Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.2% → 59.0%

Time: 31.1s

Alternatives: 11

Speedup: 151.0×

• could not determine a ground truth

  1. c0 = 3.576592467000211e+163
  2. w = -5.762267470057866e-293
  3. h = 8.066197159533724e-183
  4. D = 6.435504018288029e-298
  5. d = -2.917658357971397e+258
  6. M = 3.857776583854928e+127

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: 24.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 59.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c0 \cdot \left(d \cdot d\right)\\ t_1 := {\left(\frac{d}{D}\right)}^{2}\\ t_2 := \frac{c0}{2 \cdot w}\\ t_3 := \frac{t_0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_4 := t_2 \cdot \left(t_3 + \sqrt{t_3 \cdot t_3 - M \cdot M}\right)\\ \mathbf{if}\;t_4 \leq -4 \cdot 10^{+42}:\\ \;\;\;\;t_2 \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\left(w \cdot h\right) \cdot {D}^{2}}\right)\\ \mathbf{elif}\;t_4 \leq 0:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot D\right)\right)}{d \cdot d}\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;t_2 \cdot \left(\frac{c0}{w \cdot h} \cdot t_1 + \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, t_1, 0.5 \cdot \frac{w}{\frac{t_0}{h \cdot 0}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{D \cdot D}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* c0 (* d d)))
        (t_1 (pow (/ d D) 2.0))
        (t_2 (/ c0 (* 2.0 w)))
        (t_3 (/ t_0 (* (* w h) (* D D))))
        (t_4 (* t_2 (+ t_3 (sqrt (- (* t_3 t_3) (* M M)))))))
   (if (<= t_4 -4e+42)
     (* t_2 (* 2.0 (/ (* c0 (pow d 2.0)) (* (* w h) (pow D 2.0)))))
     (if (<= t_4 0.0)
       (* 0.25 (/ (* D (* (* M M) (* h D))) (* d d)))
       (if (<= t_4 INFINITY)
         (*
          t_2
          (+
           (* (/ c0 (* w h)) t_1)
           (fma (/ (/ c0 w) h) t_1 (* 0.5 (/ w (/ t_0 (* h 0.0)))))))
         (* 0.25 (* (/ (* D D) d) (/ (* h (* M M)) d))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 * (d * d);
	double t_1 = pow((d / D), 2.0);
	double t_2 = c0 / (2.0 * w);
	double t_3 = t_0 / ((w * h) * (D * D));
	double t_4 = t_2 * (t_3 + sqrt(((t_3 * t_3) - (M * M))));
	double tmp;
	if (t_4 <= -4e+42) {
		tmp = t_2 * (2.0 * ((c0 * pow(d, 2.0)) / ((w * h) * pow(D, 2.0))));
	} else if (t_4 <= 0.0) {
		tmp = 0.25 * ((D * ((M * M) * (h * D))) / (d * d));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_2 * (((c0 / (w * h)) * t_1) + fma(((c0 / w) / h), t_1, (0.5 * (w / (t_0 / (h * 0.0))))));
	} else {
		tmp = 0.25 * (((D * D) / d) * ((h * (M * M)) / d));
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 * Float64(d * d))
	t_1 = Float64(d / D) ^ 2.0
	t_2 = Float64(c0 / Float64(2.0 * w))
	t_3 = Float64(t_0 / Float64(Float64(w * h) * Float64(D * D)))
	t_4 = Float64(t_2 * Float64(t_3 + sqrt(Float64(Float64(t_3 * t_3) - Float64(M * M)))))
	tmp = 0.0
	if (t_4 <= -4e+42)
		tmp = Float64(t_2 * Float64(2.0 * Float64(Float64(c0 * (d ^ 2.0)) / Float64(Float64(w * h) * (D ^ 2.0)))));
	elseif (t_4 <= 0.0)
		tmp = Float64(0.25 * Float64(Float64(D * Float64(Float64(M * M) * Float64(h * D))) / Float64(d * d)));
	elseif (t_4 <= Inf)
		tmp = Float64(t_2 * Float64(Float64(Float64(c0 / Float64(w * h)) * t_1) + fma(Float64(Float64(c0 / w) / h), t_1, Float64(0.5 * Float64(w / Float64(t_0 / Float64(h * 0.0)))))));
	else
		tmp = Float64(0.25 * Float64(Float64(Float64(D * D) / d) * Float64(Float64(h * Float64(M * M)) / d)));
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 * N[(t$95$3 + N[Sqrt[N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -4e+42], N[(t$95$2 * N[(2.0 * N[(N[(c0 * N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[Power[D, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.0], N[(0.25 * N[(N[(D * N[(N[(M * M), $MachinePrecision] * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(t$95$2 * N[(N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(N[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision] * t$95$1 + N[(0.5 * N[(w / N[(t$95$0 / N[(h * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(N[(D * D), $MachinePrecision] / d), $MachinePrecision] * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

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))))) < -4.00000000000000018e42

   1. Initial program 73.8%

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

      1. times-frac 68.8%

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

      2. fma-def 68.8%

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

      3. associate-/r* 68.8%

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

      4. difference-of-squares 68.8%

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

   3. Simplified 71.3%

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

   4. Taylor expanded in c0 around inf 79.6%

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

   if -4.00000000000000018e42 < (*.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 36.4%

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

   2. Taylor expanded in c0 around -inf 67.3%

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

      1. fma-def 67.3%

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

      2. times-frac 59.3%

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

      3. unpow2 59.3%

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

      4. unpow2 59.3%

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

      5. *-commutative 59.3%

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

      6. unpow2 59.3%

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

      7. associate-*r* 59.3%

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

   4. Simplified 59.3%

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

   5. Taylor expanded in c0 around 0 67.8%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}} \]
   6. Step-by-step derivation

      1. associate-/l* 67.5%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]

      2. associate-*r/ 67.5%

         \[\leadsto \color{blue}{\frac{0.25 \cdot {D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]

      3. unpow2 67.5%

         \[\leadsto \frac{0.25 \cdot \color{blue}{\left(D \cdot D\right)}}{\frac{{d}^{2}}{h \cdot {M}^{2}}} \]

      4. unpow2 67.5%

         \[\leadsto \frac{0.25 \cdot \left(D \cdot D\right)}{\frac{\color{blue}{d \cdot d}}{h \cdot {M}^{2}}} \]

      5. unpow2 67.5%

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

   7. Simplified 67.5%

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

   8. Taylor expanded in D around 0 67.8%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. *-commutative 67.8%

         \[\leadsto 0.25 \cdot \frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{{d}^{2}} \]

      2. unpow2 67.8%

         \[\leadsto 0.25 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \]

      3. *-commutative 67.8%

         \[\leadsto 0.25 \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{{d}^{2}} \]

      4. unpow2 67.8%

         \[\leadsto 0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)}{{d}^{2}} \]

      5. unpow2 67.8%

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

   10. Simplified 67.8%

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

   11. Taylor expanded in D around 0 67.8%

       \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}} \]
   12. Step-by-step derivation

       1. unpow2 67.8%

          \[\leadsto 0.25 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}} \]

       2. unpow2 67.8%

          \[\leadsto 0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)}{{d}^{2}} \]

       3. associate-*r* 67.9%

          \[\leadsto 0.25 \cdot \frac{\color{blue}{D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}}{{d}^{2}} \]

       4. unpow2 67.9%

          \[\leadsto 0.25 \cdot \frac{D \cdot \left(D \cdot \left(h \cdot \color{blue}{{M}^{2}}\right)\right)}{{d}^{2}} \]

       5. associate-*r* 67.9%

          \[\leadsto 0.25 \cdot \frac{D \cdot \color{blue}{\left(\left(D \cdot h\right) \cdot {M}^{2}\right)}}{{d}^{2}} \]

       6. unpow2 67.9%

          \[\leadsto 0.25 \cdot \frac{D \cdot \left(\left(D \cdot h\right) \cdot \color{blue}{\left(M \cdot M\right)}\right)}{{d}^{2}} \]

       7. unpow2 67.9%

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

   13. Simplified 67.9%

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

   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 85.4%

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

      1. times-frac 85.4%

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

      2. fma-def 85.4%

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

      3. times-frac 85.3%

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

      4. difference-of-squares 85.3%

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

   3. Simplified 85.3%

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

      1. fma-udef 85.3%

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

      2. pow2 85.3%

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

      3. associate-*l* 85.3%

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

      4. div-inv 85.3%

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

      5. clear-num 85.3%

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

      6. associate-*r/ 85.3%

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

      7. *-commutative 85.3%

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

   5. Applied egg-rr 88.2%

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

      1. fma-udef 88.2%

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

      2. associate-*r/ 88.2%

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

      3. associate-/l* 88.2%

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

   7. Applied egg-rr 88.2%

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

   8. Taylor expanded in D around 0 27.6%

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

      1. +-commutative 27.6%

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

      2. times-frac 27.6%

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

      3. unpow2 27.6%

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

      4. unpow2 27.6%

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

      5. times-frac 30.5%

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

      6. unpow2 30.5%

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

      7. *-commutative 30.5%

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

      8. fma-def 30.5%

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

      9. associate-/r* 30.5%

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

   10. Simplified 91.1%

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

   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%

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

   2. Taylor expanded in c0 around -inf 1.6%

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

      1. fma-def 1.6%

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

      2. times-frac 1.0%

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

      3. unpow2 1.0%

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

      4. unpow2 1.0%

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

      5. *-commutative 1.0%

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

      6. unpow2 1.0%

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

      7. associate-*r* 1.0%

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

   4. Simplified 31.3%

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

   5. Taylor expanded in c0 around 0 44.4%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}} \]
   6. Step-by-step derivation

      1. associate-/l* 44.2%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]

      2. associate-*r/ 44.2%

         \[\leadsto \color{blue}{\frac{0.25 \cdot {D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]

      3. unpow2 44.2%

         \[\leadsto \frac{0.25 \cdot \color{blue}{\left(D \cdot D\right)}}{\frac{{d}^{2}}{h \cdot {M}^{2}}} \]

      4. unpow2 44.2%

         \[\leadsto \frac{0.25 \cdot \left(D \cdot D\right)}{\frac{\color{blue}{d \cdot d}}{h \cdot {M}^{2}}} \]

      5. unpow2 44.2%

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

   7. Simplified 44.2%

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

   8. Taylor expanded in D around 0 44.4%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. *-commutative 44.4%

         \[\leadsto 0.25 \cdot \frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{{d}^{2}} \]

      2. unpow2 44.4%

         \[\leadsto 0.25 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \]

      3. *-commutative 44.4%

         \[\leadsto 0.25 \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{{d}^{2}} \]

      4. unpow2 44.4%

         \[\leadsto 0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)}{{d}^{2}} \]

      5. unpow2 44.4%

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

   10. Simplified 44.4%

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

       1. times-frac 49.8%

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

   12. Applied egg-rr 49.8%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq -4 \cdot 10^{+42}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\left(w \cdot h\right) \cdot {D}^{2}}\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 0:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot D\right)\right)}{d \cdot d}\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2} + \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, {\left(\frac{d}{D}\right)}^{2}, 0.5 \cdot \frac{w}{\frac{c0 \cdot \left(d \cdot d\right)}{h \cdot 0}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{D \cdot D}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\\ \end{array} \]

Alternative 2: 58.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := t_0 \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\left(w \cdot h\right) \cdot {D}^{2}}\right)\\ t_2 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_3 := t_0 \cdot \left(t_2 + \sqrt{t_2 \cdot t_2 - M \cdot M}\right)\\ \mathbf{if}\;t_3 \leq -4 \cdot 10^{+42}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot D\right)\right)}{d \cdot d}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{D \cdot D}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w)))
        (t_1 (* t_0 (* 2.0 (/ (* c0 (pow d 2.0)) (* (* w h) (pow D 2.0))))))
        (t_2 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_3 (* t_0 (+ t_2 (sqrt (- (* t_2 t_2) (* M M)))))))
   (if (<= t_3 -4e+42)
     t_1
     (if (<= t_3 0.0)
       (* 0.25 (/ (* D (* (* M M) (* h D))) (* d d)))
       (if (<= t_3 INFINITY)
         t_1
         (* 0.25 (* (/ (* D D) d) (/ (* h (* M M)) d))))))))

