Toniolo and Linder, Equation (10+)

Average Error: 32.5 → 9.6

Time: 24.1s

Precision: binary64

Cost: 21064

Math

\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;t \leq -1.8693790316622432 \cdot 10^{-129} \lor \neg \left(t \leq 8.663175062679125 \cdot 10^{-147}\right):\\ \;\;\;\;\frac{1}{t \cdot \left(\sin k \cdot \frac{t}{\ell}\right)} \cdot \left(\frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{t}}{\tan k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \end{array}\]

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))

↓

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -1.8693790316622432e-129) (not (<= t 8.663175062679125e-147)))
   (*
    (/ 1.0 (* t (* (sin k) (/ t l))))
    (* (/ 2.0 (+ 2.0 (pow (/ k t) 2.0))) (/ (/ l t) (tan k))))
   (* 2.0 (/ (* (cos k) (* l l)) (* (* k k) (* t (pow (sin k) 2.0)))))))

C

double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0));
}

↓

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.8693790316622432e-129) || !(t <= 8.663175062679125e-147)) {
		tmp = (1.0 / (t * (sin(k) * (t / l)))) * ((2.0 / (2.0 + pow((k / t), 2.0))) * ((l / t) / tan(k)));
	} else {
		tmp = 2.0 * ((cos(k) * (l * l)) / ((k * k) * (t * pow(sin(k), 2.0))));
	}
	return tmp;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Alternatives

Alternative 1
Error	10.4
Cost	21250

\[\begin{array}{l} \mathbf{if}\;t \leq -1.346046074153672 \cdot 10^{-127}:\\ \;\;\;\;\frac{\frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{t}}{\left(t \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right) \cdot \tan k}\\ \mathbf{elif}\;t \leq 9.861053656754025 \cdot 10^{-146}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\ell}{t}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(t \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right)}\\ \end{array}\]

Alternative 2
Error	10.5
Cost	20936

\[\begin{array}{l} \mathbf{if}\;t \leq -9.928524417393147 \cdot 10^{-133} \lor \neg \left(t \leq 9.123335154664629 \cdot 10^{-146}\right):\\ \;\;\;\;\frac{2 \cdot \frac{\ell}{t}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(t \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \end{array}\]

Alternative 3
Error	15.7
Cost	21571

\[\begin{array}{l} \mathbf{if}\;t \leq -3.315233517806434 \cdot 10^{-84}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \frac{t \cdot t}{\ell}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 5.39381608006051 \cdot 10^{-152}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;t \leq 6.945650438831643 \cdot 10^{+140}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \frac{t \cdot t}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{t}}{\tan k \cdot \left(t \cdot \frac{k}{\frac{\ell}{t}}\right)}\\ \end{array}\]

Alternative 4
Error	16.8
Cost	21892

\[\begin{array}{l} \mathbf{if}\;k \leq -1.3130496760861736 \cdot 10^{+164}:\\ \;\;\;\;0\\ \mathbf{elif}\;k \leq -7.527115813338382 \cdot 10^{-06}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{t}{\frac{\ell}{\frac{t \cdot t}{\ell}}}\right)\right)}\\ \mathbf{elif}\;k \leq 3.778903568569646 \cdot 10^{-16}:\\ \;\;\;\;\frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{t}}{\tan k \cdot \left(t \cdot \frac{k}{\frac{\ell}{t}}\right)}\\ \mathbf{elif}\;k \leq 4.859697594468356 \cdot 10^{+58}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{t}{\frac{\ell}{\frac{t \cdot t}{\ell}}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \end{array}\]

Alternative 5
Error	16.4
Cost	20866

\[\begin{array}{l} \mathbf{if}\;k \leq -3.6657376125864176 \cdot 10^{-05}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)}}\\ \mathbf{elif}\;k \leq 6.462770160792783 \cdot 10^{-16}:\\ \;\;\;\;\frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{t}}{\tan k \cdot \left(t \cdot \frac{k}{\frac{\ell}{t}}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \end{array}\]

Alternative 6
Error	16.4
Cost	20552

\[\begin{array}{l} \mathbf{if}\;k \leq -1.6044874314052518 \cdot 10^{-06} \lor \neg \left(k \leq 6.462770160792783 \cdot 10^{-16}\right):\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{t}}{\tan k \cdot \left(t \cdot \frac{k}{\frac{\ell}{t}}\right)}\\ \end{array}\]

Alternative 7
Error	19.7
Cost	14208

\[\frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{t}}{\tan k \cdot \left(t \cdot \frac{k}{\frac{\ell}{t}}\right)}\]

Alternative 8
Error	20.8
Cost	14208

\[\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot \left(k \cdot \frac{t}{\ell}\right)\right)\right)\right)}\]

