Toniolo and Linder, Equation (13)

Average Error: 34.9 → 24.7

Time: 13.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \mathsf{fma}\left(\ell, -2, \frac{n}{\frac{Om}{\ell}} \cdot \left(U* - U\right)\right)\\ t_2 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_2 \leq 0:\\ \;\;\;\;{\left({\left(2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, t_1, t\right)\right)\right)\right)}^{0.25}\right)}^{2}\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+294}:\\ \;\;\;\;\sqrt{t_2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \mathsf{fma}\left(n, U \cdot t, \frac{t_1}{\frac{\frac{Om}{U \cdot \ell}}{n}}\right)}\\ \end{array} \]

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 (fma l -2.0 (* (/ n (/ Om l)) (- U* U))))
        (t_2
         (*
          (* (* 2.0 n) U)
          (+
           (- t (* 2.0 (/ (* l l) Om)))
           (* (* n (pow (/ l Om) 2.0)) (- U* U))))))
   (if (<= t_2 0.0)
     (pow (pow (* 2.0 (* n (* U (fma (/ l Om) t_1 t)))) 0.25) 2.0)
     (if (<= t_2 2e+294)
       (sqrt t_2)
       (sqrt (* 2.0 (fma n (* U t) (/ t_1 (/ (/ Om (* U l)) n)))))))))

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 = fma(l, -2.0, ((n / (Om / l)) * (U_42_ - U)));
	double t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + ((n * pow((l / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = pow(pow((2.0 * (n * (U * fma((l / Om), t_1, t)))), 0.25), 2.0);
	} else if (t_2 <= 2e+294) {
		tmp = sqrt(t_2);
	} else {
		tmp = sqrt((2.0 * fma(n, (U * t), (t_1 / ((Om / (U * l)) / n)))));
	}
	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 = fma(l, -2.0, Float64(Float64(n / Float64(Om / l)) * Float64(U_42_ - U)))
	t_2 = 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_42_ - U))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = (Float64(2.0 * Float64(n * Float64(U * fma(Float64(l / Om), t_1, t)))) ^ 0.25) ^ 2.0;
	elseif (t_2 <= 2e+294)
		tmp = sqrt(t_2);
	else
		tmp = sqrt(Float64(2.0 * fma(n, Float64(U * t), Float64(t_1 / Float64(Float64(Om / Float64(U * l)) / n)))));
	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[(l * -2.0 + N[(N[(n / N[(Om / l), $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[Power[N[Power[N[(2.0 * N[(n * N[(U * N[(N[(l / Om), $MachinePrecision] * t$95$1 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[t$95$2, 2e+294], N[Sqrt[t$95$2], $MachinePrecision], N[Sqrt[N[(2.0 * N[(n * N[(U * t), $MachinePrecision] + N[(t$95$1 / N[(N[(Om / N[(U * l), $MachinePrecision]), $MachinePrecision] / n), $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 3 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 51.7

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

   3. Applied egg-rr 51.7

      \[\leadsto \color{blue}{{\left({\left(2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right), t\right)\right)\right)}^{0.25}\right)}^{2}} \]

   4. Applied egg-rr 36.5

      \[\leadsto {\left({\left(2 \cdot \color{blue}{{\left(n \cdot \left(U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, \left(U* - U\right) \cdot \frac{n}{\frac{Om}{\ell}}\right), t\right)\right)\right)}^{1}}\right)}^{0.25}\right)}^{2} \]

   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*)))) < 2.00000000000000013e294

   1. Initial program 1.7

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

   if 2.00000000000000013e294 < (*.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 62.9

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

   2. Simplified 55.5

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

   3. Taylor expanded in n around 0 58.6

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

   4. Simplified 55.5

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

   5. Taylor expanded in t around inf 50.1

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

   6. Simplified 45.7

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

4. Final simplification 24.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 0:\\ \;\;\;\;{\left({\left(2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, \frac{n}{\frac{Om}{\ell}} \cdot \left(U* - U\right)\right), t\right)\right)\right)\right)}^{0.25}\right)}^{2}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 2 \cdot 10^{+294}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \mathsf{fma}\left(n, U \cdot t, \frac{\mathsf{fma}\left(\ell, -2, \frac{n}{\frac{Om}{\ell}} \cdot \left(U* - U\right)\right)}{\frac{\frac{Om}{U \cdot \ell}}{n}}\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022178 
(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*))))))