Average Error: 26.4 → 13.5
Time: 40.5s
Precision: binary64
Cost: 89932
\[ \begin{array}{c}[M, D] = \mathsf{sort}([M, D])\\ \end{array} \]
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
↓
\[\begin{array}{l} t_0 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := 1 - 0.5 \cdot {\left(\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\\ t_3 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right)\\ \mathbf{if}\;t_3 \leq -5 \cdot 10^{-214}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot t_1\right) \cdot t_2\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;t_0 \cdot \mathsf{fma}\left(\frac{h}{\ell} \cdot {\left(\frac{0.5}{\frac{d}{M \cdot D}}\right)}^{2}, -0.5, 1\right)\\ \mathbf{elif}\;t_3 \leq 5 \cdot 10^{+264}:\\ \;\;\;\;t_2 \cdot \left(t_1 \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \mathsf{fma}\left(h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}, -0.5, 1\right)\\ \end{array} \]
(FPCore (d h l M D)
 :precision binary64
 (*
  (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
  (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
↓
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (fabs (/ d (sqrt (* h l)))))
        (t_1 (sqrt (/ d l)))
        (t_2
         (- 1.0 (* 0.5 (pow (* (* M (* 0.5 (/ D d))) (sqrt (/ h l))) 2.0))))
        (t_3
         (*
          (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
          (+ 1.0 (* (/ h l) (* (pow (/ (* M D) (* d 2.0)) 2.0) -0.5))))))
   (if (<= t_3 -5e-214)
     (* (* (/ 1.0 (sqrt (/ h d))) t_1) t_2)
     (if (<= t_3 0.0)
       (* t_0 (fma (* (/ h l) (pow (/ 0.5 (/ d (* M D))) 2.0)) -0.5 1.0))
       (if (<= t_3 5e+264)
         (* t_2 (* t_1 (sqrt (/ d h))))
         (*
          t_0
          (fma (* h (/ (pow (* D (* M (/ 0.5 d))) 2.0) l)) -0.5 1.0)))))))
double code(double d, double h, double l, double M, double D) {
	return (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
}
↓
double code(double d, double h, double l, double M, double D) {
	double t_0 = fabs((d / sqrt((h * l))));
	double t_1 = sqrt((d / l));
	double t_2 = 1.0 - (0.5 * pow(((M * (0.5 * (D / d))) * sqrt((h / l))), 2.0));
	double t_3 = (pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 + ((h / l) * (pow(((M * D) / (d * 2.0)), 2.0) * -0.5)));
	double tmp;
	if (t_3 <= -5e-214) {
		tmp = ((1.0 / sqrt((h / d))) * t_1) * t_2;
	} else if (t_3 <= 0.0) {
		tmp = t_0 * fma(((h / l) * pow((0.5 / (d / (M * D))), 2.0)), -0.5, 1.0);
	} else if (t_3 <= 5e+264) {
		tmp = t_2 * (t_1 * sqrt((d / h)));
	} else {
		tmp = t_0 * fma((h * (pow((D * (M * (0.5 / d))), 2.0) / l)), -0.5, 1.0);
	}
	return tmp;
}

function code(d, h, l, M, D)
	return Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
end
↓
function code(d, h, l, M, D)
	t_0 = abs(Float64(d / sqrt(Float64(h * l))))
	t_1 = sqrt(Float64(d / l))
	t_2 = Float64(1.0 - Float64(0.5 * (Float64(Float64(M * Float64(0.5 * Float64(D / d))) * sqrt(Float64(h / l))) ^ 2.0)))
	t_3 = Float64(Float64((Float64(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5)) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0) * -0.5))))
	tmp = 0.0
	if (t_3 <= -5e-214)
		tmp = Float64(Float64(Float64(1.0 / sqrt(Float64(h / d))) * t_1) * t_2);
	elseif (t_3 <= 0.0)
		tmp = Float64(t_0 * fma(Float64(Float64(h / l) * (Float64(0.5 / Float64(d / Float64(M * D))) ^ 2.0)), -0.5, 1.0));
	elseif (t_3 <= 5e+264)
		tmp = Float64(t_2 * Float64(t_1 * sqrt(Float64(d / h))));
	else
		tmp = Float64(t_0 * fma(Float64(h * Float64((Float64(D * Float64(M * Float64(0.5 / d))) ^ 2.0) / l)), -0.5, 1.0));
	end
	return tmp
end

