Toniolo and Linder, Equation (10-)

Average Error: 48.0 → 6.2

Time: 34.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := t \cdot {\sin k}^{2}\\ t_2 := \frac{t_1}{\ell}\\ \mathbf{if}\;t \leq -1.003321514759076 \cdot 10^{-137}:\\ \;\;\;\;\begin{array}{l} t_3 := \sqrt[3]{\cos k}\\ \frac{t_3 \cdot t_3}{\frac{k}{\ell}} \cdot \frac{\frac{2}{\frac{k}{t_3}}}{t_2} \end{array}\\ \mathbf{elif}\;t \leq 1.4456610753441822 \cdot 10^{-198}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{\cos k}}}{\frac{\frac{k \cdot t_1}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell}} \cdot \frac{\cos k \cdot \frac{1}{k}}{t_2}\\ \end{array} \]

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 (* t (pow (sin k) 2.0))) (t_2 (/ t_1 l)))
   (if (<= t -1.003321514759076e-137)
     (let* ((t_3 (cbrt (cos k))))
       (* (/ (* t_3 t_3) (/ k l)) (/ (/ 2.0 (/ k t_3)) t_2)))
     (if (<= t 1.4456610753441822e-198)
       (/ (/ 2.0 (/ k (cos k))) (/ (/ (* k t_1) l) l))
       (* (/ 2.0 (/ k l)) (/ (* (cos k) (/ 1.0 k)) t_2))))))

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 = t * pow(sin(k), 2.0);
	double t_2 = t_1 / l;
	double tmp;
	if (t <= -1.003321514759076e-137) {
		double t_3_1 = cbrt(cos(k));
		tmp = ((t_3_1 * t_3_1) / (k / l)) * ((2.0 / (k / t_3_1)) / t_2);
	} else if (t <= 1.4456610753441822e-198) {
		tmp = (2.0 / (k / cos(k))) / (((k * t_1) / l) / l);
	} else {
		tmp = (2.0 / (k / l)) * ((cos(k) * (1.0 / k)) / 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 < -1.0033215147590761e-137

   1. Initial program 43.9

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

   2. Simplified 33.5

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

   3. Taylor expanded in t around 0 22.2

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

   4. Applied unpow2_binary64 22.2

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

   5. Applied associate-*l*_binary64 21.6

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

   6. Applied times-frac_binary64 19.1

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

   7. Applied associate-/r*_binary64 19.0

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

   8. Applied add-sqr-sqrt_binary64 41.6

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

   9. Applied unpow-prod-down_binary64 41.6

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

   10. Applied times-frac_binary64 35.1

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

   11. Applied add-cube-cbrt_binary64 35.2

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

   12. Applied *-un-lft-identity_binary64 35.2

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

   13. Applied times-frac_binary64 35.2

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

   14. Applied *-un-lft-identity_binary64 35.2

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

   15. Applied times-frac_binary64 35.2

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

   16. Applied times-frac_binary64 34.1

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

   17. Simplified 34.1

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

   18. Simplified 4.3

       \[\leadsto \frac{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}}{\frac{k}{\ell}} \cdot \color{blue}{\frac{\frac{2}{\frac{k}{\sqrt[3]{\cos k}}}}{\frac{t \cdot {\sin k}^{2}}{\ell}}} \]

   if -1.0033215147590761e-137 < t < 1.4456610753441822e-198

   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

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

   3. Taylor expanded in t around 0 26.9

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

   4. Applied unpow2_binary64 26.9

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

   5. Applied associate-*l*_binary64 18.8

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

   6. Applied times-frac_binary64 18.8

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

   7. Applied associate-/r*_binary64 18.6

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

   8. Applied unpow2_binary64 18.6

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

   9. Applied associate-/r*_binary64 14.1

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

   if 1.4456610753441822e-198 < t

   1. Initial program 45.3

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

   2. Simplified 36.0

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

   3. Taylor expanded in t around 0 21.1

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

   4. Applied unpow2_binary64 21.1

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

   5. Applied associate-*l*_binary64 20.2

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

   6. Applied times-frac_binary64 17.4

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

   7. Applied associate-/r*_binary64 17.3

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

   8. Applied add-sqr-sqrt_binary64 41.3

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

   9. Applied unpow-prod-down_binary64 41.3

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

   10. Applied times-frac_binary64 35.7

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

   11. Applied div-inv_binary64 35.7

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

   12. Applied times-frac_binary64 34.8

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

   13. Simplified 34.8

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

   14. Simplified 4.8

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

4. Final simplification 6.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.003321514759076 \cdot 10^{-137}:\\ \;\;\;\;\frac{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}}{\frac{k}{\ell}} \cdot \frac{\frac{2}{\frac{k}{\sqrt[3]{\cos k}}}}{\frac{t \cdot {\sin k}^{2}}{\ell}}\\ \mathbf{elif}\;t \leq 1.4456610753441822 \cdot 10^{-198}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{\cos k}}}{\frac{\frac{k \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell}} \cdot \frac{\cos k \cdot \frac{1}{k}}{\frac{t \cdot {\sin k}^{2}}{\ell}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022103 
(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))))