Average Error: 32.7 → 11.2
Time: 23.4s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
↓
\[\begin{array}{l} t_1 := 0 \cdot \tan k\\ t_2 := \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\\ t_3 := \frac{2}{\left(\mathsf{fma}\left(t_2, \tan k \cdot {t_2}^{2}, -t_1\right) + \mathsf{fma}\left(0, \tan k, t_1\right)\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{if}\;t \leq -5.24857608547832 \cdot 10^{-110}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 8.893990945635184 \cdot 10^{-99}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]
\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 := 0 \cdot \tan k\\
t_2 := \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\\
t_3 := \frac{2}{\left(\mathsf{fma}\left(t_2, \tan k \cdot {t_2}^{2}, -t_1\right) + \mathsf{fma}\left(0, \tan k, t_1\right)\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\
\mathbf{if}\;t \leq -5.24857608547832 \cdot 10^{-110}:\\
\;\;\;\;t_3\\

\mathbf{elif}\;t \leq 8.893990945635184 \cdot 10^{-99}:\\
\;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k \cdot {\ell}^{2}}}\\

\mathbf{else}:\\
\;\;\;\;t_3\\


\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 (* 0.0 (tan k)))
        (t_2 (* (/ t (pow (cbrt l) 2.0)) (cbrt (sin k))))
        (t_3
         (/
          2.0
          (*
           (+
            (fma t_2 (* (tan k) (pow t_2 2.0)) (- t_1))
            (fma 0.0 (tan k) t_1))
           (+ 2.0 (pow (/ k t) 2.0))))))
   (if (<= t -5.24857608547832e-110)
     t_3
     (if (<= t 8.893990945635184e-99)
       (/
        2.0
        (/ (* (pow k 2.0) (* t (pow (sin k) 2.0))) (* (cos k) (pow l 2.0))))
       t_3))))
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 = 0.0 * tan(k);
	double t_2 = (t / pow(cbrt(l), 2.0)) * cbrt(sin(k));
	double t_3 = 2.0 / ((fma(t_2, (tan(k) * pow(t_2, 2.0)), -t_1) + fma(0.0, tan(k), t_1)) * (2.0 + pow((k / t), 2.0)));
	double tmp;
	if (t <= -5.24857608547832e-110) {
		tmp = t_3;
	} else if (t <= 8.893990945635184e-99) {
		tmp = 2.0 / ((pow(k, 2.0) * (t * pow(sin(k), 2.0))) / (cos(k) * pow(l, 2.0)));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 2 regimes

  2. if t < -5.2485760854783201e-110 or 8.8939909456351842e-99 < t

    1. Initial program 23.6

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

    2. Simplified23.6

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

    3. Applied egg-rr6.5

      \[\leadsto \frac{2}{\color{blue}{\left(\mathsf{fma}\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}, \tan k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{2}, -0 \cdot \tan k\right) + \mathsf{fma}\left(0, \tan k, 0 \cdot \tan k\right)\right)} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

    if -5.2485760854783201e-110 < t < 8.8939909456351842e-99

    1. Initial program 63.2

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

    2. Simplified63.2

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

    3. Taylor expanded in t around 0 27.0

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

  4. Final simplification11.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.24857608547832 \cdot 10^{-110}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}, \tan k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{2}, -0 \cdot \tan k\right) + \mathsf{fma}\left(0, \tan k, 0 \cdot \tan k\right)\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{elif}\;t \leq 8.893990945635184 \cdot 10^{-99}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}, \tan k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{2}, -0 \cdot \tan k\right) + \mathsf{fma}\left(0, \tan k, 0 \cdot \tan k\right)\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022127 
(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))))