Toniolo and Linder, Equation (10+)

Average Error: 32.2 → 12.6

Time: 1.0min

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \leq -1.61988277404358 \cdot 10^{-126}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{t}{\ell} \cdot \left(\left(\sin k \cdot \frac{t}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right) \cdot \frac{t}{\sqrt[3]{\ell}}\right)\right) \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{\cos k}}\\ \mathbf{elif}\;t \leq 6.364702243051122 \cdot 10^{-151}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\ell \cdot \ell} \cdot \left(\frac{t \cdot \left(k \cdot k\right)}{\cos k} + 2 \cdot \frac{{t}^{3}}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{t}{\ell} \cdot \left(\left(\sin k \cdot \frac{t}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right) \cdot \frac{t}{\sqrt[3]{\ell}}\right)} \cdot \frac{\sqrt{2}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k}\\ \end{array}\]

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 (<= t -1.61988277404358e-126)
   (/
    2.0
    (/
     (*
      (* (/ t l) (* (* (sin k) (/ t (* (cbrt l) (cbrt l)))) (/ t (cbrt l))))
      (* (sin k) (+ 2.0 (pow (/ k t) 2.0))))
     (cos k)))
   (if (<= t 6.364702243051122e-151)
     (/
      2.0
      (*
       (/ (pow (sin k) 2.0) (* l l))
       (+ (/ (* t (* k k)) (cos k)) (* 2.0 (/ (pow t 3.0) (cos k))))))
     (*
      (/
       (sqrt 2.0)
       (* (/ t l) (* (* (sin k) (/ t (* (cbrt l) (cbrt l)))) (/ t (cbrt l)))))
      (/ (sqrt 2.0) (* (+ 2.0 (pow (/ k t) 2.0)) (tan k)))))))

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 (t <= -1.61988277404358e-126) {
		tmp = 2.0 / ((((t / l) * ((sin(k) * (t / (cbrt(l) * cbrt(l)))) * (t / cbrt(l)))) * (sin(k) * (2.0 + pow((k / t), 2.0)))) / cos(k));
	} else if (t <= 6.364702243051122e-151) {
		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 = (sqrt(2.0) / ((t / l) * ((sin(k) * (t / (cbrt(l) * cbrt(l)))) * (t / cbrt(l))))) * (sqrt(2.0) / ((2.0 + pow((k / t), 2.0)) * tan(k)));
	}
	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 < -1.6198827740435801e-126

   1. Initial program 24.4

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

   2. Simplified 24.4

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

   4. Applied cube-mult_binary64_449 24.4

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

   5. Applied times-frac_binary64_425 16.7

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

   6. Applied associate-*l*_binary64_360 14.4

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

   7. Simplified 14.4

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

   9. Applied add-cube-cbrt_binary64_454 14.6

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

   10. Applied times-frac_binary64_425 10.2

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

   11. Applied associate-*r*_binary64_359 9.2

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

   13. Applied associate-*l*_binary64_360 9.2

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

   14. Simplified 9.2

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

   16. Applied tan-quot_binary64_578 9.2

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

   17. Applied associate-*r/_binary64_361 9.2

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

   18. Applied associate-*r/_binary64_361 9.2

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

   if -1.6198827740435801e-126 < t < 6.3647022430511221e-151

   1. Initial program 64.0

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

   2. Simplified 64.0

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

   4. Applied cube-mult_binary64_449 64.0

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

   5. Applied times-frac_binary64_425 59.3

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

   6. Applied associate-*l*_binary64_360 59.3

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

   7. Simplified 59.3

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

   8. Taylor expanded around 0 41.2

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

   9. Simplified 26.5

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

   if 6.3647022430511221e-151 < t

   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. Simplified 25.1

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

   4. Applied cube-mult_binary64_449 25.1

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

   5. Applied times-frac_binary64_425 16.9

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

   6. Applied associate-*l*_binary64_360 14.7

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

   7. Simplified 14.7

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

   9. Applied add-cube-cbrt_binary64_454 14.9

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

   10. Applied times-frac_binary64_425 10.8

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

   11. Applied associate-*r*_binary64_359 9.7

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

   13. Applied associate-*l*_binary64_360 9.6

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

   14. Simplified 9.6

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

   16. Applied add-sqr-sqrt_binary64_441 9.7

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

   17. Applied times-frac_binary64_425 9.6

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

4. Final simplification 12.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.61988277404358 \cdot 10^{-126}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{t}{\ell} \cdot \left(\left(\sin k \cdot \frac{t}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right) \cdot \frac{t}{\sqrt[3]{\ell}}\right)\right) \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{\cos k}}\\ \mathbf{elif}\;t \leq 6.364702243051122 \cdot 10^{-151}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\ell \cdot \ell} \cdot \left(\frac{t \cdot \left(k \cdot k\right)}{\cos k} + 2 \cdot \frac{{t}^{3}}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{t}{\ell} \cdot \left(\left(\sin k \cdot \frac{t}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right) \cdot \frac{t}{\sqrt[3]{\ell}}\right)} \cdot \frac{\sqrt{2}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k}\\ \end{array}\]

Reproduce

herbie shell --seed 2021023 
(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))))