Alternative 9
Error	23.5
Cost	14536

\[\begin{array}{l} \mathbf{if}\;t \leq -1.8460178730588518 \cdot 10^{-148} \lor \neg \left(t \leq 2.103110743229398 \cdot 10^{-155}\right):\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(\frac{t}{\ell} \cdot \frac{k}{\frac{\ell}{t \cdot t}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Alternative 10
Error	26.6
Cost	8589

\[\begin{array}{l} \mathbf{if}\;k \leq -0.001874235989885437:\\ \;\;\;\;0\\ \mathbf{elif}\;k \leq -1.5905221359463185 \cdot 10^{-162} \lor \neg \left(k \leq 1.5874524061093742 \cdot 10^{-162}\right) \land k \leq 261511665.4993392:\\ \;\;\;\;\frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{t}}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot t\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Alternative 11
Error	29.7
Cost	7938

\[\begin{array}{l} \mathbf{if}\;t \leq -2.4152762656242185 \cdot 10^{+172}:\\ \;\;\;\;0\\ \mathbf{elif}\;t \leq -1.0808523198619037 \cdot 10^{-47}:\\ \;\;\;\;\frac{2}{2 \cdot \frac{k \cdot k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Alternative 12
Error	30.2
Cost	64

\[0\]

Alternative 13
Error	61.9
Cost	64

\[1\]

Derivation

1. Split input into 2 regimes

2. if t < -1.86937903166224318e-129 or 8.66317506267912491e-147 < t

   1. Initial program 24.9

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   2. Simplified 24.9

      \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}}\]
   3. Using strategy rm

   4. Applied cube-mult_binary64_449 24.9

      \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\]

   5. Applied times-frac_binary64_425 17.0

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\]

   6. Applied associate-*l*_binary64_360 14.7

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\]
   7. Using strategy rm

   8. Applied *-un-lft-identity_binary64_419 14.7

      \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\color{blue}{1 \cdot \ell}} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\]

   9. Applied times-frac_binary64_425 10.1

      \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(\color{blue}{\left(\frac{t}{1} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\]

   10. Applied associate-*l*_binary64_360 9.0

       \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t}{1} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\]
   11. Using strategy rm

   12. Applied *-un-lft-identity_binary64_419 9.0

       \[\leadsto \frac{\color{blue}{1 \cdot 2}}{\left(\left(\frac{t}{\ell} \cdot \left(\frac{t}{1} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\]

   13. Applied times-frac_binary64_425 9.0

       \[\leadsto \color{blue}{\frac{1}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{1} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)\right) \cdot \tan k} \cdot \frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}}}\]

   14. Simplified 7.3

       \[\leadsto \color{blue}{\frac{\frac{\ell}{t}}{\left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \tan k}} \cdot \frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}}\]
   15. Using strategy rm

   16. Applied *-un-lft-identity_binary64_419 7.3

       \[\leadsto \frac{\frac{\ell}{\color{blue}{1 \cdot t}}}{\left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \tan k} \cdot \frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}}\]

   17. Applied *-un-lft-identity_binary64_419 7.3

       \[\leadsto \frac{\frac{\color{blue}{1 \cdot \ell}}{1 \cdot t}}{\left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \tan k} \cdot \frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}}\]

   18. Applied times-frac_binary64_425 7.3

       \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{\ell}{t}}}{\left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \tan k} \cdot \frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}}\]

   19. Applied times-frac_binary64_425 6.3

       \[\leadsto \color{blue}{\left(\frac{\frac{1}{1}}{t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)} \cdot \frac{\frac{\ell}{t}}{\tan k}\right)} \cdot \frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}}\]

   20. Applied associate-*l*_binary64_360 5.0

       \[\leadsto \color{blue}{\frac{\frac{1}{1}}{t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)} \cdot \left(\frac{\frac{\ell}{t}}{\tan k} \cdot \frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}}\right)}\]

   21. Simplified 5.0

       \[\leadsto \frac{\frac{1}{1}}{t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)} \cdot \color{blue}{\left(\frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{t}}{\tan k}\right)}\]

   22. Simplified 5.0

       \[\leadsto \color{blue}{\frac{1}{t \cdot \left(\sin k \cdot \frac{t}{\ell}\right)} \cdot \left(\frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{t}}{\tan k}\right)}\]

   if -1.86937903166224318e-129 < t < 8.66317506267912491e-147

   1. Initial program 64.0

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   2. Simplified 64.0

      \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}}\]
   3. Using strategy rm

   4. Applied cube-mult_binary64_449 64.0

      \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\]

   5. Applied times-frac_binary64_425 60.0

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\]

   6. Applied associate-*l*_binary64_360 60.0

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\]
   7. Using strategy rm

   8. Applied *-un-lft-identity_binary64_419 60.0

      \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\color{blue}{1 \cdot \ell}} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\]

   9. Applied times-frac_binary64_425 48.6

      \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(\color{blue}{\left(\frac{t}{1} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\]

   10. Applied associate-*l*_binary64_360 48.6

       \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t}{1} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\]

   11. Taylor expanded around 0 28.6

       \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}}\]

   12. Simplified 28.6

       \[\leadsto \color{blue}{2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}}\]

   13. Simplified 28.6

       \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 9.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.8693790316622432 \cdot 10^{-129} \lor \neg \left(t \leq 8.663175062679125 \cdot 10^{-147}\right):\\ \;\;\;\;\frac{1}{t \cdot \left(\sin k \cdot \frac{t}{\ell}\right)} \cdot \left(\frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{t}}{\tan k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  :precision binary64
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))