Toniolo and Linder, Equation (10+)

Average Error: 32.5 → 11.0

Time: 24.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \leq -1.5330480998573135 \cdot 10^{-55}:\\ \;\;\;\;\frac{2}{\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(t \cdot \frac{t \cdot \sin k}{\ell}\right)\right)}{\frac{\ell}{t}}}\\ \mathbf{elif}\;t \leq 9.192060272633729 \cdot 10^{-156}:\\ \;\;\;\;2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{\left(t \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k\right)}\\ \end{array}\]

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 (<= t -1.5330480998573135e-55)
   (/
    2.0
    (/
     (* (+ 2.0 (pow (/ k t) 2.0)) (* (tan k) (* t (/ (* t (sin k)) l))))
     (/ l t)))
   (if (<= t 9.192060272633729e-156)
     (* 2.0 (/ (* (* l l) (cos k)) (* (* k k) (* t (pow (sin k) 2.0)))))
     (*
      (/ l t)
      (/
       2.0
       (* (* t (/ (* t (sin k)) l)) (* (+ 2.0 (pow (/ k t) 2.0)) (tan 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 <= -1.5330480998573135e-55) {
		tmp = 2.0 / (((2.0 + pow((k / t), 2.0)) * (tan(k) * (t * ((t * sin(k)) / l)))) / (l / t));
	} else if (t <= 9.192060272633729e-156) {
		tmp = 2.0 * (((l * l) * cos(k)) / ((k * k) * (t * pow(sin(k), 2.0))));
	} else {
		tmp = (l / t) * (2.0 / ((t * ((t * sin(k)) / l)) * ((2.0 + pow((k / t), 2.0)) * tan(k))));
	}
	return tmp;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if t < -1.53304809985731348e-55

   1. Initial program 22.3

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

   2. Simplified 22.3

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

   4. Applied unpow3_binary64_485 22.3

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

   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 13.8

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

   7. Simplified 13.8

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

   9. Applied associate-/l*_binary64_364 8.4

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

   11. Applied associate-*l/_binary64_362 7.2

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

   12. Applied associate-*l/_binary64_362 4.7

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

   13. Applied associate-*l/_binary64_362 4.3

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

   14. Simplified 4.2

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

   if -1.53304809985731348e-55 < t < 9.1920602726337291e-156

   1. Initial program 59.2

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

   2. Simplified 59.2

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

   4. Applied unpow3_binary64_485 59.2

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

   5. Applied times-frac_binary64_425 53.3

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

   6. Applied associate-*l*_binary64_360 52.9

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

   7. Simplified 52.9

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

   9. Applied associate-/l*_binary64_364 44.2

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

   11. Applied associate-*l*_binary64_360 43.6

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

   12. Simplified 43.6

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

   13. Taylor expanded around 0 27.2

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

   14. Simplified 27.2

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

   if 9.1920602726337291e-156 < t

   1. Initial program 26.1

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

   2. Simplified 26.1

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

   4. Applied unpow3_binary64_485 26.1

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

   5. Applied times-frac_binary64_425 18.0

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

   6. Applied associate-*l*_binary64_360 15.9

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

   7. Simplified 15.9

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

   9. Applied associate-/l*_binary64_364 11.9

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

   11. Applied associate-*l*_binary64_360 11.8

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

   12. Simplified 11.8

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

   14. Applied associate-*l/_binary64_362 10.7

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

   15. Applied associate-*l/_binary64_362 8.0

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

   16. Applied associate-/r/_binary64_365 7.9

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

   17. Simplified 7.8

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

4. Final simplification 11.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.5330480998573135 \cdot 10^{-55}:\\ \;\;\;\;\frac{2}{\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(t \cdot \frac{t \cdot \sin k}{\ell}\right)\right)}{\frac{\ell}{t}}}\\ \mathbf{elif}\;t \leq 9.192060272633729 \cdot 10^{-156}:\\ \;\;\;\;2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{\left(t \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021024 
(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))))