Toniolo and Linder, Equation (10-)

Average Error: 48.3 → 14.6

Time: 39.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt[3]{\sin k \cdot \tan k}\\ t_2 := \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\\ \mathbf{if}\;\ell \leq -4.478607638891889 \cdot 10^{+152}:\\ \;\;\;\;\frac{2}{{\left(t_2 \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\tan k}\right)\right)}^{2} \cdot \left(\frac{\frac{t \cdot t_1}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}} \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_3 := {\sin k}^{2}\\ t_4 := \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot t_3\right)}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{if}\;\ell \leq -1.0913728313287718 \cdot 10^{-156}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\ell \leq -1.4584328222929 \cdot 10^{-310}:\\ \;\;\;\;\frac{2}{{\left(t_2 \cdot t_1\right)}^{2} \cdot \left(\left(\sqrt[3]{\frac{t_3}{\cos k}} \cdot {\left(\frac{-1}{\ell}\right)}^{0.6666666666666666}\right) \cdot \frac{k \cdot k}{t \cdot {\left(\sqrt[3]{-1}\right)}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_5 := \left(t_1 \cdot \frac{t}{{\ell}^{0.6666666666666666}}\right) \cdot {\left(\sqrt[3]{\frac{k}{t}}\right)}^{2}\\ \mathbf{if}\;\ell \leq 2.570052205725406 \cdot 10^{-142}:\\ \;\;\;\;\frac{2}{t_5 \cdot {t_5}^{2}}\\ \mathbf{elif}\;\ell \leq 2.309802372999623 \cdot 10^{+92}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{t_5}^{3}}\\ \end{array}\\ \end{array}\\ \end{array} \]

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 (cbrt (* (sin k) (tan k)))) (t_2 (/ t (pow (cbrt l) 2.0))))
   (if (<= l -4.478607638891889e+152)
     (/
      2.0
      (*
       (pow (* t_2 (* (cbrt (sin k)) (cbrt (tan k)))) 2.0)
       (* (/ (/ (* t t_1) (cbrt l)) (cbrt l)) (pow (/ k t) 2.0))))
     (let* ((t_3 (pow (sin k) 2.0))
            (t_4
             (/ 2.0 (/ (* (pow k 2.0) (* t t_3)) (* (cos k) (pow l 2.0))))))
       (if (<= l -1.0913728313287718e-156)
         t_4
         (if (<= l -1.4584328222929e-310)
           (/
            2.0
            (*
             (pow (* t_2 t_1) 2.0)
             (*
              (* (cbrt (/ t_3 (cos k))) (pow (/ -1.0 l) 0.6666666666666666))
              (/ (* k k) (* t (pow (cbrt -1.0) 2.0))))))
           (let* ((t_5
                   (*
                    (* t_1 (/ t (pow l 0.6666666666666666)))
                    (pow (cbrt (/ k t)) 2.0))))
             (if (<= l 2.570052205725406e-142)
               (/ 2.0 (* t_5 (pow t_5 2.0)))
               (if (<= l 2.309802372999623e+92)
                 t_4
                 (/ 2.0 (pow t_5 3.0)))))))))))

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 t_1 = cbrt(sin(k) * tan(k));
	double t_2 = t / pow(cbrt(l), 2.0);
	double tmp;
	if (l <= -4.478607638891889e+152) {
		tmp = 2.0 / (pow((t_2 * (cbrt(sin(k)) * cbrt(tan(k)))), 2.0) * ((((t * t_1) / cbrt(l)) / cbrt(l)) * pow((k / t), 2.0)));
	} else {
		double t_3 = pow(sin(k), 2.0);
		double t_4 = 2.0 / ((pow(k, 2.0) * (t * t_3)) / (cos(k) * pow(l, 2.0)));
		double tmp_1;
		if (l <= -1.0913728313287718e-156) {
			tmp_1 = t_4;
		} else if (l <= -1.4584328222929e-310) {
			tmp_1 = 2.0 / (pow((t_2 * t_1), 2.0) * ((cbrt(t_3 / cos(k)) * pow((-1.0 / l), 0.6666666666666666)) * ((k * k) / (t * pow(cbrt(-1.0), 2.0)))));
		} else {
			double t_5 = (t_1 * (t / pow(l, 0.6666666666666666))) * pow(cbrt(k / t), 2.0);
			double tmp_2;
			if (l <= 2.570052205725406e-142) {
				tmp_2 = 2.0 / (t_5 * pow(t_5, 2.0));
			} else if (l <= 2.309802372999623e+92) {
				tmp_2 = t_4;
			} else {
				tmp_2 = 2.0 / pow(t_5, 3.0);
			}
			tmp_1 = tmp_2;
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 5 regimes