code[d_, h_, l_, M_, D_] := N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[(0.5 * N[Power[N[(N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -5e-214], N[(N[(N[(1.0 / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(t$95$0 * N[(N[(N[(h / l), $MachinePrecision] * N[Power[N[(0.5 / N[(d / N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e+264], N[(t$95$2 * N[(t$95$1 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(h * N[(N[Power[N[(D * N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
↓
\begin{array}{l}
t_0 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\
t_1 := \sqrt{\frac{d}{\ell}}\\
t_2 := 1 - 0.5 \cdot {\left(\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\\
t_3 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right)\\
\mathbf{if}\;t_3 \leq -5 \cdot 10^{-214}:\\
\;\;\;\;\left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot t_1\right) \cdot t_2\\

\mathbf{elif}\;t_3 \leq 0:\\
\;\;\;\;t_0 \cdot \mathsf{fma}\left(\frac{h}{\ell} \cdot {\left(\frac{0.5}{\frac{d}{M \cdot D}}\right)}^{2}, -0.5, 1\right)\\

\mathbf{elif}\;t_3 \leq 5 \cdot 10^{+264}:\\
\;\;\;\;t_2 \cdot \left(t_1 \cdot \sqrt{\frac{d}{h}}\right)\\

\mathbf{else}:\\
\;\;\;\;t_0 \cdot \mathsf{fma}\left(h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}, -0.5, 1\right)\\


\end{array}

Error

Derivation

  1. Split input into 4 regimes

  2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < -4.9999999999999998e-214

    1. Initial program 26.9

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

    2. Simplified27.7

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
      Proof
      (*.f64 (*.f64 (sqrt.f64 (/.f64 d h)) (sqrt.f64 (/.f64 d l))) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (Rewrite<= unpow1/2_binary64 (pow.f64 (/.f64 d h) 1/2)) (sqrt.f64 (/.f64 d l))) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (Rewrite<= metadata-eval (/.f64 1 2))) (sqrt.f64 (/.f64 d l))) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l))))): 8 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (Rewrite<= unpow1/2_binary64 (pow.f64 (/.f64 d l) 1/2))) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l))))): 8 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (Rewrite<= metadata-eval (/.f64 1 2)))) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l))))): 0 points increase in error, 8 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (Rewrite<= metadata-eval (/.f64 1 2)) (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l))))): 8 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (/.f64 1 2) (*.f64 (pow.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 M D) (*.f64 2 d))) 2) (/.f64 h l))))): 0 points increase in error, 8 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l))))): 0 points increase in error, 0 points decrease in error

    3. Applied egg-rr21.8

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

    4. Applied egg-rr22.4

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

    if -4.9999999999999998e-214 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < 0.0

