Toniolo and Linder, Equation (10+)

Average Error: 32.4 → 9.5

Time: 17.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \tan k\\ \mathbf{if}\;t \leq -1.2880254342721147 \cdot 10^{-77}:\\ \;\;\;\;\frac{2}{\frac{t_2}{\frac{\frac{\ell}{t}}{t_1}}}\\ \mathbf{elif}\;t \leq 1.0522518200796339 \cdot 10^{-72}:\\ \;\;\;\;\begin{array}{l} t_3 := {\sin k}^{2}\\ \frac{2}{\frac{\frac{\mathsf{fma}\left(2, \frac{t_3 \cdot \left(t \cdot t\right)}{\cos k}, \frac{t_3 \cdot \left(k \cdot k\right)}{\cos k}\right)}{\ell}}{\frac{\ell}{t}}} \end{array}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_2 \cdot \frac{t_1}{\frac{\ell}{t}}}\\ \end{array} \]

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 (* (/ t l) (sin k))) (tan k))))
   (if (<= t -1.2880254342721147e-77)
     (/ 2.0 (/ t_2 (/ (/ l t) t_1)))
     (if (<= t 1.0522518200796339e-72)
       (let* ((t_3 (pow (sin k) 2.0)))
         (/
          2.0
          (/
           (/
            (fma 2.0 (/ (* t_3 (* t t)) (cos k)) (/ (* t_3 (* k k)) (cos k)))
            l)
           (/ l t))))
       (/ 2.0 (* t_2 (/ t_1 (/ l t))))))))

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 * ((t / l) * sin(k))) * tan(k);
	double tmp;
	if (t <= -1.2880254342721147e-77) {
		tmp = 2.0 / (t_2 / ((l / t) / t_1));
	} else if (t <= 1.0522518200796339e-72) {
		double t_3 = pow(sin(k), 2.0);
		tmp = 2.0 / ((fma(2.0, ((t_3 * (t * t)) / cos(k)), ((t_3 * (k * k)) / cos(k))) / l) / (l / t));
	} else {
		tmp = 2.0 / (t_2 * (t_1 / (l / t)));
	}
	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 < -1.2880254342721147e-77

   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. Applied unpow3_binary64 22.5

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

   4. Applied times-frac_binary64 16.3

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

   5. Applied associate-*l*_binary64 13.8

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

   6. Applied associate-/l*_binary64 8.9

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

   7. Applied associate-*l/_binary64 7.7

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

   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 5.2

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

   10. Applied associate-/l*_binary64 5.0

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

   if -1.2880254342721147e-77 < t < 1.05225182007963393e-72

   1. Initial program 58.3

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

   2. Simplified 58.3

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

   3. Applied unpow3_binary64 58.3

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

   4. Applied times-frac_binary64 49.3

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

   5. Applied associate-*l*_binary64 49.0

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

   6. Applied associate-/l*_binary64 41.1

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

   7. Applied associate-*l/_binary64 41.1

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

   8. Applied associate-*l/_binary64 42.1

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

   9. Applied associate-*l/_binary64 38.8

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

   10. Taylor expanded in l around inf 22.4

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

   11. Simplified 22.4

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

   if 1.05225182007963393e-72 < t

   1. Initial program 23.3

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

   2. Simplified 23.3

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

   3. Applied unpow3_binary64 23.3

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

   4. Applied times-frac_binary64 17.1

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

   5. Applied associate-*l*_binary64 14.8

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

   6. Applied associate-/l*_binary64 9.1

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

   7. Applied associate-*l/_binary64 7.7

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

   8. Applied associate-*l/_binary64 5.4

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

   9. Applied associate-*l/_binary64 4.6

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

   10. Applied *-un-lft-identity_binary64 4.6

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

   11. Applied *-un-lft-identity_binary64 4.6

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

   12. Applied times-frac_binary64 4.6

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

   13. Applied times-frac_binary64 4.5

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

   14. Simplified 4.5

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

4. Final simplification 9.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.2880254342721147 \cdot 10^{-77}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \tan k}{\frac{\frac{\ell}{t}}{2 + {\left(\frac{k}{t}\right)}^{2}}}}\\ \mathbf{elif}\;t \leq 1.0522518200796339 \cdot 10^{-72}:\\ \;\;\;\;\frac{2}{\frac{\frac{\mathsf{fma}\left(2, \frac{{\sin k}^{2} \cdot \left(t \cdot t\right)}{\cos k}, \frac{{\sin k}^{2} \cdot \left(k \cdot k\right)}{\cos k}\right)}{\ell}}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \frac{2 + {\left(\frac{k}{t}\right)}^{2}}{\frac{\ell}{t}}}\\ \end{array} \]

Reproduce

herbie shell --seed 2021313 
(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))))