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 = t_0 * (2.0 * ((c0 * pow(d, 2.0)) / ((w * h) * pow(D, 2.0))));
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_3 = t_0 * (t_2 + sqrt(((t_2 * t_2) - (M * M))));
	double tmp;
	if (t_3 <= -4e+42) {
		tmp = t_1;
	} else if (t_3 <= 0.0) {
		tmp = 0.25 * ((D * ((M * M) * (h * D))) / (d * d));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = 0.25 * (((D * D) / d) * ((h * (M * M)) / d));
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = t_0 * (2.0 * ((c0 * Math.pow(d, 2.0)) / ((w * h) * Math.pow(D, 2.0))));
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_3 = t_0 * (t_2 + Math.sqrt(((t_2 * t_2) - (M * M))));
	double tmp;
	if (t_3 <= -4e+42) {
		tmp = t_1;
	} else if (t_3 <= 0.0) {
		tmp = 0.25 * ((D * ((M * M) * (h * D))) / (d * d));
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = 0.25 * (((D * D) / d) * ((h * (M * M)) / d));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	t_1 = t_0 * (2.0 * ((c0 * math.pow(d, 2.0)) / ((w * h) * math.pow(D, 2.0))))
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D))
	t_3 = t_0 * (t_2 + math.sqrt(((t_2 * t_2) - (M * M))))
	tmp = 0
	if t_3 <= -4e+42:
		tmp = t_1
	elif t_3 <= 0.0:
		tmp = 0.25 * ((D * ((M * M) * (h * D))) / (d * d))
	elif t_3 <= math.inf:
		tmp = t_1
	else:
		tmp = 0.25 * (((D * D) / d) * ((h * (M * M)) / d))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(t_0 * Float64(2.0 * Float64(Float64(c0 * (d ^ 2.0)) / Float64(Float64(w * h) * (D ^ 2.0)))))
	t_2 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	t_3 = Float64(t_0 * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M)))))
	tmp = 0.0
	if (t_3 <= -4e+42)
		tmp = t_1;
	elseif (t_3 <= 0.0)
		tmp = Float64(0.25 * Float64(Float64(D * Float64(Float64(M * M) * Float64(h * D))) / Float64(d * d)));
	elseif (t_3 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(0.25 * Float64(Float64(Float64(D * D) / d) * Float64(Float64(h * Float64(M * M)) / d)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	t_1 = t_0 * (2.0 * ((c0 * (d ^ 2.0)) / ((w * h) * (D ^ 2.0))));
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	t_3 = t_0 * (t_2 + sqrt(((t_2 * t_2) - (M * M))));
	tmp = 0.0;
	if (t_3 <= -4e+42)
		tmp = t_1;
	elseif (t_3 <= 0.0)
		tmp = 0.25 * ((D * ((M * M) * (h * D))) / (d * d));
	elseif (t_3 <= Inf)
		tmp = t_1;
	else
		tmp = 0.25 * (((D * D) / d) * ((h * (M * M)) / d));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(2.0 * N[(N[(c0 * N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[Power[D, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 * N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -4e+42], t$95$1, If[LessEqual[t$95$3, 0.0], N[(0.25 * N[(N[(D * N[(N[(M * M), $MachinePrecision] * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$1, N[(0.25 * N[(N[(N[(D * D), $MachinePrecision] / d), $MachinePrecision] * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 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))))) < -4.00000000000000018e42 or 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 79.0%

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

      1. times-frac 76.3%

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

      2. fma-def 76.3%

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

      3. associate-/r* 76.3%

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

      4. difference-of-squares 76.3%

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

   3. Simplified 79.0%

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

   4. Taylor expanded in c0 around inf 83.6%

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

   if -4.00000000000000018e42 < (*.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 36.4%

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

   2. Taylor expanded in c0 around -inf 67.3%

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

      1. fma-def 67.3%

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

      2. times-frac 59.3%

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

      3. unpow2 59.3%

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

      4. unpow2 59.3%

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

      5. *-commutative 59.3%

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

      6. unpow2 59.3%

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

      7. associate-*r* 59.3%

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

   4. Simplified 59.3%

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

   5. Taylor expanded in c0 around 0 67.8%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}} \]
   6. Step-by-step derivation

      1. associate-/l* 67.5%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]

      2. associate-*r/ 67.5%

         \[\leadsto \color{blue}{\frac{0.25 \cdot {D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]

      3. unpow2 67.5%

         \[\leadsto \frac{0.25 \cdot \color{blue}{\left(D \cdot D\right)}}{\frac{{d}^{2}}{h \cdot {M}^{2}}} \]

      4. unpow2 67.5%

         \[\leadsto \frac{0.25 \cdot \left(D \cdot D\right)}{\frac{\color{blue}{d \cdot d}}{h \cdot {M}^{2}}} \]

      5. unpow2 67.5%

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

   7. Simplified 67.5%

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

   8. Taylor expanded in D around 0 67.8%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. *-commutative 67.8%

         \[\leadsto 0.25 \cdot \frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{{d}^{2}} \]

      2. unpow2 67.8%

         \[\leadsto 0.25 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \]

      3. *-commutative 67.8%

         \[\leadsto 0.25 \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{{d}^{2}} \]

      4. unpow2 67.8%

         \[\leadsto 0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)}{{d}^{2}} \]

      5. unpow2 67.8%

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

   10. Simplified 67.8%

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

   11. Taylor expanded in D around 0 67.8%

       \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}} \]
   12. Step-by-step derivation

       1. unpow2 67.8%

          \[\leadsto 0.25 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}} \]

       2. unpow2 67.8%

          \[\leadsto 0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)}{{d}^{2}} \]

       3. associate-*r* 67.9%

          \[\leadsto 0.25 \cdot \frac{\color{blue}{D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}}{{d}^{2}} \]

       4. unpow2 67.9%

          \[\leadsto 0.25 \cdot \frac{D \cdot \left(D \cdot \left(h \cdot \color{blue}{{M}^{2}}\right)\right)}{{d}^{2}} \]

       5. associate-*r* 67.9%

          \[\leadsto 0.25 \cdot \frac{D \cdot \color{blue}{\left(\left(D \cdot h\right) \cdot {M}^{2}\right)}}{{d}^{2}} \]

       6. unpow2 67.9%

          \[\leadsto 0.25 \cdot \frac{D \cdot \left(\left(D \cdot h\right) \cdot \color{blue}{\left(M \cdot M\right)}\right)}{{d}^{2}} \]

       7. unpow2 67.9%

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

   13. Simplified 67.9%

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

   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%

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

   2. Taylor expanded in c0 around -inf 1.6%

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

      1. fma-def 1.6%

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

      2. times-frac 1.0%

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

      3. unpow2 1.0%

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

      4. unpow2 1.0%

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

      5. *-commutative 1.0%

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

      6. unpow2 1.0%

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

      7. associate-*r* 1.0%

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

   4. Simplified 31.3%

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

   5. Taylor expanded in c0 around 0 44.4%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}} \]
   6. Step-by-step derivation

      1. associate-/l* 44.2%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]

      2. associate-*r/ 44.2%

         \[\leadsto \color{blue}{\frac{0.25 \cdot {D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]

      3. unpow2 44.2%

         \[\leadsto \frac{0.25 \cdot \color{blue}{\left(D \cdot D\right)}}{\frac{{d}^{2}}{h \cdot {M}^{2}}} \]

      4. unpow2 44.2%

         \[\leadsto \frac{0.25 \cdot \left(D \cdot D\right)}{\frac{\color{blue}{d \cdot d}}{h \cdot {M}^{2}}} \]

      5. unpow2 44.2%

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

   7. Simplified 44.2%

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

   8. Taylor expanded in D around 0 44.4%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. *-commutative 44.4%

         \[\leadsto 0.25 \cdot \frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{{d}^{2}} \]

      2. unpow2 44.4%

         \[\leadsto 0.25 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \]

      3. *-commutative 44.4%

         \[\leadsto 0.25 \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{{d}^{2}} \]

      4. unpow2 44.4%

         \[\leadsto 0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)}{{d}^{2}} \]

      5. unpow2 44.4%

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

   10. Simplified 44.4%

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

       1. times-frac 49.8%

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

   12. Applied egg-rr 49.8%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq -4 \cdot 10^{+42}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\left(w \cdot h\right) \cdot {D}^{2}}\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 0:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot D\right)\right)}{d \cdot d}\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\left(w \cdot h\right) \cdot {D}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{D \cdot D}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\\ \end{array} \]

Alternative 3: 57.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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 73.0%

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

   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%

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

   2. Taylor expanded in c0 around -inf 1.6%

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

      1. fma-def 1.6%

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

      2. times-frac 1.0%

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

      3. unpow2 1.0%

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

      4. unpow2 1.0%

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

      5. *-commutative 1.0%

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

      6. unpow2 1.0%

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

      7. associate-*r* 1.0%

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

   4. Simplified 31.3%

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

   5. Taylor expanded in c0 around 0 44.4%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}} \]
   6. Step-by-step derivation

      1. associate-/l* 44.2%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]

      2. associate-*r/ 44.2%

         \[\leadsto \color{blue}{\frac{0.25 \cdot {D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]

      3. unpow2 44.2%

         \[\leadsto \frac{0.25 \cdot \color{blue}{\left(D \cdot D\right)}}{\frac{{d}^{2}}{h \cdot {M}^{2}}} \]

      4. unpow2 44.2%

         \[\leadsto \frac{0.25 \cdot \left(D \cdot D\right)}{\frac{\color{blue}{d \cdot d}}{h \cdot {M}^{2}}} \]

      5. unpow2 44.2%

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

   7. Simplified 44.2%

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

   8. Taylor expanded in D around 0 44.4%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. *-commutative 44.4%

         \[\leadsto 0.25 \cdot \frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{{d}^{2}} \]

      2. unpow2 44.4%

         \[\leadsto 0.25 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \]

      3. *-commutative 44.4%

         \[\leadsto 0.25 \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{{d}^{2}} \]

      4. unpow2 44.4%

         \[\leadsto 0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)}{{d}^{2}} \]

