

      5. associate-/l* 36.8%

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

      6. *-commutative 36.8%

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

   10. Simplified 36.8%

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

       1. *-commutative 36.8%

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

       2. associate-/l/ 36.8%

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

       3. clear-num 36.8%

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

       4. un-div-inv 36.8%

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

   12. Applied egg-rr 45.8%

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

       1. times-frac 45.8%

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

       2. metadata-eval 45.8%

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

       3. *-lft-identity 45.8%

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

       4. associate-/r/ 44.3%

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

       5. associate-/r/ 44.3%

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

       6. unpow2 44.3%

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

       7. swap-sqr 42.5%

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

       8. unpow2 42.5%

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

       9. rem-square-sqrt 46.9%

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

       10. associate-*r/ 48.6%

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

       11. *-commutative 48.6%

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

       12. associate-*r/ 48.6%

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

       13. *-commutative 48.6%

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

   14. Simplified 48.6%

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

       1. unpow2 48.6%

          \[\leadsto \frac{c0 \cdot \frac{\color{blue}{\frac{d}{D} \cdot \frac{d}{D}}}{h \cdot w}}{w} \cdot c0 \]

   16. Applied egg-rr 48.6%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.043:\\ \;\;\;\;0\\ \mathbf{elif}\;w \leq 1.7 \cdot 10^{-238}:\\ \;\;\;\;\frac{\frac{c0}{w}}{2} \cdot \left(2 \cdot \left(c0 \cdot \left(\frac{1}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;w \leq 7.5 \cdot 10^{-202}:\\ \;\;\;\;0\\ \mathbf{elif}\;w \leq 10^{+99}:\\ \;\;\;\;c0 \cdot \frac{c0 \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w \cdot h}}{w}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6: 42.1% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -1.55 \cdot 10^{-5}:\\ \;\;\;\;0\\ \mathbf{elif}\;w \leq 1.6 \cdot 10^{-249} \lor \neg \left(w \leq 7.2 \cdot 10^{-202}\right) \land w \leq 1.45 \cdot 10^{+99}:\\ \;\;\;\;c0 \cdot \frac{c0 \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w \cdot h}}{w}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= w -1.55e-5)
   0.0
   (if (or (<= w 1.6e-249) (and (not (<= w 7.2e-202)) (<= w 1.45e+99)))
     (* c0 (/ (* c0 (/ (* (/ d D) (/ d D)) (* w h))) w))
     0.0)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (w <= -1.55e-5) {
		tmp = 0.0;
	} else if ((w <= 1.6e-249) || (!(w <= 7.2e-202) && (w <= 1.45e+99))) {
		tmp = c0 * ((c0 * (((d / D) * (d / D)) / (w * h))) / w);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (w <= (-1.55d-5)) then
        tmp = 0.0d0
    else if ((w <= 1.6d-249) .or. (.not. (w <= 7.2d-202)) .and. (w <= 1.45d+99)) then
        tmp = c0 * ((c0 * (((d_1 / d) * (d_1 / d)) / (w * h))) / w)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (w <= -1.55e-5) {
		tmp = 0.0;
	} else if ((w <= 1.6e-249) || (!(w <= 7.2e-202) && (w <= 1.45e+99))) {
		tmp = c0 * ((c0 * (((d / D) * (d / D)) / (w * h))) / w);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if w <= -1.55e-5:
		tmp = 0.0
	elif (w <= 1.6e-249) or (not (w <= 7.2e-202) and (w <= 1.45e+99)):
		tmp = c0 * ((c0 * (((d / D) * (d / D)) / (w * h))) / w)
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (w <= -1.55e-5)
		tmp = 0.0;
	elseif ((w <= 1.6e-249) || (!(w <= 7.2e-202) && (w <= 1.45e+99)))
		tmp = Float64(c0 * Float64(Float64(c0 * Float64(Float64(Float64(d / D) * Float64(d / D)) / Float64(w * h))) / w));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (w <= -1.55e-5)
		tmp = 0.0;
	elseif ((w <= 1.6e-249) || (~((w <= 7.2e-202)) && (w <= 1.45e+99)))
		tmp = c0 * ((c0 * (((d / D) * (d / D)) / (w * h))) / w);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[w, -1.55e-5], 0.0, If[Or[LessEqual[w, 1.6e-249], And[N[Not[LessEqual[w, 7.2e-202]], $MachinePrecision], LessEqual[w, 1.45e+99]]], N[(c0 * N[(N[(c0 * N[(N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if w < -1.55000000000000007e-5 or 1.6000000000000001e-249 < w < 7.2000000000000003e-202 or 1.4500000000000001e99 < w

