Toniolo and Linder, Equation (13)

Average Error: 34.5 → 28.6

Time: 17.8s

Precision: binary64

Math

\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

↓

\[\begin{array}{l} t_1 := \frac{n \cdot U*}{{Om}^{2}} + \left(2 \cdot \frac{-1}{Om} - \frac{n \cdot U}{{Om}^{2}}\right)\\ t_2 := \ell \cdot \sqrt{2}\\ \mathbf{if}\;\ell \leq -5.9 \cdot 10^{+184}:\\ \;\;\;\;\left(\sqrt{\frac{n \cdot U}{t_1}} \cdot \frac{t \cdot \sqrt{2}}{\ell}\right) \cdot -0.5 - \sqrt{n \cdot \left(U \cdot t_1\right)} \cdot t_2\\ \mathbf{elif}\;\ell \leq 6.3 \cdot 10^{-170}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right), t\right)\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.05 \cdot 10^{+51}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right), t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.76 \cdot 10^{+162}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t + -2 \cdot \frac{U \cdot {\ell}^{2}}{Om}\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \sqrt{U \cdot \left(n \cdot \left(U* \cdot \frac{n}{Om \cdot Om} + \left(\frac{-2}{Om} - \frac{n \cdot U}{Om \cdot Om}\right)\right)\right)}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (sqrt
  (*
   (* (* 2.0 n) U)
   (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))

↓

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (+
          (/ (* n U*) (pow Om 2.0))
          (- (* 2.0 (/ -1.0 Om)) (/ (* n U) (pow Om 2.0)))))
        (t_2 (* l (sqrt 2.0))))
   (if (<= l -5.9e+184)
     (-
      (* (* (sqrt (/ (* n U) t_1)) (/ (* t (sqrt 2.0)) l)) -0.5)
      (* (sqrt (* n (* U t_1))) t_2))
     (if (<= l 6.3e-170)
       (pow
        (*
         2.0
         (* U (* n (fma (/ l Om) (fma l -2.0 (* (- U* U) (* n (/ l Om)))) t))))
        0.5)
       (if (<= l 1.05e+51)
         (sqrt
          (*
           2.0
           (* n (* U (fma (/ l Om) (fma l -2.0 (/ (* n (* l U*)) Om)) t)))))
         (if (<= l 1.76e+162)
           (pow
            (* 2.0 (* n (+ (* U t) (* -2.0 (/ (* U (pow l 2.0)) Om)))))
            0.5)
           (*
            t_2
            (sqrt
             (*
              U
              (*
               n
               (+
                (* U* (/ n (* Om Om)))
                (- (/ -2.0 Om) (/ (* n U) (* Om Om))))))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
}

↓

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = ((n * U_42_) / pow(Om, 2.0)) + ((2.0 * (-1.0 / Om)) - ((n * U) / pow(Om, 2.0)));
	double t_2 = l * sqrt(2.0);
	double tmp;
	if (l <= -5.9e+184) {
		tmp = ((sqrt(((n * U) / t_1)) * ((t * sqrt(2.0)) / l)) * -0.5) - (sqrt((n * (U * t_1))) * t_2);
	} else if (l <= 6.3e-170) {
		tmp = pow((2.0 * (U * (n * fma((l / Om), fma(l, -2.0, ((U_42_ - U) * (n * (l / Om)))), t)))), 0.5);
	} else if (l <= 1.05e+51) {
		tmp = sqrt((2.0 * (n * (U * fma((l / Om), fma(l, -2.0, ((n * (l * U_42_)) / Om)), t)))));
	} else if (l <= 1.76e+162) {
		tmp = pow((2.0 * (n * ((U * t) + (-2.0 * ((U * pow(l, 2.0)) / Om))))), 0.5);
	} else {
		tmp = t_2 * sqrt((U * (n * ((U_42_ * (n / (Om * Om))) + ((-2.0 / Om) - ((n * U) / (Om * Om)))))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)))))
end