      5. unpow2 44.4%

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

   10. Simplified 44.4%

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

       1. times-frac 49.8%

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

   12. Applied egg-rr 49.8%

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

4. Final simplification 57.5%

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

Alternative 4: 37.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{w \cdot \left(h \cdot D\right)} \cdot \frac{d}{\frac{D}{d}}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := t_1 \cdot \left(\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\right)\\ \mathbf{if}\;c0 \leq -2.9 \cdot 10^{-107}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c0 \leq 6.5 \cdot 10^{-84}:\\ \;\;\;\;\frac{\left(D \cdot D\right) \cdot 0.25}{\frac{d}{h} \cdot \frac{d}{M \cdot M}}\\ \mathbf{elif}\;c0 \leq 16:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c0 \leq 4.9 \cdot 10^{+228}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot D\right)\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(t_0 + \sqrt{M \cdot \left(t_0 - M\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ c0 (* w (* h D))) (/ d (/ D d))))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (* t_1 (* (/ c0 (* w h)) (pow (/ d D) 2.0)))))
   (if (<= c0 -2.9e-107)
     t_2
     (if (<= c0 6.5e-84)
       (/ (* (* D D) 0.25) (* (/ d h) (/ d (* M M))))
       (if (<= c0 16.0)
         t_2
         (if (<= c0 4.9e+228)
           (* 0.25 (/ (* D (* (* M M) (* h D))) (* d d)))
           (* t_1 (+ t_0 (sqrt (* M (- t_0 M)))))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 / (w * (h * D))) * (d / (D / d));
	double t_1 = c0 / (2.0 * w);
	double t_2 = t_1 * ((c0 / (w * h)) * pow((d / D), 2.0));
	double tmp;
	if (c0 <= -2.9e-107) {
		tmp = t_2;
	} else if (c0 <= 6.5e-84) {
		tmp = ((D * D) * 0.25) / ((d / h) * (d / (M * M)));
	} else if (c0 <= 16.0) {
		tmp = t_2;
	} else if (c0 <= 4.9e+228) {
		tmp = 0.25 * ((D * ((M * M) * (h * D))) / (d * d));
	} else {
		tmp = t_1 * (t_0 + sqrt((M * (t_0 - M))));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (c0 / (w * (h * d))) * (d_1 / (d / d_1))
    t_1 = c0 / (2.0d0 * w)
    t_2 = t_1 * ((c0 / (w * h)) * ((d_1 / d) ** 2.0d0))
    if (c0 <= (-2.9d-107)) then
        tmp = t_2
    else if (c0 <= 6.5d-84) then
        tmp = ((d * d) * 0.25d0) / ((d_1 / h) * (d_1 / (m * m)))
    else if (c0 <= 16.0d0) then
        tmp = t_2
    else if (c0 <= 4.9d+228) then
        tmp = 0.25d0 * ((d * ((m * m) * (h * d))) / (d_1 * d_1))
    else
        tmp = t_1 * (t_0 + sqrt((m * (t_0 - m))))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 / (w * (h * D))) * (d / (D / d));
	double t_1 = c0 / (2.0 * w);
	double t_2 = t_1 * ((c0 / (w * h)) * Math.pow((d / D), 2.0));
	double tmp;
	if (c0 <= -2.9e-107) {
		tmp = t_2;
	} else if (c0 <= 6.5e-84) {
		tmp = ((D * D) * 0.25) / ((d / h) * (d / (M * M)));
	} else if (c0 <= 16.0) {
		tmp = t_2;
	} else if (c0 <= 4.9e+228) {
		tmp = 0.25 * ((D * ((M * M) * (h * D))) / (d * d));
	} else {
		tmp = t_1 * (t_0 + Math.sqrt((M * (t_0 - M))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 / (w * (h * D))) * (d / (D / d))
	t_1 = c0 / (2.0 * w)
	t_2 = t_1 * ((c0 / (w * h)) * math.pow((d / D), 2.0))
	tmp = 0
	if c0 <= -2.9e-107:
		tmp = t_2
	elif c0 <= 6.5e-84:
		tmp = ((D * D) * 0.25) / ((d / h) * (d / (M * M)))
	elif c0 <= 16.0:
		tmp = t_2
	elif c0 <= 4.9e+228:
		tmp = 0.25 * ((D * ((M * M) * (h * D))) / (d * d))
	else:
		tmp = t_1 * (t_0 + math.sqrt((M * (t_0 - M))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 / Float64(w * Float64(h * D))) * Float64(d / Float64(D / d)))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(t_1 * Float64(Float64(c0 / Float64(w * h)) * (Float64(d / D) ^ 2.0)))
	tmp = 0.0
	if (c0 <= -2.9e-107)
		tmp = t_2;
	elseif (c0 <= 6.5e-84)
		tmp = Float64(Float64(Float64(D * D) * 0.25) / Float64(Float64(d / h) * Float64(d / Float64(M * M))));
	elseif (c0 <= 16.0)
		tmp = t_2;
	elseif (c0 <= 4.9e+228)
		tmp = Float64(0.25 * Float64(Float64(D * Float64(Float64(M * M) * Float64(h * D))) / Float64(d * d)));
	else
		tmp = Float64(t_1 * Float64(t_0 + sqrt(Float64(M * Float64(t_0 - M)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 / (w * (h * D))) * (d / (D / d));
	t_1 = c0 / (2.0 * w);
	t_2 = t_1 * ((c0 / (w * h)) * ((d / D) ^ 2.0));
	tmp = 0.0;
	if (c0 <= -2.9e-107)
		tmp = t_2;
	elseif (c0 <= 6.5e-84)
		tmp = ((D * D) * 0.25) / ((d / h) * (d / (M * M)));
	elseif (c0 <= 16.0)
		tmp = t_2;
	elseif (c0 <= 4.9e+228)
		tmp = 0.25 * ((D * ((M * M) * (h * D))) / (d * d));
	else
		tmp = t_1 * (t_0 + sqrt((M * (t_0 - M))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 / N[(w * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c0, -2.9e-107], t$95$2, If[LessEqual[c0, 6.5e-84], N[(N[(N[(D * D), $MachinePrecision] * 0.25), $MachinePrecision] / N[(N[(d / h), $MachinePrecision] * N[(d / N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c0, 16.0], t$95$2, If[LessEqual[c0, 4.9e+228], N[(0.25 * N[(N[(D * N[(N[(M * M), $MachinePrecision] * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(t$95$0 + N[Sqrt[N[(M * N[(t$95$0 - M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if c0 < -2.8999999999999998e-107 or 6.50000000000000022e-84 < c0 < 16

   1. Initial program 30.1%

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

      1. times-frac 29.3%

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

      2. fma-def 29.3%

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

      3. associate-/r* 29.3%

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

      4. difference-of-squares 33.0%

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

   3. Simplified 42.5%

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

   4. Taylor expanded in c0 around 0 17.3%

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

      1. div-inv 17.3%

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

   6. Applied egg-rr 17.3%

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

   7. Taylor expanded in c0 around inf 31.4%

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

      1. times-frac 32.6%

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

      2. unpow2 32.6%

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

      3. unpow2 32.6%

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

      4. times-frac 43.4%

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

      5. unpow2 43.4%

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

      6. *-commutative 43.4%

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

      7. *-commutative 43.4%

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

   9. Simplified 43.4%

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

   if -2.8999999999999998e-107 < c0 < 6.50000000000000022e-84

   1. Initial program 16.2%

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

   2. Taylor expanded in c0 around -inf 6.0%

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

      1. fma-def 6.0%

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

      2. times-frac 4.8%

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

      3. unpow2 4.8%

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

      4. unpow2 4.8%

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

      5. *-commutative 4.8%

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

      6. unpow2 4.8%

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

      7. associate-*r* 4.8%

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

   4. Simplified 35.8%

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

   5. Taylor expanded in c0 around 0 43.0%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}} \]
   6. Step-by-step derivation

      1. associate-/l* 43.0%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]

      2. associate-*r/ 43.0%

         \[\leadsto \color{blue}{\frac{0.25 \cdot {D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]

      3. unpow2 43.0%

         \[\leadsto \frac{0.25 \cdot \color{blue}{\left(D \cdot D\right)}}{\frac{{d}^{2}}{h \cdot {M}^{2}}} \]

      4. unpow2 43.0%

         \[\leadsto \frac{0.25 \cdot \left(D \cdot D\right)}{\frac{\color{blue}{d \cdot d}}{h \cdot {M}^{2}}} \]

      5. unpow2 43.0%

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

   7. Simplified 43.0%

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

   8. Taylor expanded in d around 0 43.0%

      \[\leadsto \frac{0.25 \cdot \left(D \cdot D\right)}{\color{blue}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]
   9. Step-by-step derivation

      1. unpow2 43.0%

         \[\leadsto \frac{0.25 \cdot \left(D \cdot D\right)}{\frac{\color{blue}{d \cdot d}}{h \cdot {M}^{2}}} \]

      2. times-frac 50.4%

         \[\leadsto \frac{0.25 \cdot \left(D \cdot D\right)}{\color{blue}{\frac{d}{h} \cdot \frac{d}{{M}^{2}}}} \]

      3. unpow2 50.4%

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

   10. Simplified 50.4%

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

   if 16 < c0 < 4.9000000000000002e228

   1. Initial program 16.2%

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

   2. Taylor expanded in c0 around -inf 9.4%

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

      1. fma-def 9.4%

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

      2. times-frac 9.2%

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

      3. unpow2 9.2%

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

      4. unpow2 9.2%

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

      5. *-commutative 9.2%

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

      6. unpow2 9.2%

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

      7. associate-*r* 9.2%

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

   4. Simplified 29.8%

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

   5. Taylor expanded in c0 around 0 44.6%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}} \]
   6. Step-by-step derivation

      1. associate-/l* 46.0%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]

      2. associate-*r/ 46.0%

         \[\leadsto \color{blue}{\frac{0.25 \cdot {D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]

      3. unpow2 46.0%

         \[\leadsto \frac{0.25 \cdot \color{blue}{\left(D \cdot D\right)}}{\frac{{d}^{2}}{h \cdot {M}^{2}}} \]

      4. unpow2 46.0%

         \[\leadsto \frac{0.25 \cdot \left(D \cdot D\right)}{\frac{\color{blue}{d \cdot d}}{h \cdot {M}^{2}}} \]

      5. unpow2 46.0%

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

   7. Simplified 46.0%

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

   8. Taylor expanded in D around 0 44.6%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. *-commutative 44.6%

         \[\leadsto 0.25 \cdot \frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{{d}^{2}} \]

      2. unpow2 44.6%

         \[\leadsto 0.25 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \]

      3. *-commutative 44.6%

         \[\leadsto 0.25 \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{{d}^{2}} \]

      4. unpow2 44.6%

         \[\leadsto 0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)}{{d}^{2}} \]

      5. unpow2 44.6%

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

   10. Simplified 44.6%

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

   11. Taylor expanded in D around 0 44.6%

       \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}} \]
   12. Step-by-step derivation

       1. unpow2 44.6%

          \[\leadsto 0.25 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}} \]

       2. unpow2 44.6%

          \[\leadsto 0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)}{{d}^{2}} \]

       3. associate-*r* 54.2%

          \[\leadsto 0.25 \cdot \frac{\color{blue}{D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}}{{d}^{2}} \]

       4. unpow2 54.2%

          \[\leadsto 0.25 \cdot \frac{D \cdot \left(D \cdot \left(h \cdot \color{blue}{{M}^{2}}\right)\right)}{{d}^{2}} \]

       5. associate-*r* 54.0%

          \[\leadsto 0.25 \cdot \frac{D \cdot \color{blue}{\left(\left(D \cdot h\right) \cdot {M}^{2}\right)}}{{d}^{2}} \]

       6. unpow2 54.0%

          \[\leadsto 0.25 \cdot \frac{D \cdot \left(\left(D \cdot h\right) \cdot \color{blue}{\left(M \cdot M\right)}\right)}{{d}^{2}} \]

       7. unpow2 54.0%

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

   13. Simplified 54.0%

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

   if 4.9000000000000002e228 < c0

   1. Initial program 42.9%

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

      1. times-frac 42.9%

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

      2. fma-def 42.9%

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

      3. associate-/r* 42.9%

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

      4. difference-of-squares 52.4%

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

   3. Simplified 52.9%

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

   4. Taylor expanded in c0 around 0 33.9%

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

      1. div-inv 33.9%

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

   6. Applied egg-rr 33.9%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\color{blue}{c0 \cdot \frac{1}{w \cdot h}}, \frac{\frac{d \cdot d}{D}}{D}, \sqrt{M \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{\frac{d \cdot d}{D}}{D} - M\right)}\right) \]
   7. Step-by-step derivation

      1. fma-udef 33.9%

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

      2. div-inv 33.9%

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

      3. frac-times 33.9%

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

      4. associate-/l* 33.9%

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

      5. associate-*r* 33.9%

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

      6. associate-*l/ 33.9%

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

      7. frac-times 33.7%

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

      8. frac-times 33.7%

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

      9. associate-/l* 38.4%

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

      10. associate-*r* 38.4%

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

      11. associate-*l/ 38.4%

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

      12. frac-times 38.4%

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

   8. Applied egg-rr 38.4%

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

      1. times-frac 38.4%

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

      2. *-commutative 38.4%

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

      3. times-frac 38.6%

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

      4. *-commutative 38.6%

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

   10. Simplified 38.6%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\frac{c0}{w \cdot \left(D \cdot h\right)} \cdot \frac{d}{\frac{D}{d}} + \sqrt{M \cdot \left(\frac{c0}{w \cdot \left(D \cdot h\right)} \cdot \frac{d}{\frac{D}{d}} - M\right)}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -2.9 \cdot 10^{-107}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\right)\\ \mathbf{elif}\;c0 \leq 6.5 \cdot 10^{-84}:\\ \;\;\;\;\frac{\left(D \cdot D\right) \cdot 0.25}{\frac{d}{h} \cdot \frac{d}{M \cdot M}}\\ \mathbf{elif}\;c0 \leq 16:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\right)\\ \mathbf{elif}\;c0 \leq 4.9 \cdot 10^{+228}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot D\right)\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot \left(h \cdot D\right)} \cdot \frac{d}{\frac{D}{d}} + \sqrt{M \cdot \left(\frac{c0}{w \cdot \left(h \cdot D\right)} \cdot \frac{d}{\frac{D}{d}} - M\right)}\right)\\ \end{array} \]

