Toniolo and Linder, Equation (10-)

Average Error: 48.5 → 14.3

Time: 1.1m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \le -2.015060177812317540999976002228685217361 \cdot 10^{153}:\\ \;\;\;\;\frac{\frac{2}{\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \sin k\right)\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;\ell \le 5.307433067528376064772596510757992502501 \cdot 10^{135}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\left(\frac{\cos k}{\left|\sin k\right|} \cdot \frac{\ell}{\frac{\left|\sin k\right|}{\ell}}\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot {\left(\sin k\right)}^{2}}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \frac{\cos k}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r103062 = 2.0;
        double r103063 = t;
        double r103064 = 3.0;
        double r103065 = pow(r103063, r103064);
        double r103066 = l;
        double r103067 = r103066 * r103066;
        double r103068 = r103065 / r103067;
        double r103069 = k;
        double r103070 = sin(r103069);
        double r103071 = r103068 * r103070;
        double r103072 = tan(r103069);
        double r103073 = r103071 * r103072;
        double r103074 = 1.0;
        double r103075 = r103069 / r103063;
        double r103076 = pow(r103075, r103062);
        double r103077 = r103074 + r103076;
        double r103078 = r103077 - r103074;
        double r103079 = r103073 * r103078;
        double r103080 = r103062 / r103079;
        return r103080;
}

↓

double f(double t, double l, double k) {
        double r103081 = l;
        double r103082 = -2.0150601778123175e+153;
        bool r103083 = r103081 <= r103082;
        double r103084 = 2.0;
        double r103085 = t;
        double r103086 = 3.0;
        double r103087 = 2.0;
        double r103088 = r103086 / r103087;
        double r103089 = pow(r103085, r103088);
        double r103090 = r103089 / r103081;
        double r103091 = k;
        double r103092 = sin(r103091);
        double r103093 = r103090 * r103092;
        double r103094 = r103090 * r103093;
        double r103095 = tan(r103091);
        double r103096 = r103094 * r103095;
        double r103097 = r103084 / r103096;
        double r103098 = r103091 / r103085;
        double r103099 = pow(r103098, r103084);
        double r103100 = r103097 / r103099;
        double r103101 = 5.307433067528376e+135;
        bool r103102 = r103081 <= r103101;
        double r103103 = 1.0;
        double r103104 = r103084 / r103087;
        double r103105 = pow(r103091, r103104);
        double r103106 = r103103 / r103105;
        double r103107 = 1.0;
        double r103108 = pow(r103106, r103107);
        double r103109 = cos(r103091);
        double r103110 = fabs(r103092);
        double r103111 = r103109 / r103110;
        double r103112 = r103110 / r103081;
        double r103113 = r103081 / r103112;
        double r103114 = r103111 * r103113;
        double r103115 = pow(r103085, r103107);
        double r103116 = r103105 * r103115;
        double r103117 = r103103 / r103116;
        double r103118 = pow(r103117, r103107);
        double r103119 = r103114 * r103118;
        double r103120 = r103108 * r103119;
        double r103121 = r103084 * r103120;
        double r103122 = pow(r103085, r103086);
        double r103123 = r103122 / r103081;
        double r103124 = r103123 / r103081;
        double r103125 = pow(r103092, r103087);
        double r103126 = r103124 * r103125;
        double r103127 = r103084 / r103126;
        double r103128 = pow(r103098, r103104);
        double r103129 = r103127 / r103128;
        double r103130 = r103109 / r103128;
        double r103131 = r103129 * r103130;
        double r103132 = r103102 ? r103121 : r103131;
        double r103133 = r103083 ? r103100 : r103132;
        return r103133;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if l < -2.0150601778123175e+153

   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

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

   4. Applied sqr-pow 64.0

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

   5. Applied times-frac 52.3

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

   6. Applied associate-*l* 52.3

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

   if -2.0150601778123175e+153 < l < 5.307433067528376e+135

   1. Initial program 45.6

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

   2. Simplified 36.2

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

   3. Taylor expanded around inf 14.6

      \[\leadsto \color{blue}{2 \cdot \left({\left(\frac{1}{{k}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)}\]
   4. Using strategy rm

   5. Applied sqr-pow 14.6

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

   6. Applied associate-*l* 12.2

      \[\leadsto 2 \cdot \left({\left(\frac{1}{\color{blue}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\]
   7. Using strategy rm

   8. Applied *-un-lft-identity 12.2

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

   9. Applied times-frac 12.0

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

   10. Applied unpow-prod-down 12.0

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

   11. Applied associate-*l* 10.0

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

   12. Simplified 10.0

       \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \color{blue}{\left(\frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}} \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1}\right)}\right)\]
   13. Using strategy rm

   14. Applied add-sqr-sqrt 10.0

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

   15. Applied times-frac 10.0

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

   16. Simplified 10.0

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

   17. Simplified 7.1

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

   if 5.307433067528376e+135 < l

   1. Initial program 62.4

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

   2. Simplified 61.8

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

   4. Applied sqr-pow 61.8

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

   5. Applied tan-quot 61.8

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

   6. Applied associate-*r/ 61.8

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

   7. Applied associate-/r/ 61.8

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

   8. Applied times-frac 61.2

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

   9. Simplified 51.2

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

4. Final simplification 14.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -2.015060177812317540999976002228685217361 \cdot 10^{153}:\\ \;\;\;\;\frac{\frac{2}{\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \sin k\right)\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;\ell \le 5.307433067528376064772596510757992502501 \cdot 10^{135}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\left(\frac{\cos k}{\left|\sin k\right|} \cdot \frac{\ell}{\frac{\left|\sin k\right|}{\ell}}\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot {\left(\sin k\right)}^{2}}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \frac{\cos k}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  :precision binary64
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1))))