   1. Initial program 13.6%

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

   3. Simplified 11.5%

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

   4. Taylor expanded in c0 around -inf 11.7%

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

      1. mul-1-neg 11.7%

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

      2. distribute-lft-in 11.7%

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

   6. Simplified 52.1%

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

   7. Taylor expanded in c0 around 0 55.2%

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

   if -1.55000000000000007e-5 < w < 1.6000000000000001e-249 or 7.2000000000000003e-202 < w < 1.4500000000000001e99

   1. Initial program 30.7%

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

   3. Simplified 41.1%

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

      1. sqrt-prod 40.2%

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

      2. fma-udef 40.2%

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

      3. *-commutative 40.2%

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

      4. fma-udef 40.2%

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

      5. associate-/r* 40.2%

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

      6. pow2 40.2%

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

      7. pow2 40.2%

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

   5. Applied egg-rr 40.2%

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

      1. *-commutative 40.2%

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

      2. fma-udef 40.2%

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

      3. unsub-neg 40.2%

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

      4. *-commutative 40.2%

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

      5. *-commutative 40.2%

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

      6. associate-*r* 41.4%

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

   7. Simplified 41.4%

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

   8. Taylor expanded in c0 around inf 40.2%

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

      1. associate-/r* 40.9%

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

      2. *-commutative 40.9%

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

      3. *-commutative 40.9%

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

      4. associate-/r* 40.2%

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

      5. associate-/l* 40.3%

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

      6. *-commutative 40.3%

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

   10. Simplified 40.3%

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

       1. *-commutative 40.3%

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

       2. associate-/l/ 40.3%

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

       3. clear-num 40.3%

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

       4. un-div-inv 40.3%

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

   12. Applied egg-rr 49.5%

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

       1. times-frac 49.5%

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

       2. metadata-eval 49.5%

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

       3. *-lft-identity 49.5%

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

       4. associate-/r/ 47.8%

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

       5. associate-/r/ 47.8%

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

       6. unpow2 47.8%

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

       7. swap-sqr 47.1%

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

       8. unpow2 47.1%

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

       9. rem-square-sqrt 50.6%

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

       10. associate-*r/ 51.3%

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

       11. *-commutative 51.3%

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

       12. associate-*r/ 51.9%

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

       13. *-commutative 51.9%

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

   14. Simplified 51.9%

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

       1. unpow2 51.9%

          \[\leadsto \frac{c0 \cdot \frac{\color{blue}{\frac{d}{D} \cdot \frac{d}{D}}}{h \cdot w}}{w} \cdot c0 \]

   16. Applied egg-rr 51.9%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -1.55 \cdot 10^{-5}:\\ \;\;\;\;0\\ \mathbf{elif}\;w \leq 1.6 \cdot 10^{-249} \lor \neg \left(w \leq 7.2 \cdot 10^{-202}\right) \land w \leq 1.45 \cdot 10^{+99}:\\ \;\;\;\;c0 \cdot \frac{c0 \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w \cdot h}}{w}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7: 33.9% accurate, 151.0× 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 24.1%

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

3. Simplified 22.6%

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

4. Taylor expanded in c0 around -inf 5.5%

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

   1. mul-1-neg 5.5%

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

   2. distribute-lft-in 5.5%

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

6. Simplified 30.2%

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

7. Taylor expanded in c0 around 0 34.9%

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

8. Final simplification 34.9%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023322 
(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))))))