Toniolo and Linder, Equation (10-)

Average Error: 48.4 → 13.7

Time: 31.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 7.66 \cdot 10^{-322}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{\mathsf{fma}\left(\frac{\ell}{t}, \frac{\ell}{k \cdot k}, \frac{\ell \cdot \ell}{t} \cdot -0.16666666666666666\right)}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 4.847791631987853 \cdot 10^{+303}:\\ \;\;\;\;\frac{\frac{2}{k} \cdot \left(\left(\ell \cdot \ell\right) \cdot \cos k\right)}{\frac{k}{\frac{1}{t \cdot \left(\sin k \cdot \sin k\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]

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
 (if (<= (* l l) 7.66e-322)
   (/
    2.0
    (/
     (pow k 2.0)
     (fma (/ l t) (/ l (* k k)) (* (/ (* l l) t) -0.16666666666666666))))
   (if (<= (* l l) 4.847791631987853e+303)
     (/
      (* (/ 2.0 k) (* (* l l) (cos k)))
      (/ k (/ 1.0 (* t (* (sin k) (sin k))))))
     (/
      2.0
      (*
       (* (* (/ (* t t) l) (* (sin k) (/ t l))) (tan k))
       (pow (/ k t) 2.0))))))

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 ((l * l) <= 7.66e-322) {
		tmp = 2.0 / (pow(k, 2.0) / fma((l / t), (l / (k * k)), (((l * l) / t) * -0.16666666666666666)));
	} else if ((l * l) <= 4.847791631987853e+303) {
		tmp = ((2.0 / k) * ((l * l) * cos(k))) / (k / (1.0 / (t * (sin(k) * sin(k)))));
	} else {
		tmp = 2.0 / (((((t * t) / l) * (sin(k) * (t / l))) * tan(k)) * pow((k / t), 2.0));
	}
	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) < 7.65802e-322

   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 37.5

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

   3. Taylor expanded in t around 0 21.0

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

   4. Applied associate-/l*_binary64 20.7

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

   5. Taylor expanded in k around 0 20.8

      \[\leadsto \frac{2}{\frac{{k}^{2}}{\color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t} - 0.16666666666666666 \cdot \frac{{\ell}^{2}}{t}}}} \]

   6. Simplified 11.9

      \[\leadsto \frac{2}{\frac{{k}^{2}}{\color{blue}{\mathsf{fma}\left(\frac{\ell}{t}, \frac{\ell}{k \cdot k}, \frac{\ell \cdot \ell}{t} \cdot -0.16666666666666666\right)}}} \]

   if 7.65802e-322 < (*.f64 l l) < 4.84779163198785305e303

   1. Initial program 45.5

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

   2. Simplified 35.9

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

   3. Taylor expanded in t around 0 12.7

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

   4. Applied associate-/l*_binary64 11.1

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

   5. Applied div-inv_binary64 11.2

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

   6. Applied unpow2_binary64 11.2

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

   7. Applied times-frac_binary64 4.8

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

   8. Applied associate-/r*_binary64 4.6

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

   9. Simplified 4.5

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

   10. Applied add-sqr-sqrt_binary64 34.3

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

   11. Applied unpow-prod-down_binary64 34.3

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

   12. Simplified 34.2

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

   13. Simplified 4.5

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

   if 4.84779163198785305e303 < (*.f64 l l)

   1. Initial program 63.6

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

   2. Simplified 63.5

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

   3. Applied unpow3_binary64 63.5

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

   4. Applied times-frac_binary64 47.9

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

   5. Applied associate-*l*_binary64 47.9

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

4. Final simplification 13.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 7.66 \cdot 10^{-322}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{\mathsf{fma}\left(\frac{\ell}{t}, \frac{\ell}{k \cdot k}, \frac{\ell \cdot \ell}{t} \cdot -0.16666666666666666\right)}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 4.847791631987853 \cdot 10^{+303}:\\ \;\;\;\;\frac{\frac{2}{k} \cdot \left(\left(\ell \cdot \ell\right) \cdot \cos k\right)}{\frac{k}{\frac{1}{t \cdot \left(\sin k \cdot \sin k\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]

Reproduce

herbie shell --seed 2021329 
(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))))