↓

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(Float64(n * U_42_) / (Om ^ 2.0)) + Float64(Float64(2.0 * Float64(-1.0 / Om)) - Float64(Float64(n * U) / (Om ^ 2.0))))
	t_2 = Float64(l * sqrt(2.0))
	tmp = 0.0
	if (l <= -5.9e+184)
		tmp = Float64(Float64(Float64(sqrt(Float64(Float64(n * U) / t_1)) * Float64(Float64(t * sqrt(2.0)) / l)) * -0.5) - Float64(sqrt(Float64(n * Float64(U * t_1))) * t_2));
	elseif (l <= 6.3e-170)
		tmp = Float64(2.0 * Float64(U * Float64(n * fma(Float64(l / Om), fma(l, -2.0, Float64(Float64(U_42_ - U) * Float64(n * Float64(l / Om)))), t)))) ^ 0.5;
	elseif (l <= 1.05e+51)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * fma(Float64(l / Om), fma(l, -2.0, Float64(Float64(n * Float64(l * U_42_)) / Om)), t)))));
	elseif (l <= 1.76e+162)
		tmp = Float64(2.0 * Float64(n * Float64(Float64(U * t) + Float64(-2.0 * Float64(Float64(U * (l ^ 2.0)) / Om))))) ^ 0.5;
	else
		tmp = Float64(t_2 * sqrt(Float64(U * Float64(n * Float64(Float64(U_42_ * Float64(n / Float64(Om * Om))) + Float64(Float64(-2.0 / Om) - Float64(Float64(n * U) / Float64(Om * Om))))))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

↓

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(N[(n * U$42$), $MachinePrecision] / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(-1.0 / Om), $MachinePrecision]), $MachinePrecision] - N[(N[(n * U), $MachinePrecision] / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5.9e+184], N[(N[(N[(N[Sqrt[N[(N[(n * U), $MachinePrecision] / t$95$1), $MachinePrecision]], $MachinePrecision] * N[(N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] - N[(N[Sqrt[N[(n * N[(U * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.3e-170], N[Power[N[(2.0 * N[(U * N[(n * N[(N[(l / Om), $MachinePrecision] * N[(l * -2.0 + N[(N[(U$42$ - U), $MachinePrecision] * N[(n * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l, 1.05e+51], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(N[(l / Om), $MachinePrecision] * N[(l * -2.0 + N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 1.76e+162], N[Power[N[(2.0 * N[(n * N[(N[(U * t), $MachinePrecision] + N[(-2.0 * N[(N[(U * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[(t$95$2 * N[Sqrt[N[(U * N[(n * N[(N[(U$42$ * N[(n / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 / Om), $MachinePrecision] - N[(N[(n * U), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Error

Bits error versus n

Bits error versus U

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus U*

Derivation

1. Split input into 5 regimes

2. if l < -5.9000000000000001e184

   1. Initial program 64.0

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Simplified 50.0

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)}} \]

   3. Taylor expanded in l around -inf 36.1

      \[\leadsto \color{blue}{-\left(0.5 \cdot \left(\sqrt{\frac{n \cdot U}{\frac{n \cdot U*}{{Om}^{2}} - \left(2 \cdot \frac{1}{Om} + \frac{n \cdot U}{{Om}^{2}}\right)}} \cdot \frac{t \cdot \sqrt{2}}{\ell}\right) + \sqrt{n \cdot \left(\left(\frac{n \cdot U*}{{Om}^{2}} - \left(2 \cdot \frac{1}{Om} + \frac{n \cdot U}{{Om}^{2}}\right)\right) \cdot U\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\right)} \]

   if -5.9000000000000001e184 < l < 6.3000000000000002e-170

   1. Initial program 28.7

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Simplified 26.6

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)}} \]

   3. Applied egg-rr 26.6

      \[\leadsto \sqrt{\color{blue}{{\left(2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right), t\right)\right)\right)}^{1}}} \]

   4. Applied egg-rr 29.9

      \[\leadsto \sqrt{{\left(2 \cdot \color{blue}{{\left(n \cdot \left(U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right), t\right)\right)\right)}^{1}}\right)}^{1}} \]

   5. Applied egg-rr 26.8

      \[\leadsto \color{blue}{{\left(2 \cdot \left(U \cdot \left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right), t\right) \cdot n\right)\right)\right)}^{0.5}} \]

   if 6.3000000000000002e-170 < l < 1.0500000000000001e51

   1. Initial program 27.3

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Simplified 25.0

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)}} \]

