Average Error: 32.5 → 9.8
Time: 15.3s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
↓
\[\begin{array}{l} t_1 := \frac{\ell}{t} \cdot \cos k\\ t_2 := \left(\sin k \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\\ \mathbf{if}\;t \leq -3.182538400401493 \cdot 10^{-132}:\\ \;\;\;\;\frac{2}{\frac{t_2}{t_1}}\\ \mathbf{elif}\;t \leq 4.886482039548544 \cdot 10^{-48}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\sin k}^{2} \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}}{t_1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{t_1}{t_2}}}\\ \end{array} \]
\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 := \frac{\ell}{t} \cdot \cos k\\
t_2 := \left(\sin k \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\\
\mathbf{if}\;t \leq -3.182538400401493 \cdot 10^{-132}:\\
\;\;\;\;\frac{2}{\frac{t_2}{t_1}}\\

\mathbf{elif}\;t \leq 4.886482039548544 \cdot 10^{-48}:\\
\;\;\;\;\frac{2}{\frac{\frac{{\sin k}^{2} \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}}{t_1}}\\

\mathbf{else}:\\
\;\;\;\;\frac{2}{\frac{1}{\frac{t_1}{t_2}}}\\


\end{array}

(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 (* (/ l t) (cos k)))
        (t_2
         (* (* (sin k) (* t (* (/ t l) (sin k)))) (+ 2.0 (pow (/ k t) 2.0)))))
   (if (<= t -3.182538400401493e-132)
     (/ 2.0 (/ t_2 t_1))
     (if (<= t 4.886482039548544e-48)
       (/ 2.0 (/ (/ (* (pow (sin k) 2.0) (fma k k (* 2.0 (* t t)))) l) t_1))
       (/ 2.0 (/ 1.0 (/ t_1 t_2)))))))
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 = (l / t) * cos(k);
	double t_2 = (sin(k) * (t * ((t / l) * sin(k)))) * (2.0 + pow((k / t), 2.0));
	double tmp;
	if (t <= -3.182538400401493e-132) {
		tmp = 2.0 / (t_2 / t_1);
	} else if (t <= 4.886482039548544e-48) {
		tmp = 2.0 / (((pow(sin(k), 2.0) * fma(k, k, (2.0 * (t * t)))) / l) / t_1);
	} else {
		tmp = 2.0 / (1.0 / (t_1 / 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 t < -3.18253840040149309e-132

    1. Initial program 25.1

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

    2. Simplified25.1

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

    3. Applied unpow3_binary6425.1

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

    4. Applied times-frac_binary6417.2

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

    5. Applied associate-*l*_binary6414.9

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

    6. Applied associate-/l*_binary6410.2

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

    7. Applied tan-quot_binary6410.3

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

    8. Applied associate-*l/_binary649.2

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

    9. Applied frac-times_binary647.6

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

    10. Applied associate-*l/_binary646.4

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

    if -3.18253840040149309e-132 < t < 4.88648203954854404e-48

    1. Initial program 58.5

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

    2. Simplified58.5

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

    3. Applied unpow3_binary6458.5

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

    4. Applied times-frac_binary6451.2

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

    5. Applied associate-*l*_binary6450.7

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

    6. Applied associate-/l*_binary6442.7

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

    7. Applied tan-quot_binary6442.7

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

    8. Applied associate-*l/_binary6442.7

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

    9. Applied frac-times_binary6443.7

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

    10. Applied associate-*l/_binary6439.9

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

    11. Taylor expanded in l around inf 23.0

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

    12. Simplified23.0

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

    if 4.88648203954854404e-48 < 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. Simplified22.3

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

    3. Applied unpow3_binary6422.3

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

    4. Applied times-frac_binary6415.8

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

    5. Applied associate-*l*_binary6413.8

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

    6. Applied associate-/l*_binary648.0

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

    7. Applied tan-quot_binary648.0

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

    8. Applied associate-*l/_binary647.0

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

    9. Applied frac-times_binary644.7

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

    10. Applied associate-*l/_binary644.1

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

    11. Applied clear-num_binary644.1

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

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.182538400401493 \cdot 10^{-132}:\\ \;\;\;\;\frac{2}{\frac{\left(\sin k \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\frac{\ell}{t} \cdot \cos k}}\\ \mathbf{elif}\;t \leq 4.886482039548544 \cdot 10^{-48}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\sin k}^{2} \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}}{\frac{\ell}{t} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\frac{\ell}{t} \cdot \cos k}{\left(\sin k \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022160 
(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))))