Alternative 5: 39.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\right)\\ \mathbf{if}\;c0 \leq -1.55 \cdot 10^{-106}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c0 \leq 1.9 \cdot 10^{-82}:\\ \;\;\;\;\frac{\left(D \cdot D\right) \cdot 0.25}{\frac{d}{h} \cdot \frac{d}{M \cdot M}}\\ \mathbf{elif}\;c0 \leq 18000 \lor \neg \left(c0 \leq 2.6 \cdot 10^{+228}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot D\right)\right)}{d \cdot d}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ c0 (* 2.0 w)) (* (/ c0 (* w h)) (pow (/ d D) 2.0)))))
   (if (<= c0 -1.55e-106)
     t_0
     (if (<= c0 1.9e-82)
       (/ (* (* D D) 0.25) (* (/ d h) (/ d (* M M))))
       (if (or (<= c0 18000.0) (not (<= c0 2.6e+228)))
         t_0
         (* 0.25 (/ (* D (* (* M M) (* h D))) (* d d))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 / (2.0 * w)) * ((c0 / (w * h)) * pow((d / D), 2.0));
	double tmp;
	if (c0 <= -1.55e-106) {
		tmp = t_0;
	} else if (c0 <= 1.9e-82) {
		tmp = ((D * D) * 0.25) / ((d / h) * (d / (M * M)));
	} else if ((c0 <= 18000.0) || !(c0 <= 2.6e+228)) {
		tmp = t_0;
	} else {
		tmp = 0.25 * ((D * ((M * M) * (h * D))) / (d * d));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (c0 / (2.0d0 * w)) * ((c0 / (w * h)) * ((d_1 / d) ** 2.0d0))
    if (c0 <= (-1.55d-106)) then
        tmp = t_0
    else if (c0 <= 1.9d-82) then
        tmp = ((d * d) * 0.25d0) / ((d_1 / h) * (d_1 / (m * m)))
    else if ((c0 <= 18000.0d0) .or. (.not. (c0 <= 2.6d+228))) then
        tmp = t_0
    else
        tmp = 0.25d0 * ((d * ((m * m) * (h * d))) / (d_1 * d_1))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 / (2.0 * w)) * ((c0 / (w * h)) * Math.pow((d / D), 2.0));
	double tmp;
	if (c0 <= -1.55e-106) {
		tmp = t_0;
	} else if (c0 <= 1.9e-82) {
		tmp = ((D * D) * 0.25) / ((d / h) * (d / (M * M)));
	} else if ((c0 <= 18000.0) || !(c0 <= 2.6e+228)) {
		tmp = t_0;
	} else {
		tmp = 0.25 * ((D * ((M * M) * (h * D))) / (d * d));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 / (2.0 * w)) * ((c0 / (w * h)) * math.pow((d / D), 2.0))
	tmp = 0
	if c0 <= -1.55e-106:
		tmp = t_0
	elif c0 <= 1.9e-82:
		tmp = ((D * D) * 0.25) / ((d / h) * (d / (M * M)))
	elif (c0 <= 18000.0) or not (c0 <= 2.6e+228):
		tmp = t_0
	else:
		tmp = 0.25 * ((D * ((M * M) * (h * D))) / (d * d))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(Float64(c0 / Float64(w * h)) * (Float64(d / D) ^ 2.0)))
	tmp = 0.0
	if (c0 <= -1.55e-106)
		tmp = t_0;
	elseif (c0 <= 1.9e-82)
		tmp = Float64(Float64(Float64(D * D) * 0.25) / Float64(Float64(d / h) * Float64(d / Float64(M * M))));
	elseif ((c0 <= 18000.0) || !(c0 <= 2.6e+228))
		tmp = t_0;
	else
		tmp = Float64(0.25 * Float64(Float64(D * Float64(Float64(M * M) * Float64(h * D))) / Float64(d * d)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 / (2.0 * w)) * ((c0 / (w * h)) * ((d / D) ^ 2.0));
	tmp = 0.0;
	if (c0 <= -1.55e-106)
		tmp = t_0;
	elseif (c0 <= 1.9e-82)
		tmp = ((D * D) * 0.25) / ((d / h) * (d / (M * M)));
	elseif ((c0 <= 18000.0) || ~((c0 <= 2.6e+228)))
		tmp = t_0;
	else
		tmp = 0.25 * ((D * ((M * M) * (h * D))) / (d * d));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c0, -1.55e-106], t$95$0, If[LessEqual[c0, 1.9e-82], N[(N[(N[(D * D), $MachinePrecision] * 0.25), $MachinePrecision] / N[(N[(d / h), $MachinePrecision] * N[(d / N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[c0, 18000.0], N[Not[LessEqual[c0, 2.6e+228]], $MachinePrecision]], t$95$0, N[(0.25 * N[(N[(D * N[(N[(M * M), $MachinePrecision] * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if c0 < -1.54999999999999993e-106 or 1.9000000000000001e-82 < c0 < 18000 or 2.60000000000000007e228 < c0

   1. Initial program 32.2%

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

      1. times-frac 31.5%

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

      2. fma-def 31.5%

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

      3. associate-/r* 31.5%

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

      4. difference-of-squares 36.2%

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

   3. Simplified 44.2%

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

   4. Taylor expanded in c0 around 0 20.0%

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

      1. div-inv 20.0%

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

   6. Applied egg-rr 20.0%

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

   7. Taylor expanded in c0 around inf 35.0%

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

      1. times-frac 36.0%

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

      2. unpow2 36.0%

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

      3. unpow2 36.0%

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

      4. times-frac 46.7%

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

      5. unpow2 46.7%

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

      6. *-commutative 46.7%

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

      7. *-commutative 46.7%

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

   9. Simplified 46.7%

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

   if -1.54999999999999993e-106 < c0 < 1.9000000000000001e-82

   1. Initial program 16.2%

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

   2. Taylor expanded in c0 around -inf 6.0%

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

      1. fma-def 6.0%

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

      2. times-frac 4.8%

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

      3. unpow2 4.8%

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

      4. unpow2 4.8%

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

      5. *-commutative 4.8%

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

      6. unpow2 4.8%

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

      7. associate-*r* 4.8%

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

   4. Simplified 35.8%

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

   5. Taylor expanded in c0 around 0 43.0%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}} \]
   6. Step-by-step derivation

      1. associate-/l* 43.0%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]

      2. associate-*r/ 43.0%

         \[\leadsto \color{blue}{\frac{0.25 \cdot {D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]

      3. unpow2 43.0%

         \[\leadsto \frac{0.25 \cdot \color{blue}{\left(D \cdot D\right)}}{\frac{{d}^{2}}{h \cdot {M}^{2}}} \]

      4. unpow2 43.0%

         \[\leadsto \frac{0.25 \cdot \left(D \cdot D\right)}{\frac{\color{blue}{d \cdot d}}{h \cdot {M}^{2}}} \]

      5. unpow2 43.0%

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

   7. Simplified 43.0%

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

   8. Taylor expanded in d around 0 43.0%

      \[\leadsto \frac{0.25 \cdot \left(D \cdot D\right)}{\color{blue}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]
   9. Step-by-step derivation

      1. unpow2 43.0%

         \[\leadsto \frac{0.25 \cdot \left(D \cdot D\right)}{\frac{\color{blue}{d \cdot d}}{h \cdot {M}^{2}}} \]

      2. times-frac 50.4%

         \[\leadsto \frac{0.25 \cdot \left(D \cdot D\right)}{\color{blue}{\frac{d}{h} \cdot \frac{d}{{M}^{2}}}} \]

      3. unpow2 50.4%

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

   10. Simplified 50.4%

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

   if 18000 < c0 < 2.60000000000000007e228

   1. Initial program 16.2%

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

   2. Taylor expanded in c0 around -inf 9.4%

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

      1. fma-def 9.4%

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

      2. times-frac 9.2%

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

      3. unpow2 9.2%

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

      4. unpow2 9.2%

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

      5. *-commutative 9.2%

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

      6. unpow2 9.2%

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

      7. associate-*r* 9.2%

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

   4. Simplified 29.8%

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

   5. Taylor expanded in c0 around 0 44.6%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}} \]
   6. Step-by-step derivation

      1. associate-/l* 46.0%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]

      2. associate-*r/ 46.0%

         \[\leadsto \color{blue}{\frac{0.25 \cdot {D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]

      3. unpow2 46.0%

         \[\leadsto \frac{0.25 \cdot \color{blue}{\left(D \cdot D\right)}}{\frac{{d}^{2}}{h \cdot {M}^{2}}} \]

      4. unpow2 46.0%

         \[\leadsto \frac{0.25 \cdot \left(D \cdot D\right)}{\frac{\color{blue}{d \cdot d}}{h \cdot {M}^{2}}} \]

      5. unpow2 46.0%

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

   7. Simplified 46.0%

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

   8. Taylor expanded in D around 0 44.6%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. *-commutative 44.6%

         \[\leadsto 0.25 \cdot \frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{{d}^{2}} \]

      2. unpow2 44.6%

         \[\leadsto 0.25 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \]

      3. *-commutative 44.6%

         \[\leadsto 0.25 \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{{d}^{2}} \]

      4. unpow2 44.6%

         \[\leadsto 0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)}{{d}^{2}} \]

      5. unpow2 44.6%

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

   10. Simplified 44.6%

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

   11. Taylor expanded in D around 0 44.6%

       \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}} \]
   12. Step-by-step derivation

       1. unpow2 44.6%

          \[\leadsto 0.25 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}} \]

       2. unpow2 44.6%

          \[\leadsto 0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)}{{d}^{2}} \]

       3. associate-*r* 54.2%

          \[\leadsto 0.25 \cdot \frac{\color{blue}{D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}}{{d}^{2}} \]

       4. unpow2 54.2%

          \[\leadsto 0.25 \cdot \frac{D \cdot \left(D \cdot \left(h \cdot \color{blue}{{M}^{2}}\right)\right)}{{d}^{2}} \]

       5. associate-*r* 54.0%

          \[\leadsto 0.25 \cdot \frac{D \cdot \color{blue}{\left(\left(D \cdot h\right) \cdot {M}^{2}\right)}}{{d}^{2}} \]

       6. unpow2 54.0%

          \[\leadsto 0.25 \cdot \frac{D \cdot \left(\left(D \cdot h\right) \cdot \color{blue}{\left(M \cdot M\right)}\right)}{{d}^{2}} \]

       7. unpow2 54.0%

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

   13. Simplified 54.0%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -1.55 \cdot 10^{-106}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\right)\\ \mathbf{elif}\;c0 \leq 1.9 \cdot 10^{-82}:\\ \;\;\;\;\frac{\left(D \cdot D\right) \cdot 0.25}{\frac{d}{h} \cdot \frac{d}{M \cdot M}}\\ \mathbf{elif}\;c0 \leq 18000 \lor \neg \left(c0 \leq 2.6 \cdot 10^{+228}\right):\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot D\right)\right)}{d \cdot d}\\ \end{array} \]

