Toniolo and Linder, Equation (10+)

Average Error: 32.4 → 9.2

Time: 15.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \leq -1.4869480221444148 \cdot 10^{-75} \lor \neg \left(t \leq 1005420263.9279583\right):\\ \;\;\;\;\frac{2}{\frac{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right) \cdot \left(t \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right)}{\frac{\ell}{t} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\ell \cdot \ell} \cdot \left(\frac{k \cdot \left(t \cdot k\right)}{\cos k} + 2 \cdot \frac{{t}^{3}}{\cos k}\right)}\\ \end{array}\]

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 -1.4869480221444148e-75) (not (<= t 1005420263.9279583)))
   (/
    2.0
    (/
     (* (* (+ 2.0 (pow (/ k t) 2.0)) (sin k)) (* t (* (sin k) (/ t l))))
     (* (/ l t) (cos k))))
   (/
    2.0
    (*
     (/ (pow (sin k) 2.0) (* l l))
     (+ (/ (* k (* t k)) (cos k)) (* 2.0 (/ (pow t 3.0) (cos k))))))))

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 <= -1.4869480221444148e-75) || !(t <= 1005420263.9279583)) {
		tmp = 2.0 / ((((2.0 + pow((k / t), 2.0)) * sin(k)) * (t * (sin(k) * (t / l)))) / ((l / t) * cos(k)));
	} else {
		tmp = 2.0 / ((pow(sin(k), 2.0) / (l * l)) * (((k * (t * k)) / cos(k)) + (2.0 * (pow(t, 3.0) / cos(k)))));
	}
	return tmp;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if t < -1.4869480221444148e-75 or 1005420263.9279583 < t

   1. Initial program 22.5

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

   2. Simplified 22.5

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

   4. Applied unpow3_binary64_485 22.5

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

   5. Applied times-frac_binary64_425 16.0

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

   6. Applied associate-*l*_binary64_360 13.9

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

   7. Simplified 13.9

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

   9. Applied associate-/l*_binary64_364 8.7

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

   11. Applied tan-quot_binary64_578 8.7

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

   12. Applied associate-*l/_binary64_362 7.5

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

   13. Applied frac-times_binary64_429 4.7

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

   14. Applied associate-*l/_binary64_362 3.9

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

   15. Simplified 3.9

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

   17. Applied associate-*r*_binary64_359 3.9

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

   if -1.4869480221444148e-75 < t < 1005420263.9279583

   1. Initial program 52.2

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

   2. Simplified 52.2

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

   4. Applied unpow3_binary64_485 52.2

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

   5. Applied times-frac_binary64_425 44.3

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

   6. Applied associate-*l*_binary64_360 43.2

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

   7. Simplified 43.2

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

   9. Applied associate-/l*_binary64_364 37.4

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

   11. Applied tan-quot_binary64_578 37.4

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

   12. Applied associate-*l/_binary64_362 37.4

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

   13. Applied frac-times_binary64_429 38.5

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

   14. Applied associate-*l/_binary64_362 35.3

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

   15. Simplified 35.3

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

   16. Taylor expanded around inf 35.0

       \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k} + 2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}}}\]

   17. Simplified 19.8

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

4. Final simplification 9.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.4869480221444148 \cdot 10^{-75} \lor \neg \left(t \leq 1005420263.9279583\right):\\ \;\;\;\;\frac{2}{\frac{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right) \cdot \left(t \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right)}{\frac{\ell}{t} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\ell \cdot \ell} \cdot \left(\frac{k \cdot \left(t \cdot k\right)}{\cos k} + 2 \cdot \frac{{t}^{3}}{\cos k}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020339 
(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))))