    1. Initial program 40.1

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

    2. Applied egg-rr41.0

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

    3. Simplified35.5

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \mathsf{fma}\left({\left(\frac{0.5}{\frac{d}{D \cdot M}}\right)}^{2} \cdot \frac{h}{\ell}, -0.5, 1\right)} \]
      Proof
      (*.f64 (sqrt.f64 (*.f64 (/.f64 d (*.f64 h l)) d)) (fma.f64 (*.f64 (pow.f64 (/.f64 1/2 (/.f64 d (*.f64 D M))) 2) (/.f64 h l)) -1/2 1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (*.f64 (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 d h) l)) d)) (fma.f64 (*.f64 (pow.f64 (/.f64 1/2 (/.f64 d (*.f64 D M))) 2) (/.f64 h l)) -1/2 1)): 1 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (Rewrite<= associate-/r/_binary64 (/.f64 (/.f64 d h) (/.f64 l d)))) (fma.f64 (*.f64 (pow.f64 (/.f64 1/2 (/.f64 d (*.f64 D M))) 2) (/.f64 h l)) -1/2 1)): 10 points increase in error, 1 points decrease in error
      (*.f64 (sqrt.f64 (Rewrite<= associate-/r*_binary64 (/.f64 d (*.f64 h (/.f64 l d))))) (fma.f64 (*.f64 (pow.f64 (/.f64 1/2 (/.f64 d (*.f64 D M))) 2) (/.f64 h l)) -1/2 1)): 12 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (/.f64 d (*.f64 h (/.f64 l d)))) (fma.f64 (*.f64 (pow.f64 (/.f64 1/2 (/.f64 d (Rewrite<= *-commutative_binary64 (*.f64 M D)))) 2) (/.f64 h l)) -1/2 1)): 0 points increase in error, 12 points decrease in error
      (*.f64 (sqrt.f64 (/.f64 d (*.f64 h (/.f64 l d)))) (fma.f64 (*.f64 (pow.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 1/2 (*.f64 M D)) d)) 2) (/.f64 h l)) -1/2 1)): 12 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (/.f64 d (*.f64 h (/.f64 l d)))) (fma.f64 (*.f64 (pow.f64 (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 (*.f64 M D) d))) 2) (/.f64 h l)) -1/2 1)): 0 points increase in error, 12 points decrease in error
      (*.f64 (sqrt.f64 (/.f64 d (*.f64 h (/.f64 l d)))) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (*.f64 (pow.f64 (*.f64 1/2 (/.f64 (*.f64 M D) d)) 2) (/.f64 h l)) -1/2) 1))): 0 points increase in error, 9 points decrease in error
      (*.f64 (sqrt.f64 (/.f64 d (*.f64 h (/.f64 l d)))) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1/2 (*.f64 (pow.f64 (*.f64 1/2 (/.f64 (*.f64 M D) d)) 2) (/.f64 h l)))) 1)): 12 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (/.f64 d (*.f64 h (/.f64 l d)))) (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 -1/2 (*.f64 (pow.f64 (*.f64 1/2 (/.f64 (*.f64 M D) d)) 2) (/.f64 h l)))))): 0 points increase in error, 12 points decrease in error
      (Rewrite=> distribute-lft-in_binary64 (+.f64 (*.f64 (sqrt.f64 (/.f64 d (*.f64 h (/.f64 l d)))) 1) (*.f64 (sqrt.f64 (/.f64 d (*.f64 h (/.f64 l d)))) (*.f64 -1/2 (*.f64 (pow.f64 (*.f64 1/2 (/.f64 (*.f64 M D) d)) 2) (/.f64 h l)))))): 0 points increase in error, 12 points decrease in error
      (+.f64 (Rewrite=> *-rgt-identity_binary64 (sqrt.f64 (/.f64 d (*.f64 h (/.f64 l d))))) (*.f64 (sqrt.f64 (/.f64 d (*.f64 h (/.f64 l d)))) (*.f64 -1/2 (*.f64 (pow.f64 (*.f64 1/2 (/.f64 (*.f64 M D) d)) 2) (/.f64 h l))))): 0 points increase in error, 9 points decrease in error

    4. Applied egg-rr12.8

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

    if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < 5.00000000000000033e264

    1. Initial program 0.9

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

    2. Simplified1.3

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
      Proof
      (*.f64 (*.f64 (sqrt.f64 (/.f64 d h)) (sqrt.f64 (/.f64 d l))) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (Rewrite<= unpow1/2_binary64 (pow.f64 (/.f64 d h) 1/2)) (sqrt.f64 (/.f64 d l))) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (Rewrite<= metadata-eval (/.f64 1 2))) (sqrt.f64 (/.f64 d l))) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l))))): 8 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (Rewrite<= unpow1/2_binary64 (pow.f64 (/.f64 d l) 1/2))) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l))))): 8 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (Rewrite<= metadata-eval (/.f64 1 2)))) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l))))): 0 points increase in error, 8 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (Rewrite<= metadata-eval (/.f64 1 2)) (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l))))): 8 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (/.f64 1 2) (*.f64 (pow.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 M D) (*.f64 2 d))) 2) (/.f64 h l))))): 0 points increase in error, 8 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l))))): 0 points increase in error, 0 points decrease in error