Alternative 6: 41.5% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := h \cdot \left(M \cdot M\right)\\ t_1 := \frac{D \cdot D}{d}\\ \mathbf{if}\;d \cdot d \leq 4 \cdot 10^{-164}:\\ \;\;\;\;0.25 \cdot \left(t_1 \cdot \frac{t_0}{d}\right)\\ \mathbf{elif}\;d \cdot d \leq 2 \cdot 10^{-126}:\\ \;\;\;\;\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}\\ \mathbf{elif}\;d \cdot d \leq 5 \cdot 10^{-54}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \frac{0.25}{d}\right)\\ \mathbf{elif}\;d \cdot d \leq 5 \cdot 10^{+307}:\\ \;\;\;\;0.25 \cdot \left(\left(D \cdot \left(D \cdot t_0\right)\right) \cdot \frac{1}{d \cdot d}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* h (* M M))) (t_1 (/ (* D D) d)))
   (if (<= (* d d) 4e-164)
     (* 0.25 (* t_1 (/ t_0 d)))
     (if (<= (* d d) 2e-126)
       (/ (* (* d d) (* c0 c0)) (* (* D D) (* h (* w w))))
       (if (<= (* d d) 5e-54)
         (* t_0 (* t_1 (/ 0.25 d)))
         (if (<= (* d d) 5e+307)
           (* 0.25 (* (* D (* D t_0)) (/ 1.0 (* d d))))
           0.0))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = h * (M * M);
	double t_1 = (D * D) / d;
	double tmp;
	if ((d * d) <= 4e-164) {
		tmp = 0.25 * (t_1 * (t_0 / d));
	} else if ((d * d) <= 2e-126) {
		tmp = ((d * d) * (c0 * c0)) / ((D * D) * (h * (w * w)));
	} else if ((d * d) <= 5e-54) {
		tmp = t_0 * (t_1 * (0.25 / d));
	} else if ((d * d) <= 5e+307) {
		tmp = 0.25 * ((D * (D * t_0)) * (1.0 / (d * d)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = h * (m * m)
    t_1 = (d * d) / d_1
    if ((d_1 * d_1) <= 4d-164) then
        tmp = 0.25d0 * (t_1 * (t_0 / d_1))
    else if ((d_1 * d_1) <= 2d-126) then
        tmp = ((d_1 * d_1) * (c0 * c0)) / ((d * d) * (h * (w * w)))
    else if ((d_1 * d_1) <= 5d-54) then
        tmp = t_0 * (t_1 * (0.25d0 / d_1))
    else if ((d_1 * d_1) <= 5d+307) then
        tmp = 0.25d0 * ((d * (d * t_0)) * (1.0d0 / (d_1 * d_1)))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = h * (M * M);
	double t_1 = (D * D) / d;
	double tmp;
	if ((d * d) <= 4e-164) {
		tmp = 0.25 * (t_1 * (t_0 / d));
	} else if ((d * d) <= 2e-126) {
		tmp = ((d * d) * (c0 * c0)) / ((D * D) * (h * (w * w)));
	} else if ((d * d) <= 5e-54) {
		tmp = t_0 * (t_1 * (0.25 / d));
	} else if ((d * d) <= 5e+307) {
		tmp = 0.25 * ((D * (D * t_0)) * (1.0 / (d * d)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = h * (M * M)
	t_1 = (D * D) / d
	tmp = 0
	if (d * d) <= 4e-164:
		tmp = 0.25 * (t_1 * (t_0 / d))
	elif (d * d) <= 2e-126:
		tmp = ((d * d) * (c0 * c0)) / ((D * D) * (h * (w * w)))
	elif (d * d) <= 5e-54:
		tmp = t_0 * (t_1 * (0.25 / d))
	elif (d * d) <= 5e+307:
		tmp = 0.25 * ((D * (D * t_0)) * (1.0 / (d * d)))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(h * Float64(M * M))
	t_1 = Float64(Float64(D * D) / d)
	tmp = 0.0
	if (Float64(d * d) <= 4e-164)
		tmp = Float64(0.25 * Float64(t_1 * Float64(t_0 / d)));
	elseif (Float64(d * d) <= 2e-126)
		tmp = Float64(Float64(Float64(d * d) * Float64(c0 * c0)) / Float64(Float64(D * D) * Float64(h * Float64(w * w))));
	elseif (Float64(d * d) <= 5e-54)
		tmp = Float64(t_0 * Float64(t_1 * Float64(0.25 / d)));
	elseif (Float64(d * d) <= 5e+307)
		tmp = Float64(0.25 * Float64(Float64(D * Float64(D * t_0)) * Float64(1.0 / Float64(d * d))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = h * (M * M);
	t_1 = (D * D) / d;
	tmp = 0.0;
	if ((d * d) <= 4e-164)
		tmp = 0.25 * (t_1 * (t_0 / d));
	elseif ((d * d) <= 2e-126)
		tmp = ((d * d) * (c0 * c0)) / ((D * D) * (h * (w * w)));
	elseif ((d * d) <= 5e-54)
		tmp = t_0 * (t_1 * (0.25 / d));
	elseif ((d * d) <= 5e+307)
		tmp = 0.25 * ((D * (D * t_0)) * (1.0 / (d * d)));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(D * D), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[N[(d * d), $MachinePrecision], 4e-164], N[(0.25 * N[(t$95$1 * N[(t$95$0 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(d * d), $MachinePrecision], 2e-126], N[(N[(N[(d * d), $MachinePrecision] * N[(c0 * c0), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(h * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(d * d), $MachinePrecision], 5e-54], N[(t$95$0 * N[(t$95$1 * N[(0.25 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(d * d), $MachinePrecision], 5e+307], N[(0.25 * N[(N[(D * N[(D * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 d d) < 3.99999999999999985e-164

   1. Initial program 8.9%

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

   2. Taylor expanded in c0 around -inf 6.4%

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

      1. fma-def 6.4%

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

      2. times-frac 6.3%

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

      3. unpow2 6.3%

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

      4. unpow2 6.3%

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

      5. *-commutative 6.3%

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

      6. unpow2 6.3%

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

      7. associate-*r* 6.3%

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

   4. Simplified 23.6%

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

   5. Taylor expanded in c0 around 0 31.2%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}} \]
   6. Step-by-step derivation

      1. associate-/l* 32.5%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]

      2. associate-*r/ 32.5%

         \[\leadsto \color{blue}{\frac{0.25 \cdot {D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]

      3. unpow2 32.5%

         \[\leadsto \frac{0.25 \cdot \color{blue}{\left(D \cdot D\right)}}{\frac{{d}^{2}}{h \cdot {M}^{2}}} \]

      4. unpow2 32.5%

         \[\leadsto \frac{0.25 \cdot \left(D \cdot D\right)}{\frac{\color{blue}{d \cdot d}}{h \cdot {M}^{2}}} \]

      5. unpow2 32.5%

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

   7. Simplified 32.5%

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

   8. Taylor expanded in D around 0 31.2%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. *-commutative 31.2%

         \[\leadsto 0.25 \cdot \frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{{d}^{2}} \]

      2. unpow2 31.2%

         \[\leadsto 0.25 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \]

      3. *-commutative 31.2%

         \[\leadsto 0.25 \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{{d}^{2}} \]

      4. unpow2 31.2%

         \[\leadsto 0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)}{{d}^{2}} \]

      5. unpow2 31.2%

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

   10. Simplified 31.2%

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

       1. times-frac 45.5%

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

   12. Applied egg-rr 45.5%

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

   if 3.99999999999999985e-164 < (*.f64 d d) < 1.9999999999999999e-126

   1. Initial program 83.3%

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

      1. times-frac 66.7%

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

      2. fma-def 66.7%

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

      3. associate-/r* 66.7%

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

      4. difference-of-squares 83.3%

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

   3. Simplified 83.3%

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

   4. Taylor expanded in c0 around inf 99.7%

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

      1. unpow2 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. *-commutative 99.7%

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

      5. unpow2 99.7%

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

   6. Simplified 99.7%

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

   if 1.9999999999999999e-126 < (*.f64 d d) < 5.00000000000000015e-54

   1. Initial program 15.4%

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

   2. Taylor expanded in c0 around -inf 16.0%

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

      1. fma-def 16.0%

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

      2. times-frac 1.2%

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

      3. unpow2 1.2%

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

      4. unpow2 1.2%

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

      5. *-commutative 1.2%

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

      6. unpow2 1.2%

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

      7. associate-*r* 1.2%

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

   4. Simplified 58.3%

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

   5. Taylor expanded in c0 around 0 73.1%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}} \]
   6. Step-by-step derivation

      1. associate-/l* 73.1%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]

      2. associate-*r/ 73.1%

         \[\leadsto \color{blue}{\frac{0.25 \cdot {D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]

      3. unpow2 73.1%

         \[\leadsto \frac{0.25 \cdot \color{blue}{\left(D \cdot D\right)}}{\frac{{d}^{2}}{h \cdot {M}^{2}}} \]

      4. unpow2 73.1%

         \[\leadsto \frac{0.25 \cdot \left(D \cdot D\right)}{\frac{\color{blue}{d \cdot d}}{h \cdot {M}^{2}}} \]

      5. unpow2 73.1%

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

   7. Simplified 73.1%

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

   8. Taylor expanded in D around 0 73.1%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. associate-/l* 73.1%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]

      2. unpow2 73.1%

         \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{{d}^{2}}{h \cdot \color{blue}{\left(M \cdot M\right)}}} \]

      3. unpow2 73.1%

         \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\color{blue}{d \cdot d}}{h \cdot \left(M \cdot M\right)}} \]

      4. associate-*r/ 73.1%

         \[\leadsto \color{blue}{\frac{0.25 \cdot {D}^{2}}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}} \]

      5. unpow2 73.1%

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

      6. unpow2 73.1%

         \[\leadsto \frac{0.25 \cdot \left(D \cdot D\right)}{\frac{\color{blue}{{d}^{2}}}{h \cdot \left(M \cdot M\right)}} \]

      7. associate-/r/ 73.7%

         \[\leadsto \color{blue}{\frac{0.25 \cdot \left(D \cdot D\right)}{{d}^{2}} \cdot \left(h \cdot \left(M \cdot M\right)\right)} \]

      8. unpow2 73.7%

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

      9. *-commutative 73.7%

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

      10. unpow2 73.7%

          \[\leadsto \left(h \cdot \left(M \cdot M\right)\right) \cdot \frac{0.25 \cdot \color{blue}{{D}^{2}}}{d \cdot d} \]

      11. times-frac 73.7%

          \[\leadsto \left(h \cdot \left(M \cdot M\right)\right) \cdot \color{blue}{\left(\frac{0.25}{d} \cdot \frac{{D}^{2}}{d}\right)} \]

      12. unpow2 73.7%

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

   10. Simplified 73.7%

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

   if 5.00000000000000015e-54 < (*.f64 d d) < 5e307

   1. Initial program 28.0%

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

   2. Taylor expanded in c0 around -inf 9.8%

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

      1. fma-def 9.8%

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

      2. times-frac 8.7%

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

      3. unpow2 8.7%

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

      4. unpow2 8.7%

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

      5. *-commutative 8.7%

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

      6. unpow2 8.7%

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

      7. associate-*r* 8.7%

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

   4. Simplified 29.2%

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

   5. Taylor expanded in c0 around 0 41.1%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}} \]
   6. Step-by-step derivation

      1. associate-/l* 40.0%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]

      2. associate-*r/ 40.0%

         \[\leadsto \color{blue}{\frac{0.25 \cdot {D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]

      3. unpow2 40.0%

         \[\leadsto \frac{0.25 \cdot \color{blue}{\left(D \cdot D\right)}}{\frac{{d}^{2}}{h \cdot {M}^{2}}} \]

      4. unpow2 40.0%

         \[\leadsto \frac{0.25 \cdot \left(D \cdot D\right)}{\frac{\color{blue}{d \cdot d}}{h \cdot {M}^{2}}} \]

      5. unpow2 40.0%

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

   7. Simplified 40.0%

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

   8. Taylor expanded in D around 0 41.1%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. *-commutative 41.1%

         \[\leadsto 0.25 \cdot \frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{{d}^{2}} \]

      2. unpow2 41.1%

         \[\leadsto 0.25 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \]

      3. *-commutative 41.1%

         \[\leadsto 0.25 \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{{d}^{2}} \]

      4. unpow2 41.1%

         \[\leadsto 0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)}{{d}^{2}} \]

      5. unpow2 41.1%

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

   10. Simplified 41.1%

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

       1. div-inv 41.1%

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

       2. associate-*l* 48.2%

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

   12. Applied egg-rr 48.2%

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

   if 5e307 < (*.f64 d d)

   1. Initial program 25.2%

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

      1. times-frac 25.2%

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

      2. fma-def 25.2%

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

      3. associate-/r* 25.2%

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

      4. difference-of-squares 33.1%

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

   3. Simplified 36.3%

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

   4. Taylor expanded in c0 around -inf 0.0%

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

      1. associate-*r* 0.0%

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

      2. distribute-rgt1-in 0.0%

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

      3. metadata-eval 0.0%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)\right) \cdot c0\right) \]

      4. mul0-lft 36.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot \color{blue}{0}\right) \cdot c0\right) \]

      5. metadata-eval 36.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{0} \cdot c0\right) \]

      6. mul0-lft 0.0%

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

      7. metadata-eval 0.0%

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

      8. distribute-lft1-in 0.0%

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

      9. *-commutative 0.0%

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

      10. distribute-lft1-in 0.0%

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

      11. metadata-eval 0.0%

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

      12. mul0-lft 36.9%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{0}\right) \]

   6. Simplified 36.9%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]

   7. Taylor expanded in c0 around 0 39.9%

      \[\leadsto \color{blue}{0} \]
3. Recombined 5 regimes into one program.

