Toniolo and Linder, Equation (13)

Average Error: 34.3 → 25.8

Time: 12.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := t_1 \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_2 \leq 0:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{n \cdot \left(U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, \left(U* - U\right) \cdot \frac{n \cdot \ell}{Om}\right), t\right)\right)}\\ \mathbf{elif}\;t_2 \leq 10^{+305}:\\ \;\;\;\;\sqrt{t_2}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right), t\right)} \cdot \sqrt{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om \cdot Om} + \left(\frac{-2}{Om} - \frac{n \cdot U}{Om \cdot Om}\right)\right)\right)} \cdot \left(\ell \cdot \left(-\sqrt{2}\right)\right)\\ \end{array} \]

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 (* (* 2.0 n) U))
        (t_2
         (*
          t_1
          (+
           (+ t (* (/ (* l l) Om) -2.0))
           (* (* n (pow (/ l Om) 2.0)) (- U* U))))))
   (if (<= t_2 0.0)
     (*
      (sqrt 2.0)
      (sqrt
       (* n (* U (fma (/ l Om) (fma l -2.0 (* (- U* U) (/ (* n l) Om))) t)))))
     (if (<= t_2 1e+305)
       (sqrt t_2)
       (if (<= t_2 INFINITY)
         (*
          (sqrt (fma (/ l Om) (fma l -2.0 (* (- U* U) (* n (/ l Om)))) t))
          (sqrt t_1))
         (*
          (sqrt
           (*
            n
            (*
             U
             (+
              (/ (* n U*) (* Om Om))
              (- (/ -2.0 Om) (/ (* n U) (* Om Om)))))))
          (* l (- (sqrt 2.0)))))))))

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 = (2.0 * n) * U;
	double t_2 = t_1 * ((t + (((l * l) / Om) * -2.0)) + ((n * pow((l / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt(2.0) * sqrt((n * (U * fma((l / Om), fma(l, -2.0, ((U_42_ - U) * ((n * l) / Om))), t))));
	} else if (t_2 <= 1e+305) {
		tmp = sqrt(t_2);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(fma((l / Om), fma(l, -2.0, ((U_42_ - U) * (n * (l / Om)))), t)) * sqrt(t_1);
	} else {
		tmp = sqrt((n * (U * (((n * U_42_) / (Om * Om)) + ((-2.0 / Om) - ((n * U) / (Om * Om))))))) * (l * -sqrt(2.0));
	}
	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(2.0 * n) * U)
	t_2 = Float64(t_1 * Float64(Float64(t + Float64(Float64(Float64(l * l) / Om) * -2.0)) + Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(sqrt(2.0) * sqrt(Float64(n * Float64(U * fma(Float64(l / Om), fma(l, -2.0, Float64(Float64(U_42_ - U) * Float64(Float64(n * l) / Om))), t)))));
	elseif (t_2 <= 1e+305)
		tmp = sqrt(t_2);
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(fma(Float64(l / Om), fma(l, -2.0, Float64(Float64(U_42_ - U) * Float64(n * Float64(l / Om)))), t)) * sqrt(t_1));
	else
		tmp = Float64(sqrt(Float64(n * Float64(U * Float64(Float64(Float64(n * U_42_) / Float64(Om * Om)) + Float64(Float64(-2.0 / Om) - Float64(Float64(n * U) / Float64(Om * Om))))))) * Float64(l * Float64(-sqrt(2.0))));
	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[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(t + N[(N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(n * N[(U * N[(N[(l / Om), $MachinePrecision] * N[(l * -2.0 + N[(N[(U$42$ - U), $MachinePrecision] * N[(N[(n * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+305], N[Sqrt[t$95$2], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[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] * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(n * N[(U * N[(N[(N[(n * U$42$), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 / Om), $MachinePrecision] - N[(N[(n * U), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l * (-N[Sqrt[2.0], $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 4 regimes

2. if (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < 0.0

   1. Initial program 57.8

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

   2. Simplified 51.5

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

   3. Applied egg-rr 51.6

      \[\leadsto \sqrt{\color{blue}{{\left(\sqrt[3]{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right), t\right)\right)}\right)}^{3}}} \]

   4. Applied egg-rr 37.3

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

   if 0.0 < (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < 9.9999999999999994e304

   1. Initial program 1.5

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

   if 9.9999999999999994e304 < (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < +inf.0

   1. Initial program 63.5

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

   2. Simplified 53.6

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

   3. Applied egg-rr 49.8

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

   if +inf.0 < (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))

   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 63.5

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

   3. Applied egg-rr 63.5

      \[\leadsto \sqrt{\color{blue}{{\left(\sqrt[3]{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right), t\right)\right)}\right)}^{3}}} \]

   4. Taylor expanded in l around -inf 51.2

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

   5. Simplified 51.2

      \[\leadsto \color{blue}{\sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om \cdot Om} - \left(\frac{2}{Om} + \frac{n \cdot U}{Om \cdot Om}\right)\right)\right)} \cdot \left(-\ell \cdot \sqrt{2}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 25.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 0:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{n \cdot \left(U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, \left(U* - U\right) \cdot \frac{n \cdot \ell}{Om}\right), t\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 10^{+305}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right), t\right)} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om \cdot Om} + \left(\frac{-2}{Om} - \frac{n \cdot U}{Om \cdot Om}\right)\right)\right)} \cdot \left(\ell \cdot \left(-\sqrt{2}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022160 
(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*))))))