Toniolo and Linder, Equation (10+)

Average Error: 32.2 → 13.5

Time: 21.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_2 := t \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\\ t_3 := t_2 \cdot \tan k\\ \mathbf{if}\;t \leq -6.360285793829747 \cdot 10^{+21}:\\ \;\;\;\;\frac{2}{\left(t \cdot \frac{t_3}{\ell}\right) \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_4 := \frac{t}{\ell} \cdot t_2\\ \mathbf{if}\;t \leq -1.5979410545040742 \cdot 10^{-68}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot t_4}}{t_1}\\ \mathbf{elif}\;t \leq -4.599803275105722 \cdot 10^{-125}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(t_3 \cdot t_1\right)}\\ \mathbf{elif}\;t \leq 5.299016542242219 \cdot 10^{-177}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_4 \cdot \left(\tan k \cdot t_1\right)}\\ \end{array}\\ \end{array} \]

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 (+ 2.0 (pow (/ k t) 2.0)))
        (t_2 (* t (* (sin k) (/ t l))))
        (t_3 (* t_2 (tan k))))
   (if (<= t -6.360285793829747e+21)
     (/ 2.0 (* (* t (/ t_3 l)) t_1))
     (let* ((t_4 (* (/ t l) t_2)))
       (if (<= t -1.5979410545040742e-68)
         (/ (/ 2.0 (* (tan k) t_4)) t_1)
         (if (<= t -4.599803275105722e-125)
           (/ 2.0 (* (/ t l) (* t_3 t_1)))
           (if (<= t 5.299016542242219e-177)
             (*
              2.0
              (/
               (* (cos k) (pow l 2.0))
               (* (pow k 2.0) (* t (pow (sin k) 2.0)))))
             (/ 2.0 (* t_4 (* (tan k) t_1))))))))))

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 = 2.0 + pow((k / t), 2.0);
	double t_2 = t * (sin(k) * (t / l));
	double t_3 = t_2 * tan(k);
	double tmp;
	if (t <= -6.360285793829747e+21) {
		tmp = 2.0 / ((t * (t_3 / l)) * t_1);
	} else {
		double t_4 = (t / l) * t_2;
		double tmp_1;
		if (t <= -1.5979410545040742e-68) {
			tmp_1 = (2.0 / (tan(k) * t_4)) / t_1;
		} else if (t <= -4.599803275105722e-125) {
			tmp_1 = 2.0 / ((t / l) * (t_3 * t_1));
		} else if (t <= 5.299016542242219e-177) {
			tmp_1 = 2.0 * ((cos(k) * pow(l, 2.0)) / (pow(k, 2.0) * (t * pow(sin(k), 2.0))));
		} else {
			tmp_1 = 2.0 / (t_4 * (tan(k) * t_1));
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 5 regimes

2. if t < -6.3602857938297469e21

   1. Initial program 22.1

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

   2. Simplified 22.1

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

   3. Applied cube-mult_binary64 22.1

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

   4. Applied times-frac_binary64 15.4

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

   5. Applied associate-*l*_binary64 13.6

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

   6. Applied *-un-lft-identity_binary64 13.6

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

   7. Applied times-frac_binary64 7.2

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

   8. Applied associate-*l*_binary64 5.7

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

   9. Applied associate-*l*_binary64 2.6

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

   10. Applied div-inv_binary64 2.6

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

   11. Applied associate-*l*_binary64 5.3

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

   12. Simplified 5.3

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

   if -6.3602857938297469e21 < t < -1.597941054504074e-68

   1. Initial program 23.4

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

   2. Simplified 23.4

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

   3. Applied cube-mult_binary64 23.5

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

   4. Applied times-frac_binary64 21.3

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

   5. Applied associate-*l*_binary64 16.7

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

   6. Applied *-un-lft-identity_binary64 16.7

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

   7. Applied times-frac_binary64 16.7

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

   8. Applied associate-*l*_binary64 16.7

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

   9. Applied associate-/r*_binary64 17.1

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

   if -1.597941054504074e-68 < t < -4.599803275105722e-125

   1. Initial program 45.7

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

   2. Simplified 45.7

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

   3. Applied cube-mult_binary64 45.7

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

   4. Applied times-frac_binary64 27.1

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

   5. Applied associate-*l*_binary64 25.0

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

   6. Applied *-un-lft-identity_binary64 25.0

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

   7. Applied times-frac_binary64 24.9

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

   8. Applied associate-*l*_binary64 25.0

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

   9. Applied associate-*l*_binary64 27.6

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

   10. Applied associate-*l*_binary64 23.8

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

   11. Simplified 23.8

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

   if -4.599803275105722e-125 < t < 5.29901654224221922e-177

   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. Applied cube-mult_binary64 64.0

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

   4. Applied times-frac_binary64 60.6

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

   5. Applied associate-*l*_binary64 60.6

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

   6. Applied *-un-lft-identity_binary64 60.6

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

   7. Applied times-frac_binary64 50.6

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

   8. Applied associate-*l*_binary64 50.5

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

   9. Taylor expanded in t around 0 29.6

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

   if 5.29901654224221922e-177 < t

   1. Initial program 27.0

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

   2. Simplified 27.0

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

   3. Applied cube-mult_binary64 27.0

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

   4. Applied times-frac_binary64 19.0

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

   5. Applied associate-*l*_binary64 17.1

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

   6. Applied *-un-lft-identity_binary64 17.1

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

   7. Applied times-frac_binary64 12.6

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

   8. Applied associate-*l*_binary64 11.6

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

   9. Applied associate-*l*_binary64 11.4

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{1} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 13.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.360285793829747 \cdot 10^{+21}:\\ \;\;\;\;\frac{2}{\left(t \cdot \frac{\left(t \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right) \cdot \tan k}{\ell}\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{elif}\;t \leq -1.5979410545040742 \cdot 10^{-68}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right)\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq -4.599803275105722 \cdot 10^{-125}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(t \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;t \leq 5.299016542242219 \cdot 10^{-177}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right)\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022082 
(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))))