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \cdot d \leq 4 \cdot 10^{-164}:\\ \;\;\;\;0.25 \cdot \left(\frac{D \cdot D}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\\ \mathbf{elif}\;d \cdot d \leq 2 \cdot 10^{-126}:\\ \;\;\;\;\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}\\ \mathbf{elif}\;d \cdot d \leq 5 \cdot 10^{-54}:\\ \;\;\;\;\left(h \cdot \left(M \cdot M\right)\right) \cdot \left(\frac{D \cdot D}{d} \cdot \frac{0.25}{d}\right)\\ \mathbf{elif}\;d \cdot d \leq 5 \cdot 10^{+307}:\\ \;\;\;\;0.25 \cdot \left(\left(D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right) \cdot \frac{1}{d \cdot d}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7: 38.7% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 2 \cdot 10^{-161}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 1.9 \cdot 10^{+51}:\\ \;\;\;\;\left(h \cdot \left(M \cdot M\right)\right) \cdot \left(\frac{D \cdot D}{d} \cdot \frac{0.25}{d}\right)\\ \mathbf{elif}\;M \leq 1.25 \cdot 10^{+76} \lor \neg \left(M \leq 9.6 \cdot 10^{+138}\right):\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot D\right)\right)}{d \cdot d}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= M 2e-161)
   0.0
   (if (<= M 1.9e+51)
     (* (* h (* M M)) (* (/ (* D D) d) (/ 0.25 d)))
     (if (or (<= M 1.25e+76) (not (<= M 9.6e+138)))
       (* (/ c0 (* 2.0 w)) (* 2.0 (* (/ c0 (* w h)) (/ (* d d) (* D D)))))
       (* 0.25 (/ (* D (* (* M M) (* h D))) (* d d)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 2e-161) {
		tmp = 0.0;
	} else if (M <= 1.9e+51) {
		tmp = (h * (M * M)) * (((D * D) / d) * (0.25 / d));
	} else if ((M <= 1.25e+76) || !(M <= 9.6e+138)) {
		tmp = (c0 / (2.0 * w)) * (2.0 * ((c0 / (w * h)) * ((d * d) / (D * D))));
	} else {
		tmp = 0.25 * ((D * ((M * M) * (h * D))) / (d * d));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 2d-161) then
        tmp = 0.0d0
    else if (m <= 1.9d+51) then
        tmp = (h * (m * m)) * (((d * d) / d_1) * (0.25d0 / d_1))
    else if ((m <= 1.25d+76) .or. (.not. (m <= 9.6d+138))) then
        tmp = (c0 / (2.0d0 * w)) * (2.0d0 * ((c0 / (w * h)) * ((d_1 * d_1) / (d * d))))
    else
        tmp = 0.25d0 * ((d * ((m * m) * (h * d))) / (d_1 * d_1))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 2e-161) {
		tmp = 0.0;
	} else if (M <= 1.9e+51) {
		tmp = (h * (M * M)) * (((D * D) / d) * (0.25 / d));
	} else if ((M <= 1.25e+76) || !(M <= 9.6e+138)) {
		tmp = (c0 / (2.0 * w)) * (2.0 * ((c0 / (w * h)) * ((d * d) / (D * D))));
	} else {
		tmp = 0.25 * ((D * ((M * M) * (h * D))) / (d * d));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if M <= 2e-161:
		tmp = 0.0
	elif M <= 1.9e+51:
		tmp = (h * (M * M)) * (((D * D) / d) * (0.25 / d))
	elif (M <= 1.25e+76) or not (M <= 9.6e+138):
		tmp = (c0 / (2.0 * w)) * (2.0 * ((c0 / (w * h)) * ((d * d) / (D * D))))
	else:
		tmp = 0.25 * ((D * ((M * M) * (h * D))) / (d * d))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (M <= 2e-161)
		tmp = 0.0;
	elseif (M <= 1.9e+51)
		tmp = Float64(Float64(h * Float64(M * M)) * Float64(Float64(Float64(D * D) / d) * Float64(0.25 / d)));
	elseif ((M <= 1.25e+76) || !(M <= 9.6e+138))
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(2.0 * Float64(Float64(c0 / Float64(w * h)) * Float64(Float64(d * d) / Float64(D * D)))));
	else
		tmp = Float64(0.25 * Float64(Float64(D * Float64(Float64(M * M) * Float64(h * D))) / Float64(d * d)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (M <= 2e-161)
		tmp = 0.0;
	elseif (M <= 1.9e+51)
		tmp = (h * (M * M)) * (((D * D) / d) * (0.25 / d));
	elseif ((M <= 1.25e+76) || ~((M <= 9.6e+138)))
		tmp = (c0 / (2.0 * w)) * (2.0 * ((c0 / (w * h)) * ((d * d) / (D * D))));
	else
		tmp = 0.25 * ((D * ((M * M) * (h * D))) / (d * d));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[M, 2e-161], 0.0, If[LessEqual[M, 1.9e+51], N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(D * D), $MachinePrecision] / d), $MachinePrecision] * N[(0.25 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[M, 1.25e+76], N[Not[LessEqual[M, 9.6e+138]], $MachinePrecision]], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(D * N[(N[(M * M), $MachinePrecision] * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if M < 2.00000000000000006e-161

   1. Initial program 26.1%

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

      1. times-frac 24.4%

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

      2. fma-def 24.4%

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

      3. associate-/r* 24.5%

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

      4. difference-of-squares 28.5%

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

   3. Simplified 33.9%

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

   4. Taylor expanded in c0 around -inf 3.2%

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

      1. associate-*r* 3.2%

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

      2. distribute-rgt1-in 3.2%

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

      3. metadata-eval 3.2%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)\right) \cdot c0\right) \]

      4. mul0-lft 31.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot \color{blue}{0}\right) \cdot c0\right) \]

      5. metadata-eval 31.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{0} \cdot c0\right) \]

      6. mul0-lft 4.4%

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

      7. metadata-eval 4.4%

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

      8. distribute-lft1-in 4.4%

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

      9. *-commutative 4.4%

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

      10. distribute-lft1-in 4.4%

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

      11. metadata-eval 4.4%

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

      12. mul0-lft 31.9%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{0}\right) \]

   6. Simplified 31.9%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]

   7. Taylor expanded in c0 around 0 36.6%

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

   if 2.00000000000000006e-161 < M < 1.8999999999999999e51

   1. Initial program 25.5%

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

   2. Taylor expanded in c0 around -inf 8.2%

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

      1. fma-def 8.2%

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

      2. times-frac 8.1%

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

      3. unpow2 8.1%

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

      4. unpow2 8.1%

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

      5. *-commutative 8.1%

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

      6. unpow2 8.1%

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

      7. associate-*r* 8.1%

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

   4. Simplified 31.7%

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

   5. Taylor expanded in c0 around 0 45.6%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}} \]
   6. Step-by-step derivation

      1. associate-/l* 43.5%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]

      2. associate-*r/ 43.5%

         \[\leadsto \color{blue}{\frac{0.25 \cdot {D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]

      3. unpow2 43.5%

         \[\leadsto \frac{0.25 \cdot \color{blue}{\left(D \cdot D\right)}}{\frac{{d}^{2}}{h \cdot {M}^{2}}} \]

      4. unpow2 43.5%

         \[\leadsto \frac{0.25 \cdot \left(D \cdot D\right)}{\frac{\color{blue}{d \cdot d}}{h \cdot {M}^{2}}} \]

      5. unpow2 43.5%

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

   7. Simplified 43.5%

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

   8. Taylor expanded in D around 0 45.6%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. associate-/l* 43.5%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]

      2. unpow2 43.5%

         \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{{d}^{2}}{h \cdot \color{blue}{\left(M \cdot M\right)}}} \]

      3. unpow2 43.5%

         \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\color{blue}{d \cdot d}}{h \cdot \left(M \cdot M\right)}} \]

      4. associate-*r/ 43.5%

         \[\leadsto \color{blue}{\frac{0.25 \cdot {D}^{2}}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}} \]

      5. unpow2 43.5%

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

      6. unpow2 43.5%

         \[\leadsto \frac{0.25 \cdot \left(D \cdot D\right)}{\frac{\color{blue}{{d}^{2}}}{h \cdot \left(M \cdot M\right)}} \]

      7. associate-/r/ 45.7%

         \[\leadsto \color{blue}{\frac{0.25 \cdot \left(D \cdot D\right)}{{d}^{2}} \cdot \left(h \cdot \left(M \cdot M\right)\right)} \]

      8. unpow2 45.7%

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

      9. *-commutative 45.7%

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

      10. unpow2 45.7%

          \[\leadsto \left(h \cdot \left(M \cdot M\right)\right) \cdot \frac{0.25 \cdot \color{blue}{{D}^{2}}}{d \cdot d} \]

      11. times-frac 52.4%

          \[\leadsto \left(h \cdot \left(M \cdot M\right)\right) \cdot \color{blue}{\left(\frac{0.25}{d} \cdot \frac{{D}^{2}}{d}\right)} \]

      12. unpow2 52.4%

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

   10. Simplified 52.4%

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

   if 1.8999999999999999e51 < M < 1.24999999999999998e76 or 9.6000000000000003e138 < M

   1. Initial program 16.4%

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

      1. times-frac 16.4%

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

      2. fma-def 16.4%

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

      3. associate-/r* 16.4%

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

      4. difference-of-squares 36.9%

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

   3. Simplified 41.8%

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

   4. Taylor expanded in c0 around inf 41.6%

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

      1. times-frac 41.9%

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

      2. unpow2 41.9%

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

      3. unpow2 41.9%

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

   6. Simplified 41.9%

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

   if 1.24999999999999998e76 < M < 9.6000000000000003e138

   1. Initial program 8.9%

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

   2. Taylor expanded in c0 around -inf 0.0%

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

      1. fma-def 0.0%

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

      2. times-frac 0.0%

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

      3. unpow2 0.0%

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

      4. unpow2 0.0%

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

      5. *-commutative 0.0%

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

      6. unpow2 0.0%

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

      7. associate-*r* 0.0%

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

   4. Simplified 17.1%

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

   5. Taylor expanded in c0 around 0 42.3%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}} \]
   6. Step-by-step derivation

      1. associate-/l* 42.3%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]

      2. associate-*r/ 42.3%

         \[\leadsto \color{blue}{\frac{0.25 \cdot {D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]

      3. unpow2 42.3%

         \[\leadsto \frac{0.25 \cdot \color{blue}{\left(D \cdot D\right)}}{\frac{{d}^{2}}{h \cdot {M}^{2}}} \]

      4. unpow2 42.3%

         \[\leadsto \frac{0.25 \cdot \left(D \cdot D\right)}{\frac{\color{blue}{d \cdot d}}{h \cdot {M}^{2}}} \]

      5. unpow2 42.3%

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

   7. Simplified 42.3%

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

   8. Taylor expanded in D around 0 42.3%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. *-commutative 42.3%

         \[\leadsto 0.25 \cdot \frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{{d}^{2}} \]