2. if l < -4.4786076388918893e152

   1. Initial program 63.7

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

   2. Simplified 63.7

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

   3. Applied egg 36.0

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

   4. Applied egg 36.1

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

   5. Applied egg 36.1

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

   if -4.4786076388918893e152 < l < -1.0913728313287718e-156 or 2.570052205725406e-142 < l < 2.30980237299962287e92

   1. Initial program 44.7

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

   2. Simplified 34.9

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

   3. Taylor expanded in t around 0 11.8

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

   if -1.0913728313287718e-156 < l < -1.45843282229288e-310

   1. Initial program 45.1

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

   2. Simplified 36.6

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

   3. Applied egg 17.7

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

   4. Taylor expanded in l around -inf 31.3

      \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2} \cdot \color{blue}{\frac{{k}^{2} \cdot e^{0.3333333333333333 \cdot \left(\log \left(\frac{{\sin k}^{2}}{\cos k}\right) + 2 \cdot \log \left(\frac{-1}{\ell}\right)\right)}}{t \cdot {\left(\sqrt[3]{-1}\right)}^{2}}}} \]

   5. Simplified 13.1

      \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2} \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{{\sin k}^{2}}{\cos k}} \cdot {\left(\frac{-1}{\ell}\right)}^{0.6666666666666666}\right) \cdot \frac{k \cdot k}{t \cdot {\left(\sqrt[3]{-1}\right)}^{2}}\right)}} \]

   if -1.45843282229288e-310 < l < 2.570052205725406e-142

   1. Initial program 47.0

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

   2. Simplified 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. Applied egg 20.3

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

   4. Applied egg 11.2

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{t}{{\ell}^{0.6666666666666666}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot {\left(\sqrt[3]{\frac{k}{t}}\right)}^{2}\right) \cdot {\left(\left(\frac{t}{{\ell}^{0.6666666666666666}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot {\left(\sqrt[3]{\frac{k}{t}}\right)}^{2}\right)}^{2}}} \]

   if 2.30980237299962287e92 < l

   1. Initial program 59.7

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

   2. Simplified 57.6

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

   3. Applied egg 34.4

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

   4. Applied egg 19.6

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\frac{t}{{\ell}^{0.6666666666666666}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot {\left(\sqrt[3]{\frac{k}{t}}\right)}^{2}\right)}^{3}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 14.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.478607638891889 \cdot 10^{+152}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\tan k}\right)\right)}^{2} \cdot \left(\frac{\frac{t \cdot \sqrt[3]{\sin k \cdot \tan k}}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}} \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{elif}\;\ell \leq -1.0913728313287718 \cdot 10^{-156}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{elif}\;\ell \leq -1.4584328222929 \cdot 10^{-310}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2} \cdot \left(\left(\sqrt[3]{\frac{{\sin k}^{2}}{\cos k}} \cdot {\left(\frac{-1}{\ell}\right)}^{0.6666666666666666}\right) \cdot \frac{k \cdot k}{t \cdot {\left(\sqrt[3]{-1}\right)}^{2}}\right)}\\ \mathbf{elif}\;\ell \leq 2.570052205725406 \cdot 10^{-142}:\\ \;\;\;\;\frac{2}{\left(\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\ell}^{0.6666666666666666}}\right) \cdot {\left(\sqrt[3]{\frac{k}{t}}\right)}^{2}\right) \cdot {\left(\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\ell}^{0.6666666666666666}}\right) \cdot {\left(\sqrt[3]{\frac{k}{t}}\right)}^{2}\right)}^{2}}\\ \mathbf{elif}\;\ell \leq 2.309802372999623 \cdot 10^{+92}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\ell}^{0.6666666666666666}}\right) \cdot {\left(\sqrt[3]{\frac{k}{t}}\right)}^{2}\right)}^{3}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022125 
(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))))