Toniolo and Linder, Equation (10-)

Average Error: 48.5 → 6.5

Time: 32.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := t \cdot {\sin k}^{2}\\ t_2 := \frac{t_1}{\ell}\\ \mathbf{if}\;\ell \cdot \ell \leq 3.7157829738020827 \cdot 10^{-118}:\\ \;\;\;\;\frac{1}{\frac{k}{\ell}} \cdot \frac{\frac{2}{\frac{k}{\cos k}}}{t_2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 6.255295199890223 \cdot 10^{+273}:\\ \;\;\;\;\frac{2}{k \cdot t_1} \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(\cos k \cdot \frac{1}{k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \frac{1}{k}\right) \cdot \frac{\cos k \cdot \frac{2}{k}}{t_2}\\ \end{array} \]

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
 (let* ((t_1 (* t (pow (sin k) 2.0))) (t_2 (/ t_1 l)))
   (if (<= (* l l) 3.7157829738020827e-118)
     (* (/ 1.0 (/ k l)) (/ (/ 2.0 (/ k (cos k))) t_2))
     (if (<= (* l l) 6.255295199890223e+273)
       (* (/ 2.0 (* k t_1)) (* (* l l) (* (cos k) (/ 1.0 k))))
       (* (* l (/ 1.0 k)) (/ (* (cos k) (/ 2.0 k)) t_2))))))

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 t_1 = t * pow(sin(k), 2.0);
	double t_2 = t_1 / l;
	double tmp;
	if ((l * l) <= 3.7157829738020827e-118) {
		tmp = (1.0 / (k / l)) * ((2.0 / (k / cos(k))) / t_2);
	} else if ((l * l) <= 6.255295199890223e+273) {
		tmp = (2.0 / (k * t_1)) * ((l * l) * (cos(k) * (1.0 / k)));
	} else {
		tmp = (l * (1.0 / k)) * ((cos(k) * (2.0 / k)) / t_2);
	}
	return tmp;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 3.7157829738020827e-118

   1. Initial program 45.2

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

   2. Simplified 36.3

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

   3. Taylor expanded in t around 0 16.2

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

   4. Applied unpow2_binary64 16.2

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

   5. Applied associate-*l*_binary64 15.8

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

   6. Applied times-frac_binary64 14.9

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

   7. Applied associate-/r*_binary64 14.8

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

   8. Applied unpow2_binary64 14.8

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

   9. Applied times-frac_binary64 8.0

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

   10. Applied *-un-lft-identity_binary64 8.0

       \[\leadsto \frac{\color{blue}{1 \cdot \frac{2}{\frac{k}{\cos k}}}}{\frac{k}{\ell} \cdot \frac{t \cdot {\sin k}^{2}}{\ell}} \]

   11. Applied times-frac_binary64 6.9

       \[\leadsto \color{blue}{\frac{1}{\frac{k}{\ell}} \cdot \frac{\frac{2}{\frac{k}{\cos k}}}{\frac{t \cdot {\sin k}^{2}}{\ell}}} \]

   if 3.7157829738020827e-118 < (*.f64 l l) < 6.25529519989022328e273

   1. Initial program 46.0

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

   2. Simplified 36.2

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

   3. Taylor expanded in t around 0 14.0

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

   4. Applied unpow2_binary64 14.0

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

   5. Applied associate-*l*_binary64 8.3

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

   6. Applied times-frac_binary64 3.5

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

   7. Applied associate-/r*_binary64 3.1

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

   8. Applied div-inv_binary64 3.1

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

   9. Applied div-inv_binary64 3.1

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

   10. Applied times-frac_binary64 2.3

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

   11. Simplified 2.3

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

   if 6.25529519989022328e273 < (*.f64 l l)

   1. Initial program 62.5

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

   2. Simplified 61.5

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

   3. Taylor expanded in t around 0 59.7

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

   4. Applied unpow2_binary64 59.7

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

   5. Applied associate-*l*_binary64 58.8

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

   6. Applied times-frac_binary64 57.4

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

   7. Applied associate-/r*_binary64 57.4

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

   8. Applied add-sqr-sqrt_binary64 61.1

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

   9. Applied unpow-prod-down_binary64 61.1

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

   10. Applied times-frac_binary64 42.1

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

   11. Applied *-un-lft-identity_binary64 42.1

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

   12. Applied *-un-lft-identity_binary64 42.1

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

   13. Applied times-frac_binary64 42.1

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

   14. Applied *-un-lft-identity_binary64 42.1

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

   15. Applied times-frac_binary64 42.1

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

   16. Applied times-frac_binary64 39.0

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

   17. Simplified 39.0

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

   18. Simplified 12.6

       \[\leadsto \left(\frac{1}{k} \cdot \ell\right) \cdot \color{blue}{\frac{\frac{2}{k} \cdot \cos k}{\frac{t \cdot {\sin k}^{2}}{\ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 6.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 3.7157829738020827 \cdot 10^{-118}:\\ \;\;\;\;\frac{1}{\frac{k}{\ell}} \cdot \frac{\frac{2}{\frac{k}{\cos k}}}{\frac{t \cdot {\sin k}^{2}}{\ell}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 6.255295199890223 \cdot 10^{+273}:\\ \;\;\;\;\frac{2}{k \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(\cos k \cdot \frac{1}{k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \frac{1}{k}\right) \cdot \frac{\cos k \cdot \frac{2}{k}}{\frac{t \cdot {\sin k}^{2}}{\ell}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022077 
(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))))