    3. Applied egg-rr1.3

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

    if 5.00000000000000033e264 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l))))

    1. Initial program 62.7

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

    2. Applied egg-rr60.9

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

    3. Simplified60.9

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \mathsf{fma}\left({\left(\frac{0.5}{\frac{d}{D \cdot M}}\right)}^{2} \cdot \frac{h}{\ell}, -0.5, 1\right)} \]
      Proof
      (*.f64 (sqrt.f64 (*.f64 (/.f64 d (*.f64 h l)) d)) (fma.f64 (*.f64 (pow.f64 (/.f64 1/2 (/.f64 d (*.f64 D M))) 2) (/.f64 h l)) -1/2 1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (*.f64 (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 d h) l)) d)) (fma.f64 (*.f64 (pow.f64 (/.f64 1/2 (/.f64 d (*.f64 D M))) 2) (/.f64 h l)) -1/2 1)): 1 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (Rewrite<= associate-/r/_binary64 (/.f64 (/.f64 d h) (/.f64 l d)))) (fma.f64 (*.f64 (pow.f64 (/.f64 1/2 (/.f64 d (*.f64 D M))) 2) (/.f64 h l)) -1/2 1)): 10 points increase in error, 1 points decrease in error
      (*.f64 (sqrt.f64 (Rewrite<= associate-/r*_binary64 (/.f64 d (*.f64 h (/.f64 l d))))) (fma.f64 (*.f64 (pow.f64 (/.f64 1/2 (/.f64 d (*.f64 D M))) 2) (/.f64 h l)) -1/2 1)): 12 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (/.f64 d (*.f64 h (/.f64 l d)))) (fma.f64 (*.f64 (pow.f64 (/.f64 1/2 (/.f64 d (Rewrite<= *-commutative_binary64 (*.f64 M D)))) 2) (/.f64 h l)) -1/2 1)): 0 points increase in error, 12 points decrease in error
      (*.f64 (sqrt.f64 (/.f64 d (*.f64 h (/.f64 l d)))) (fma.f64 (*.f64 (pow.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 1/2 (*.f64 M D)) d)) 2) (/.f64 h l)) -1/2 1)): 12 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (/.f64 d (*.f64 h (/.f64 l d)))) (fma.f64 (*.f64 (pow.f64 (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 (*.f64 M D) d))) 2) (/.f64 h l)) -1/2 1)): 0 points increase in error, 12 points decrease in error
      (*.f64 (sqrt.f64 (/.f64 d (*.f64 h (/.f64 l d)))) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (*.f64 (pow.f64 (*.f64 1/2 (/.f64 (*.f64 M D) d)) 2) (/.f64 h l)) -1/2) 1))): 0 points increase in error, 9 points decrease in error
      (*.f64 (sqrt.f64 (/.f64 d (*.f64 h (/.f64 l d)))) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1/2 (*.f64 (pow.f64 (*.f64 1/2 (/.f64 (*.f64 M D) d)) 2) (/.f64 h l)))) 1)): 12 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (/.f64 d (*.f64 h (/.f64 l d)))) (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 -1/2 (*.f64 (pow.f64 (*.f64 1/2 (/.f64 (*.f64 M D) d)) 2) (/.f64 h l)))))): 0 points increase in error, 12 points decrease in error
      (Rewrite=> distribute-lft-in_binary64 (+.f64 (*.f64 (sqrt.f64 (/.f64 d (*.f64 h (/.f64 l d)))) 1) (*.f64 (sqrt.f64 (/.f64 d (*.f64 h (/.f64 l d)))) (*.f64 -1/2 (*.f64 (pow.f64 (*.f64 1/2 (/.f64 (*.f64 M D) d)) 2) (/.f64 h l)))))): 0 points increase in error, 12 points decrease in error
      (+.f64 (Rewrite=> *-rgt-identity_binary64 (sqrt.f64 (/.f64 d (*.f64 h (/.f64 l d))))) (*.f64 (sqrt.f64 (/.f64 d (*.f64 h (/.f64 l d)))) (*.f64 -1/2 (*.f64 (pow.f64 (*.f64 1/2 (/.f64 (*.f64 M D) d)) 2) (/.f64 h l))))): 0 points increase in error, 9 points decrease in error