   3. Applied egg-rr 25.0

      \[\leadsto \sqrt{\color{blue}{{\left(2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right), t\right)\right)\right)}^{1}}} \]

   4. Applied egg-rr 26.9

      \[\leadsto \sqrt{{\left(2 \cdot \color{blue}{{\left(n \cdot \left(U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right), t\right)\right)\right)}^{1}}\right)}^{1}} \]

   5. Taylor expanded in U* around inf 25.5

      \[\leadsto \sqrt{{\left(2 \cdot {\left(n \cdot \left(U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, \color{blue}{\frac{n \cdot \left(\ell \cdot U*\right)}{Om}}\right), t\right)\right)\right)}^{1}\right)}^{1}} \]

   if 1.0500000000000001e51 < l < 1.76e162

   1. Initial program 35.5

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Simplified 31.8

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)}} \]

   3. Applied egg-rr 31.8

      \[\leadsto \sqrt{\color{blue}{{\left(2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right), t\right)\right)\right)}^{1}}} \]

   4. Applied egg-rr 30.8

      \[\leadsto \sqrt{{\left(2 \cdot \color{blue}{{\left(n \cdot \left(U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right), t\right)\right)\right)}^{1}}\right)}^{1}} \]

   5. Applied egg-rr 28.9

      \[\leadsto \color{blue}{{\left(2 \cdot \left(U \cdot \left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right), t\right) \cdot n\right)\right)\right)}^{0.5}} \]

   6. Taylor expanded in n around 0 36.6

      \[\leadsto {\left(2 \cdot \color{blue}{\left(n \cdot \left(t \cdot U - 2 \cdot \frac{{\ell}^{2} \cdot U}{Om}\right)\right)}\right)}^{0.5} \]

   if 1.76e162 < l

   1. Initial program 64.0

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Simplified 49.1

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)}} \]

   3. Applied egg-rr 49.1

      \[\leadsto \sqrt{\color{blue}{{\left(2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right), t\right)\right)\right)}^{1}}} \]

   4. Applied egg-rr 48.8

      \[\leadsto \sqrt{{\left(2 \cdot \color{blue}{{\left(n \cdot \left(U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right), t\right)\right)\right)}^{1}}\right)}^{1}} \]

   5. Applied egg-rr 47.9

      \[\leadsto \color{blue}{{\left(2 \cdot \left(U \cdot \left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right), t\right) \cdot n\right)\right)\right)}^{0.5}} \]

   6. Taylor expanded in l around inf 32.0

      \[\leadsto \color{blue}{\sqrt{n \cdot \left(\left(\frac{n \cdot U*}{{Om}^{2}} - \left(2 \cdot \frac{1}{Om} + \frac{n \cdot U}{{Om}^{2}}\right)\right) \cdot U\right)} \cdot \left(\ell \cdot \sqrt{2}\right)} \]

   7. Simplified 33.7

      \[\leadsto \color{blue}{\sqrt{\left(n \cdot \left(\frac{n}{Om \cdot Om} \cdot U* - \left(\frac{2}{Om} + \frac{n \cdot U}{Om \cdot Om}\right)\right)\right) \cdot U} \cdot \left(\ell \cdot \sqrt{2}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 28.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.9 \cdot 10^{+184}:\\ \;\;\;\;\left(\sqrt{\frac{n \cdot U}{\frac{n \cdot U*}{{Om}^{2}} + \left(2 \cdot \frac{-1}{Om} - \frac{n \cdot U}{{Om}^{2}}\right)}} \cdot \frac{t \cdot \sqrt{2}}{\ell}\right) \cdot -0.5 - \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{{Om}^{2}} + \left(2 \cdot \frac{-1}{Om} - \frac{n \cdot U}{{Om}^{2}}\right)\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \mathbf{elif}\;\ell \leq 6.3 \cdot 10^{-170}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right), t\right)\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.05 \cdot 10^{+51}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right), t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.76 \cdot 10^{+162}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t + -2 \cdot \frac{U \cdot {\ell}^{2}}{Om}\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \left(U* \cdot \frac{n}{Om \cdot Om} + \left(\frac{-2}{Om} - \frac{n \cdot U}{Om \cdot Om}\right)\right)\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022155 
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  :precision binary64
  (sqrt (* (* (* 2.0 n) U) (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))