      2. unpow2 42.3%

         \[\leadsto 0.25 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \]

      3. *-commutative 42.3%

         \[\leadsto 0.25 \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{{d}^{2}} \]

      4. unpow2 42.3%

         \[\leadsto 0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)}{{d}^{2}} \]

      5. unpow2 42.3%

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

   10. Simplified 42.3%

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

   11. Taylor expanded in D around 0 42.3%

       \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}} \]
   12. Step-by-step derivation

       1. unpow2 42.3%

          \[\leadsto 0.25 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}} \]

       2. unpow2 42.3%

          \[\leadsto 0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)}{{d}^{2}} \]

       3. associate-*r* 43.3%

          \[\leadsto 0.25 \cdot \frac{\color{blue}{D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}}{{d}^{2}} \]

       4. unpow2 43.3%

          \[\leadsto 0.25 \cdot \frac{D \cdot \left(D \cdot \left(h \cdot \color{blue}{{M}^{2}}\right)\right)}{{d}^{2}} \]

       5. associate-*r* 59.1%

          \[\leadsto 0.25 \cdot \frac{D \cdot \color{blue}{\left(\left(D \cdot h\right) \cdot {M}^{2}\right)}}{{d}^{2}} \]

       6. unpow2 59.1%

          \[\leadsto 0.25 \cdot \frac{D \cdot \left(\left(D \cdot h\right) \cdot \color{blue}{\left(M \cdot M\right)}\right)}{{d}^{2}} \]

       7. unpow2 59.1%

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

   13. Simplified 59.1%

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

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2 \cdot 10^{-161}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 1.9 \cdot 10^{+51}:\\ \;\;\;\;\left(h \cdot \left(M \cdot M\right)\right) \cdot \left(\frac{D \cdot D}{d} \cdot \frac{0.25}{d}\right)\\ \mathbf{elif}\;M \leq 1.25 \cdot 10^{+76} \lor \neg \left(M \leq 9.6 \cdot 10^{+138}\right):\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot D\right)\right)}{d \cdot d}\\ \end{array} \]

Alternative 8: 41.8% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := h \cdot \left(M \cdot M\right)\\ \mathbf{if}\;d \cdot d \leq 5 \cdot 10^{-50}:\\ \;\;\;\;0.25 \cdot \left(\frac{D \cdot D}{d} \cdot \frac{t_0}{d}\right)\\ \mathbf{elif}\;d \cdot d \leq 5 \cdot 10^{+307}:\\ \;\;\;\;0.25 \cdot \left(\left(D \cdot \left(D \cdot t_0\right)\right) \cdot \frac{1}{d \cdot d}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* h (* M M))))
   (if (<= (* d d) 5e-50)
     (* 0.25 (* (/ (* D D) d) (/ t_0 d)))
     (if (<= (* d d) 5e+307)
       (* 0.25 (* (* D (* D t_0)) (/ 1.0 (* d d))))
       0.0))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = h * (M * M);
	double tmp;
	if ((d * d) <= 5e-50) {
		tmp = 0.25 * (((D * D) / d) * (t_0 / d));
	} else if ((d * d) <= 5e+307) {
		tmp = 0.25 * ((D * (D * t_0)) * (1.0 / (d * d)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = h * (m * m)
    if ((d_1 * d_1) <= 5d-50) then
        tmp = 0.25d0 * (((d * d) / d_1) * (t_0 / d_1))
    else if ((d_1 * d_1) <= 5d+307) then
        tmp = 0.25d0 * ((d * (d * t_0)) * (1.0d0 / (d_1 * d_1)))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = h * (M * M);
	double tmp;
	if ((d * d) <= 5e-50) {
		tmp = 0.25 * (((D * D) / d) * (t_0 / d));
	} else if ((d * d) <= 5e+307) {
		tmp = 0.25 * ((D * (D * t_0)) * (1.0 / (d * d)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = h * (M * M)
	tmp = 0
	if (d * d) <= 5e-50:
		tmp = 0.25 * (((D * D) / d) * (t_0 / d))
	elif (d * d) <= 5e+307:
		tmp = 0.25 * ((D * (D * t_0)) * (1.0 / (d * d)))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(h * Float64(M * M))
	tmp = 0.0
	if (Float64(d * d) <= 5e-50)
		tmp = Float64(0.25 * Float64(Float64(Float64(D * D) / d) * Float64(t_0 / d)));
	elseif (Float64(d * d) <= 5e+307)
		tmp = Float64(0.25 * Float64(Float64(D * Float64(D * t_0)) * Float64(1.0 / Float64(d * d))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = h * (M * M);
	tmp = 0.0;
	if ((d * d) <= 5e-50)
		tmp = 0.25 * (((D * D) / d) * (t_0 / d));
	elseif ((d * d) <= 5e+307)
		tmp = 0.25 * ((D * (D * t_0)) * (1.0 / (d * d)));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(d * d), $MachinePrecision], 5e-50], N[(0.25 * N[(N[(N[(D * D), $MachinePrecision] / d), $MachinePrecision] * N[(t$95$0 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(d * d), $MachinePrecision], 5e+307], N[(0.25 * N[(N[(D * N[(D * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 d d) < 4.99999999999999968e-50

   1. Initial program 16.5%

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

   2. Taylor expanded in c0 around -inf 8.4%

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

      1. fma-def 8.4%

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

      2. times-frac 6.7%

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

      3. unpow2 6.7%

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

      4. unpow2 6.7%

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

      5. *-commutative 6.7%

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

      6. unpow2 6.7%

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

      7. associate-*r* 6.7%

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

   4. Simplified 27.8%

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

   5. Taylor expanded in c0 around 0 36.7%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}} \]
   6. Step-by-step derivation

      1. associate-/l* 37.7%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]

      2. associate-*r/ 37.7%

         \[\leadsto \color{blue}{\frac{0.25 \cdot {D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]

      3. unpow2 37.7%

         \[\leadsto \frac{0.25 \cdot \color{blue}{\left(D \cdot D\right)}}{\frac{{d}^{2}}{h \cdot {M}^{2}}} \]

      4. unpow2 37.7%

         \[\leadsto \frac{0.25 \cdot \left(D \cdot D\right)}{\frac{\color{blue}{d \cdot d}}{h \cdot {M}^{2}}} \]

      5. unpow2 37.7%

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

   7. Simplified 37.7%

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

   8. Taylor expanded in D around 0 36.7%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. *-commutative 36.7%

         \[\leadsto 0.25 \cdot \frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{{d}^{2}} \]

      2. unpow2 36.7%

         \[\leadsto 0.25 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \]

      3. *-commutative 36.7%

         \[\leadsto 0.25 \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{{d}^{2}} \]

      4. unpow2 36.7%

         \[\leadsto 0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)}{{d}^{2}} \]

      5. unpow2 36.7%

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

   10. Simplified 36.7%

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

       1. times-frac 47.7%

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

   12. Applied egg-rr 47.7%

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

   if 4.99999999999999968e-50 < (*.f64 d d) < 5e307

   1. Initial program 28.6%

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

   2. Taylor expanded in c0 around -inf 8.8%

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

      1. fma-def 8.8%

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

      2. times-frac 7.7%

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

      3. unpow2 7.7%

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

      4. unpow2 7.7%

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

      5. *-commutative 7.7%

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

      6. unpow2 7.7%

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

      7. associate-*r* 7.7%

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

   4. Simplified 28.6%

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

   5. Taylor expanded in c0 around 0 39.8%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}} \]
   6. Step-by-step derivation

      1. associate-/l* 38.6%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]

      2. associate-*r/ 38.6%

         \[\leadsto \color{blue}{\frac{0.25 \cdot {D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]

      3. unpow2 38.6%

         \[\leadsto \frac{0.25 \cdot \color{blue}{\left(D \cdot D\right)}}{\frac{{d}^{2}}{h \cdot {M}^{2}}} \]

      4. unpow2 38.6%

         \[\leadsto \frac{0.25 \cdot \left(D \cdot D\right)}{\frac{\color{blue}{d \cdot d}}{h \cdot {M}^{2}}} \]

      5. unpow2 38.6%

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

   7. Simplified 38.6%

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

   8. Taylor expanded in D around 0 39.8%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. *-commutative 39.8%

         \[\leadsto 0.25 \cdot \frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{{d}^{2}} \]

      2. unpow2 39.8%

         \[\leadsto 0.25 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \]

      3. *-commutative 39.8%

         \[\leadsto 0.25 \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{{d}^{2}} \]

      4. unpow2 39.8%

         \[\leadsto 0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)}{{d}^{2}} \]

      5. unpow2 39.8%

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

   10. Simplified 39.8%

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

       1. div-inv 39.8%

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

       2. associate-*l* 47.1%

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

   12. Applied egg-rr 47.1%

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

   if 5e307 < (*.f64 d d)

   1. Initial program 25.2%

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

      1. times-frac 25.2%

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

      2. fma-def 25.2%

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

      3. associate-/r* 25.2%

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

      4. difference-of-squares 33.1%

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

   3. Simplified 36.3%

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

   4. Taylor expanded in c0 around -inf 0.0%

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

      1. associate-*r* 0.0%

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

      2. distribute-rgt1-in 0.0%

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

      3. metadata-eval 0.0%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)\right) \cdot c0\right) \]

      4. mul0-lft 36.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot \color{blue}{0}\right) \cdot c0\right) \]

      5. metadata-eval 36.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{0} \cdot c0\right) \]

      6. mul0-lft 0.0%

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

      7. metadata-eval 0.0%

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

      8. distribute-lft1-in 0.0%

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

      9. *-commutative 0.0%

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

      10. distribute-lft1-in 0.0%

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

      11. metadata-eval 0.0%

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

      12. mul0-lft 36.9%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{0}\right) \]

   6. Simplified 36.9%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]

   7. Taylor expanded in c0 around 0 39.9%

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

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \cdot d \leq 5 \cdot 10^{-50}:\\ \;\;\;\;0.25 \cdot \left(\frac{D \cdot D}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\\ \mathbf{elif}\;d \cdot d \leq 5 \cdot 10^{+307}:\\ \;\;\;\;0.25 \cdot \left(\left(D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right) \cdot \frac{1}{d \cdot d}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9: 41.9% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \cdot d \leq 2 \cdot 10^{-98}:\\ \;\;\;\;0.25 \cdot \left(\frac{D \cdot D}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\\ \mathbf{elif}\;d \cdot d \leq 5 \cdot 10^{+307}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot D\right)\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= (* d d) 2e-98)
   (* 0.25 (* (/ (* D D) d) (/ (* h (* M M)) d)))
   (if (<= (* d d) 5e+307)
     (* 0.25 (/ (* D (* (* M M) (* h D))) (* d d)))
     0.0)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((d * d) <= 2e-98) {
		tmp = 0.25 * (((D * D) / d) * ((h * (M * M)) / d));
	} else if ((d * d) <= 5e+307) {
		tmp = 0.25 * ((D * ((M * M) * (h * D))) / (d * d));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((d_1 * d_1) <= 2d-98) then
        tmp = 0.25d0 * (((d * d) / d_1) * ((h * (m * m)) / d_1))
    else if ((d_1 * d_1) <= 5d+307) then
        tmp = 0.25d0 * ((d * ((m * m) * (h * d))) / (d_1 * d_1))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((d * d) <= 2e-98) {
		tmp = 0.25 * (((D * D) / d) * ((h * (M * M)) / d));
	} else if ((d * d) <= 5e+307) {
		tmp = 0.25 * ((D * ((M * M) * (h * D))) / (d * d));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (d * d) <= 2e-98:
		tmp = 0.25 * (((D * D) / d) * ((h * (M * M)) / d))
	elif (d * d) <= 5e+307:
		tmp = 0.25 * ((D * ((M * M) * (h * D))) / (d * d))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (Float64(d * d) <= 2e-98)
		tmp = Float64(0.25 * Float64(Float64(Float64(D * D) / d) * Float64(Float64(h * Float64(M * M)) / d)));
	elseif (Float64(d * d) <= 5e+307)
		tmp = Float64(0.25 * Float64(Float64(D * Float64(Float64(M * M) * Float64(h * D))) / Float64(d * d)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((d * d) <= 2e-98)
		tmp = 0.25 * (((D * D) / d) * ((h * (M * M)) / d));
	elseif ((d * d) <= 5e+307)
		tmp = 0.25 * ((D * ((M * M) * (h * D))) / (d * d));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[N[(d * d), $MachinePrecision], 2e-98], N[(0.25 * N[(N[(N[(D * D), $MachinePrecision] / d), $MachinePrecision] * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(d * d), $MachinePrecision], 5e+307], N[(0.25 * N[(N[(D * N[(N[(M * M), $MachinePrecision] * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 d d) < 1.99999999999999988e-98