    4. Applied egg-rr40.0

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

    5. Applied egg-rr40.0

      \[\leadsto \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \mathsf{fma}\left(\color{blue}{e^{\mathsf{log1p}\left({\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)} - 1}, -0.5, 1\right) \]

    6. Simplified27.8

      \[\leadsto \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \mathsf{fma}\left(\color{blue}{h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}, -0.5, 1\right) \]
      Proof
      (*.f64 (fabs.f64 (/.f64 d (sqrt.f64 (*.f64 h l)))) (fma.f64 (*.f64 h (/.f64 (pow.f64 (*.f64 D (*.f64 M (/.f64 1/2 d))) 2) l)) -1/2 1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (fabs.f64 (/.f64 d (sqrt.f64 (*.f64 h l)))) (fma.f64 (*.f64 h (/.f64 (pow.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 D M) (/.f64 1/2 d))) 2) l)) -1/2 1)): 7 points increase in error, 0 points decrease in error
      (*.f64 (fabs.f64 (/.f64 d (sqrt.f64 (*.f64 h l)))) (fma.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (pow.f64 (*.f64 (*.f64 D M) (/.f64 1/2 d)) 2) l) h)) -1/2 1)): 0 points increase in error, 7 points decrease in error
      (*.f64 (fabs.f64 (/.f64 d (sqrt.f64 (*.f64 h l)))) (fma.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (pow.f64 (*.f64 (*.f64 D M) (/.f64 1/2 d)) 2) h) l)) -1/2 1)): 7 points increase in error, 0 points decrease in error
      (*.f64 (fabs.f64 (/.f64 d (sqrt.f64 (*.f64 h l)))) (fma.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (pow.f64 (*.f64 (*.f64 D M) (/.f64 1/2 d)) 2) (/.f64 h l))) -1/2 1)): 0 points increase in error, 7 points decrease in error
      (*.f64 (fabs.f64 (/.f64 d (sqrt.f64 (*.f64 h l)))) (fma.f64 (Rewrite<= expm1-log1p_binary64 (expm1.f64 (log1p.f64 (*.f64 (pow.f64 (*.f64 (*.f64 D M) (/.f64 1/2 d)) 2) (/.f64 h l))))) -1/2 1)): 0 points increase in error, 7 points decrease in error
      (*.f64 (fabs.f64 (/.f64 d (sqrt.f64 (*.f64 h l)))) (fma.f64 (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (log1p.f64 (*.f64 (pow.f64 (*.f64 (*.f64 D M) (/.f64 1/2 d)) 2) (/.f64 h l)))) 1)) -1/2 1)): 0 points increase in error, 7 points decrease in error
  3. Recombined 4 regimes into one program.

  4. Final simplification13.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \leq -5 \cdot 10^{-214}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \leq 0:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \mathsf{fma}\left(\frac{h}{\ell} \cdot {\left(\frac{0.5}{\frac{d}{M \cdot D}}\right)}^{2}, -0.5, 1\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \leq 5 \cdot 10^{+264}:\\ \;\;\;\;\left(1 - 0.5 \cdot {\left(\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \mathsf{fma}\left(h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}, -0.5, 1\right)\\ \end{array} \]

