Toniolo and Linder, Equation (10+)

Average Error: 32.3 → 9.4

Time: 17.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -1.0509227468799622 \cdot 10^{-82}:\\ \;\;\;\;\frac{1}{\left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \tan k} \cdot \left(\frac{2}{t_1} \cdot \frac{\ell}{t}\right)\\ \mathbf{elif}\;t \leq 1.7790546427720134 \cdot 10^{-91}:\\ \;\;\;\;\begin{array}{l} t_2 := \ell \cdot \cos k\\ t_3 := {\sin k}^{2}\\ \frac{\ell}{t} \cdot \frac{2}{\mathsf{fma}\left(2, \frac{t_3 \cdot \left(t \cdot t\right)}{t_2}, \frac{t_3 \cdot \left(k \cdot k\right)}{t_2}\right)} \end{array}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{t_1 \cdot \left(\tan k \cdot \left(t \cdot \frac{t \cdot \sin k}{\ell}\right)\right)}\\ \end{array} \]

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
 (let* ((t_1 (+ 2.0 (pow (/ k t) 2.0))))
   (if (<= t -1.0509227468799622e-82)
     (* (/ 1.0 (* (* t (* (/ t l) (sin k))) (tan k))) (* (/ 2.0 t_1) (/ l t)))
     (if (<= t 1.7790546427720134e-91)
       (let* ((t_2 (* l (cos k))) (t_3 (pow (sin k) 2.0)))
         (*
          (/ l t)
          (/ 2.0 (fma 2.0 (/ (* t_3 (* t t)) t_2) (/ (* t_3 (* k k)) t_2)))))
       (* (/ l t) (/ 2.0 (* t_1 (* (tan k) (* t (/ (* t (sin k)) l))))))))))

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 tmp;
	if (t <= -1.0509227468799622e-82) {
		tmp = (1.0 / ((t * ((t / l) * sin(k))) * tan(k))) * ((2.0 / t_1) * (l / t));
	} else if (t <= 1.7790546427720134e-91) {
		double t_2 = l * cos(k);
		double t_3 = pow(sin(k), 2.0);
		tmp = (l / t) * (2.0 / fma(2.0, ((t_3 * (t * t)) / t_2), ((t_3 * (k * k)) / t_2)));
	} else {
		tmp = (l / t) * (2.0 / (t_1 * (tan(k) * (t * ((t * sin(k)) / l)))));
	}
	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 < -1.0509227468799622e-82

   1. Initial program 23.0

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

   2. Simplified 23.0

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

   3. Applied unpow3_binary64 23.0

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

   4. Applied times-frac_binary64 16.6

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

   5. Applied associate-*l*_binary64 14.3

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

   6. Applied associate-/l*_binary64 9.2

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

   7. Applied associate-*l/_binary64 7.9

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

   8. Applied associate-*l/_binary64 5.8

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

   9. Applied associate-*l/_binary64 5.0

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

   10. Applied associate-/r/_binary64 4.9

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

   11. Applied *-un-lft-identity_binary64 4.9

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

   12. Applied times-frac_binary64 4.8

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

   13. Applied associate-*l*_binary64 4.8

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

   if -1.0509227468799622e-82 < t < 1.7790546427720134e-91

   1. Initial program 60.7

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

   2. Simplified 60.7

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

   3. Applied unpow3_binary64 60.7

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

   4. Applied times-frac_binary64 51.9

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

   5. Applied associate-*l*_binary64 51.9

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

   6. Applied associate-/l*_binary64 43.5

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

   7. Applied associate-*l/_binary64 43.5

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

   8. Applied associate-*l/_binary64 44.5

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

   9. Applied associate-*l/_binary64 41.1

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

   10. Applied associate-/r/_binary64 41.2

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

   11. Taylor expanded in t around 0 22.4

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

   12. Simplified 22.4

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

   if 1.7790546427720134e-91 < t

   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. Applied unpow3_binary64 22.3

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

   4. Applied times-frac_binary64 16.2

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

   5. Applied associate-*l*_binary64 13.5

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

   6. Applied associate-/l*_binary64 8.7

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

   7. Applied associate-*l/_binary64 7.8

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

   8. Applied associate-*l/_binary64 5.9

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

   9. Applied associate-*l/_binary64 5.3

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

   10. Applied associate-/r/_binary64 5.2

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

   11. Applied pow1_binary64 5.2

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

   12. Applied pow1_binary64 5.2

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

   13. Applied pow-prod-down_binary64 5.2

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

   14. Simplified 5.1

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

4. Final simplification 9.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.0509227468799622 \cdot 10^{-82}:\\ \;\;\;\;\frac{1}{\left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \tan k} \cdot \left(\frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{t}\right)\\ \mathbf{elif}\;t \leq 1.7790546427720134 \cdot 10^{-91}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{\mathsf{fma}\left(2, \frac{{\sin k}^{2} \cdot \left(t \cdot t\right)}{\ell \cdot \cos k}, \frac{{\sin k}^{2} \cdot \left(k \cdot k\right)}{\ell \cdot \cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(t \cdot \frac{t \cdot \sin k}{\ell}\right)\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022100 
(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))))