   1. Initial program 16.7%

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

   2. Taylor expanded in c0 around -inf 7.4%

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

      1. fma-def 7.4%

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

      2. times-frac 5.6%

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

      3. unpow2 5.6%

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

      4. unpow2 5.6%

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

      5. *-commutative 5.6%

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

      6. unpow2 5.6%

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

      7. associate-*r* 5.6%

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

   4. Simplified 22.2%

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

   5. Taylor expanded in c0 around 0 30.6%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}} \]
   6. Step-by-step derivation

      1. associate-/l* 31.7%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]

      2. associate-*r/ 31.7%

         \[\leadsto \color{blue}{\frac{0.25 \cdot {D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]

      3. unpow2 31.7%

         \[\leadsto \frac{0.25 \cdot \color{blue}{\left(D \cdot D\right)}}{\frac{{d}^{2}}{h \cdot {M}^{2}}} \]

      4. unpow2 31.7%

         \[\leadsto \frac{0.25 \cdot \left(D \cdot D\right)}{\frac{\color{blue}{d \cdot d}}{h \cdot {M}^{2}}} \]

      5. unpow2 31.7%

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

   7. Simplified 31.7%

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

   8. Taylor expanded in D around 0 30.6%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. *-commutative 30.6%

         \[\leadsto 0.25 \cdot \frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{{d}^{2}} \]

      2. unpow2 30.6%

         \[\leadsto 0.25 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \]

      3. *-commutative 30.6%

         \[\leadsto 0.25 \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{{d}^{2}} \]

      4. unpow2 30.6%

         \[\leadsto 0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)}{{d}^{2}} \]

      5. unpow2 30.6%

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

   10. Simplified 30.6%

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

       1. times-frac 42.9%

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

   12. Applied egg-rr 42.9%

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

   if 1.99999999999999988e-98 < (*.f64 d d) < 5e307

   1. Initial program 27.6%

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

   2. Taylor expanded in c0 around -inf 9.3%

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

      1. fma-def 9.3%

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

      2. times-frac 8.2%

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

      3. unpow2 8.2%

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

      4. unpow2 8.2%

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

      5. *-commutative 8.2%

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

      6. unpow2 8.2%

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

      7. associate-*r* 8.2%

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

   4. Simplified 31.8%

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

   5. Taylor expanded in c0 around 0 43.1%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}} \]
   6. Step-by-step derivation

      1. associate-/l* 42.1%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]

      2. associate-*r/ 42.1%

         \[\leadsto \color{blue}{\frac{0.25 \cdot {D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]

      3. unpow2 42.1%

         \[\leadsto \frac{0.25 \cdot \color{blue}{\left(D \cdot D\right)}}{\frac{{d}^{2}}{h \cdot {M}^{2}}} \]

      4. unpow2 42.1%

         \[\leadsto \frac{0.25 \cdot \left(D \cdot D\right)}{\frac{\color{blue}{d \cdot d}}{h \cdot {M}^{2}}} \]

      5. unpow2 42.1%

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

   7. Simplified 42.1%

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

   8. Taylor expanded in D around 0 43.1%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. *-commutative 43.1%

         \[\leadsto 0.25 \cdot \frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{{d}^{2}} \]

      2. unpow2 43.1%

         \[\leadsto 0.25 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \]

      3. *-commutative 43.1%

         \[\leadsto 0.25 \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{{d}^{2}} \]

      4. unpow2 43.1%

         \[\leadsto 0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)}{{d}^{2}} \]

      5. unpow2 43.1%

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

   10. Simplified 43.1%

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

   11. Taylor expanded in D around 0 43.1%

       \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}} \]
   12. Step-by-step derivation

       1. unpow2 43.1%

          \[\leadsto 0.25 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}} \]

       2. unpow2 43.1%

          \[\leadsto 0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)}{{d}^{2}} \]

       3. associate-*r* 50.0%

          \[\leadsto 0.25 \cdot \frac{\color{blue}{D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}}{{d}^{2}} \]

       4. unpow2 50.0%

          \[\leadsto 0.25 \cdot \frac{D \cdot \left(D \cdot \left(h \cdot \color{blue}{{M}^{2}}\right)\right)}{{d}^{2}} \]

       5. associate-*r* 49.8%

          \[\leadsto 0.25 \cdot \frac{D \cdot \color{blue}{\left(\left(D \cdot h\right) \cdot {M}^{2}\right)}}{{d}^{2}} \]

       6. unpow2 49.8%

          \[\leadsto 0.25 \cdot \frac{D \cdot \left(\left(D \cdot h\right) \cdot \color{blue}{\left(M \cdot M\right)}\right)}{{d}^{2}} \]

       7. unpow2 49.8%

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

   13. Simplified 49.8%

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

   if 5e307 < (*.f64 d d)

   1. Initial program 25.2%

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

      1. times-frac 25.2%

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

      2. fma-def 25.2%

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

      3. associate-/r* 25.2%

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

      4. difference-of-squares 33.1%

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

   3. Simplified 36.3%

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

   4. Taylor expanded in c0 around -inf 0.0%

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

      1. associate-*r* 0.0%

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

      2. distribute-rgt1-in 0.0%

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

      3. metadata-eval 0.0%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)\right) \cdot c0\right) \]

      4. mul0-lft 36.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot \color{blue}{0}\right) \cdot c0\right) \]

      5. metadata-eval 36.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{0} \cdot c0\right) \]

      6. mul0-lft 0.0%

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

      7. metadata-eval 0.0%

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

      8. distribute-lft1-in 0.0%

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

      9. *-commutative 0.0%

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

      10. distribute-lft1-in 0.0%

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

      11. metadata-eval 0.0%

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

      12. mul0-lft 36.9%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{0}\right) \]

   6. Simplified 36.9%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]

   7. Taylor expanded in c0 around 0 39.9%

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

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \cdot d \leq 2 \cdot 10^{-98}:\\ \;\;\;\;0.25 \cdot \left(\frac{D \cdot D}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\\ \mathbf{elif}\;d \cdot d \leq 5 \cdot 10^{+307}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot D\right)\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 10: 39.2% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \cdot d \leq 10^{+277}:\\ \;\;\;\;0.25 \cdot \left(\frac{D \cdot D}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= (* d d) 1e+277) (* 0.25 (* (/ (* D D) d) (/ (* h (* M M)) d))) 0.0))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((d * d) <= 1e+277) {
		tmp = 0.25 * (((D * D) / d) * ((h * (M * M)) / d));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((d_1 * d_1) <= 1d+277) then
        tmp = 0.25d0 * (((d * d) / d_1) * ((h * (m * m)) / d_1))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((d * d) <= 1e+277) {
		tmp = 0.25 * (((D * D) / d) * ((h * (M * M)) / d));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (d * d) <= 1e+277:
		tmp = 0.25 * (((D * D) / d) * ((h * (M * M)) / d))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (Float64(d * d) <= 1e+277)
		tmp = Float64(0.25 * Float64(Float64(Float64(D * D) / d) * Float64(Float64(h * Float64(M * M)) / d)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((d * d) <= 1e+277)
		tmp = 0.25 * (((D * D) / d) * ((h * (M * M)) / d));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[N[(d * d), $MachinePrecision], 1e+277], N[(0.25 * N[(N[(N[(D * D), $MachinePrecision] / d), $MachinePrecision] * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 d d) < 1e277

   1. Initial program 24.0%

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

   2. Taylor expanded in c0 around -inf 9.0%

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

      1. fma-def 9.0%

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

      2. times-frac 7.6%

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

      3. unpow2 7.6%

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

      4. unpow2 7.6%

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

      5. *-commutative 7.6%

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

      6. unpow2 7.6%

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

      7. associate-*r* 7.6%

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

   4. Simplified 28.9%

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

   5. Taylor expanded in c0 around 0 39.6%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}} \]
   6. Step-by-step derivation

      1. associate-/l* 39.4%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]

      2. associate-*r/ 39.4%

         \[\leadsto \color{blue}{\frac{0.25 \cdot {D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]

      3. unpow2 39.4%

         \[\leadsto \frac{0.25 \cdot \color{blue}{\left(D \cdot D\right)}}{\frac{{d}^{2}}{h \cdot {M}^{2}}} \]

      4. unpow2 39.4%

         \[\leadsto \frac{0.25 \cdot \left(D \cdot D\right)}{\frac{\color{blue}{d \cdot d}}{h \cdot {M}^{2}}} \]

      5. unpow2 39.4%

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

   7. Simplified 39.4%

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

   8. Taylor expanded in D around 0 39.6%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. *-commutative 39.6%

         \[\leadsto 0.25 \cdot \frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{{d}^{2}} \]

      2. unpow2 39.6%

         \[\leadsto 0.25 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \]

      3. *-commutative 39.6%

         \[\leadsto 0.25 \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{{d}^{2}} \]

      4. unpow2 39.6%

         \[\leadsto 0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)}{{d}^{2}} \]

      5. unpow2 39.6%

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

   10. Simplified 39.6%

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

       1. times-frac 43.0%

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

   12. Applied egg-rr 43.0%

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

   if 1e277 < (*.f64 d d)

   1. Initial program 24.6%

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

      1. times-frac 24.6%

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

      2. fma-def 24.6%

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

      3. associate-/r* 24.6%

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

      4. difference-of-squares 32.8%

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

   3. Simplified 37.7%

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

   4. Taylor expanded in c0 around -inf 0.1%

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

      1. associate-*r* 0.1%

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

      2. distribute-rgt1-in 0.1%

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

      3. metadata-eval 0.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)\right) \cdot c0\right) \]

      4. mul0-lft 35.6%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot \color{blue}{0}\right) \cdot c0\right) \]

      5. metadata-eval 35.6%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{0} \cdot c0\right) \]

      6. mul0-lft 1.0%

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

      7. metadata-eval 1.0%

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

      8. distribute-lft1-in 1.0%

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

      9. *-commutative 1.0%

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

      10. distribute-lft1-in 1.0%

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

      11. metadata-eval 1.0%

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

      12. mul0-lft 35.6%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{0}\right) \]

   6. Simplified 35.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]

   7. Taylor expanded in c0 around 0 38.4%

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

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \cdot d \leq 10^{+277}:\\ \;\;\;\;0.25 \cdot \left(\frac{D \cdot D}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 11: 33.3% 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 24.2%

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

   1. times-frac 22.7%

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

   2. fma-def 22.7%

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

   3. associate-/r* 22.7%

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

   4. difference-of-squares 27.5%

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

3. Simplified 32.4%

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

4. Taylor expanded in c0 around -inf 2.3%

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

   1. associate-*r* 2.3%

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

   2. distribute-rgt1-in 2.3%

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

   3. metadata-eval 2.3%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)\right) \cdot c0\right) \]

   4. mul0-lft 29.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot \color{blue}{0}\right) \cdot c0\right) \]

   5. metadata-eval 29.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{0} \cdot c0\right) \]

   6. mul0-lft 3.5%

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

   7. metadata-eval 3.5%

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

   8. distribute-lft1-in 3.5%

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

   9. *-commutative 3.5%

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

   10. distribute-lft1-in 3.5%

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

   11. metadata-eval 3.5%

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

   12. mul0-lft 29.5%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{0}\right) \]

6. Simplified 29.5%

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]

7. Taylor expanded in c0 around 0 33.6%

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

8. Final simplification 33.6%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023199 
(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))))))