Alternatives

Alternative 1
Error13.4
Cost89932
\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right)\\ t_1 := \left(1 - 0.5 \cdot {\left(\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ t_2 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-214}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;t_2 \cdot \mathsf{fma}\left(\frac{h}{\ell} \cdot {\left(\frac{0.5}{\frac{d}{M \cdot D}}\right)}^{2}, -0.5, 1\right)\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+264}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \mathsf{fma}\left(h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}, -0.5, 1\right)\\ \end{array} \]
Alternative 2
Error13.8
Cost89677
\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \frac{M}{\frac{2}{\frac{D}{d}}}\\ t_2 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right)\\ \mathbf{if}\;t_2 \leq -5 \cdot 10^{-214}:\\ \;\;\;\;\frac{t_0}{\frac{\sqrt{\frac{h}{d}}}{1 - \frac{t_1}{\frac{\frac{\ell}{h}}{0.5 \cdot t_1}}}}\\ \mathbf{elif}\;t_2 \leq 0 \lor \neg \left(t_2 \leq 5 \cdot 10^{+264}\right):\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \mathsf{fma}\left(h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \end{array} \]
Alternative 3
Error13.6
Cost89676
\[\begin{array}{l} t_0 := \frac{M}{\frac{2}{\frac{D}{d}}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right)\\ t_2 := \sqrt{\frac{d}{\ell}}\\ t_3 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-214}:\\ \;\;\;\;\frac{t_2}{\frac{\sqrt{\frac{h}{d}}}{1 - \frac{t_0}{\frac{\frac{\ell}{h}}{0.5 \cdot t_0}}}}\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;t_3 \cdot \mathsf{fma}\left(\frac{h}{\ell} \cdot {\left(\frac{0.5}{\frac{d}{M \cdot D}}\right)}^{2}, -0.5, 1\right)\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+264}:\\ \;\;\;\;\left(t_2 \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \mathsf{fma}\left(h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}, -0.5, 1\right)\\ \end{array} \]
Alternative 4
Error15.8
Cost83596
\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \frac{M}{\frac{2}{\frac{D}{d}}}\\ t_2 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right)\\ \mathbf{if}\;t_2 \leq -1 \cdot 10^{-243}:\\ \;\;\;\;\frac{t_0}{\frac{\sqrt{\frac{h}{d}}}{1 - \frac{t_1}{\frac{\frac{\ell}{h}}{0.5 \cdot t_1}}}}\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{h}{\ell} \cdot {\left(\frac{0.5}{\frac{d}{M \cdot D}}\right)}^{2}, -0.5, 1\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right)\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+264}:\\ \;\;\;\;\left(t_0 \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \mathsf{fma}\left(0.25 \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\frac{d \cdot \left(d \cdot \ell\right)}{h}}, -0.5, 1\right)\\ \end{array} \]
Alternative 5
Error19.6
Cost21136
\[\begin{array}{l} t_0 := \frac{M}{\frac{2}{\frac{D}{d}}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \frac{t_1}{\frac{\sqrt{\frac{h}{d}}}{1 - \frac{t_0}{\frac{\frac{\ell}{h}}{0.5 \cdot t_0}}}}\\ \mathbf{if}\;h \leq -2.1 \cdot 10^{+59}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;h \leq -6.2 \cdot 10^{-99}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;h \leq -5.5 \cdot 10^{-147}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{t_1}{\sqrt{-h}} \cdot \sqrt{-d}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
Alternative 6
Error19.5
Cost21132
\[\begin{array}{l} t_0 := \frac{M}{\frac{2}{\frac{D}{d}}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\\ \mathbf{if}\;\ell \leq -1.8 \cdot 10^{+95}:\\ \;\;\;\;\frac{t_1}{\frac{\sqrt{\frac{h}{d}}}{1 - \frac{t_0}{\frac{\frac{\ell}{h}}{0.5 \cdot t_0}}}}\\ \mathbf{elif}\;\ell \leq -4.2 \cdot 10^{-97}:\\ \;\;\;\;\mathsf{fma}\left(\frac{h}{\ell} \cdot {\left(\frac{0.5}{\frac{d}{M \cdot D}}\right)}^{2}, -0.5, 1\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right)\\ \mathbf{elif}\;\ell \leq 5.2 \cdot 10^{-288}:\\ \;\;\;\;\left(t_1 \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot t_2}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + t_2 \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
