Toniolo and Linder, Equation (10+)

Average Error: 32.6 → 10.4

Time: 20.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt[3]{\sin k}\\ t_2 := \left(\frac{t}{\sqrt[3]{\ell}} \cdot \frac{t_1}{\sqrt[3]{\ell}}\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\\ \mathbf{if}\;t \leq -1.2869095709499647 \cdot 10^{-59}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot t_1\right)}^{2} \cdot t_2}\\ \mathbf{elif}\;t \leq 1.6537578212974721 \cdot 10^{-108}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\ell \cdot \ell} \cdot \left(\frac{t \cdot \left(k \cdot k\right)}{\cos k} + 2 \cdot \frac{{t}^{3}}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_2 \cdot {\left(\frac{\frac{t \cdot t_1}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{2}}\\ \end{array} \]

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
 (let* ((t_1 (cbrt (sin k)))
        (t_2
         (*
          (* (/ t (cbrt l)) (/ t_1 (cbrt l)))
          (* (tan k) (+ 2.0 (pow (/ k t) 2.0))))))
   (if (<= t -1.2869095709499647e-59)
     (/ 2.0 (* (pow (* (/ t (pow (cbrt l) 2.0)) t_1) 2.0) t_2))
     (if (<= t 1.6537578212974721e-108)
       (/
        2.0
        (*
         (/ (pow (sin k) 2.0) (* l l))
         (+ (/ (* t (* k k)) (cos k)) (* 2.0 (/ (pow t 3.0) (cos k))))))
       (/ 2.0 (* t_2 (pow (/ (/ (* t t_1) (cbrt l)) (cbrt l)) 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 t_1 = cbrt(sin(k));
	double t_2 = ((t / cbrt(l)) * (t_1 / cbrt(l))) * (tan(k) * (2.0 + pow((k / t), 2.0)));
	double tmp;
	if (t <= -1.2869095709499647e-59) {
		tmp = 2.0 / (pow(((t / pow(cbrt(l), 2.0)) * t_1), 2.0) * t_2);
	} else if (t <= 1.6537578212974721e-108) {
		tmp = 2.0 / ((pow(sin(k), 2.0) / (l * l)) * (((t * (k * k)) / cos(k)) + (2.0 * (pow(t, 3.0) / cos(k)))));
	} else {
		tmp = 2.0 / (t_2 * pow((((t * t_1) / cbrt(l)) / cbrt(l)), 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 t < -1.28690957094996471e-59

   1. Initial program 23.2

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

   2. Simplified 23.2

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

   3. Applied egg 4.9

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

   4. Applied egg 4.9

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

   if -1.28690957094996471e-59 < t < 1.6537578212974721e-108

   1. Initial program 59.3

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

   2. Simplified 59.3

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

   3. Applied egg 35.4

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

   4. Applied egg 35.4

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

   5. Taylor expanded in t around 0 38.0

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

   6. Simplified 25.2

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

   if 1.6537578212974721e-108 < t

   1. Initial program 24.3

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

   2. Simplified 24.3

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

   3. Applied egg 6.0

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

   4. Applied egg 6.0

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

   5. Applied egg 6.0

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

4. Final simplification 10.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.2869095709499647 \cdot 10^{-59}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{2} \cdot \left(\left(\frac{t}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{\sin k}}{\sqrt[3]{\ell}}\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 1.6537578212974721 \cdot 10^{-108}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\ell \cdot \ell} \cdot \left(\frac{t \cdot \left(k \cdot k\right)}{\cos k} + 2 \cdot \frac{{t}^{3}}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{\sin k}}{\sqrt[3]{\ell}}\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot {\left(\frac{\frac{t \cdot \sqrt[3]{\sin k}}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{2}}\\ \end{array} \]

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))))