Alternative 7
Error18.6
Cost21004
\[\begin{array}{l} t_0 := \frac{M}{\frac{2}{\frac{D}{d}}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq -5.6 \cdot 10^{+87}:\\ \;\;\;\;\frac{t_1}{\frac{\sqrt{\frac{h}{d}}}{1 - \frac{t_0}{\frac{\frac{\ell}{h}}{0.5 \cdot t_0}}}}\\ \mathbf{elif}\;h \leq -2.7 \cdot 10^{-121}:\\ \;\;\;\;\mathsf{fma}\left(\frac{h}{\ell} \cdot {\left(\frac{0.5}{\frac{d}{M \cdot D}}\right)}^{2}, -0.5, 1\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{t_1}{\sqrt{-h}} \cdot \sqrt{-d}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
Alternative 8
Error21.5
Cost20304
\[\begin{array}{l} t_0 := \frac{M}{\frac{2}{\frac{D}{d}}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \frac{t_1}{\frac{\sqrt{\frac{h}{d}}}{1 - \frac{t_0}{\frac{\frac{\ell}{h}}{0.5 \cdot t_0}}}}\\ \mathbf{if}\;h \leq -2.1 \cdot 10^{+59}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;h \leq -6.3 \cdot 10^{-99}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;h \leq -1.4 \cdot 10^{-148}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{t_1}{\sqrt{-h}} \cdot \sqrt{-d}\\ \mathbf{elif}\;h \leq 2.9 \cdot 10^{-65}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 9
Error20.2
Cost15048
\[\begin{array}{l} t_0 := \frac{M}{\frac{2}{\frac{D}{d}}}\\ \mathbf{if}\;d \leq -8.8 \cdot 10^{+99}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq 6.8 \cdot 10^{+62}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\frac{\sqrt{\frac{h}{d}}}{1 - \frac{t_0}{\frac{\frac{\ell}{h}}{0.5 \cdot t_0}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
Alternative 10
Error21.4
Cost14792
\[\begin{array}{l} t_0 := M \cdot \frac{D}{d}\\ \mathbf{if}\;d \leq -6.8 \cdot 10^{+98}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq 3.3 \cdot 10^{+69}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\frac{\sqrt{\frac{h}{d}}}{1 + \left(\frac{h}{\ell} \cdot \left(t_0 \cdot t_0\right)\right) \cdot -0.125}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
Alternative 11
Error23.2
Cost14732
\[\begin{array}{l} \mathbf{if}\;d \leq -5.35 \cdot 10^{+98}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq -1.22 \cdot 10^{-91}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -4.2 \cdot 10^{-247}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d \cdot \frac{2}{M}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 6.5 \cdot 10^{-305}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
Alternative 12
Error23.2
Cost14732
\[\begin{array}{l} \mathbf{if}\;d \leq -7.5 \cdot 10^{+98}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq -1.7 \cdot 10^{-91}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-249}:\\ \;\;\;\;\left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(M \cdot \frac{0.5 \cdot D}{d}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 6.5 \cdot 10^{-305}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
Alternative 13
Error23.2
Cost13512
\[\begin{array}{l} \mathbf{if}\;d \leq -7.2 \cdot 10^{+97}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq 6.5 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
Alternative 14
Error23.7
Cost13252
\[\begin{array}{l} \mathbf{if}\;\ell \leq 10^{-292}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
Alternative 15
Error32.9
Cost7113
\[\begin{array}{l} \mathbf{if}\;h \leq -4.2 \cdot 10^{-299} \lor \neg \left(h \leq 1.95 \cdot 10^{+152}\right):\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]
Alternative 16
Error27.3
Cost7112
\[\begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;h \leq 2.7 \cdot 10^{+152}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \end{array} \]
Alternative 17
Error27.1
Cost7112
\[\begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;h \leq 2.6 \cdot 10^{+153}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \end{array} \]
Alternative 18
Error43.9
Cost6720
\[\frac{d}{\sqrt{h \cdot \ell}} \]

Error

Reproduce

herbie shell --seed 2022343 
(FPCore (d h l M D)
  :name "Henrywood and Agarwal, Equation (12)"
  :precision binary64
  (* (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))) (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))