VandenBroeck and Keller, Equation (23)

Average Error: 13.7 → 0.2

Time: 39.4s

Precision: 64

Math

\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;F \le -11125333120695831522967552:\\ \;\;\;\;\frac{\frac{1}{F \cdot F} - 1}{\sin B} - \frac{1 \cdot x}{\tan B}\\ \mathbf{elif}\;F \le 117877.5913479041191749274730682373046875:\\ \;\;\;\;\frac{F}{{\left(\left(x \cdot 2 + 2\right) + F \cdot F\right)}^{\left(\frac{1}{2}\right)} \cdot \sin B} - \frac{1 \cdot x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{1}{F \cdot F}}{\sin B} - \frac{1 \cdot x}{\tan B}\\ \end{array}\]

C

double f(double F, double B, double x) {
        double r2452245 = x;
        double r2452246 = 1.0;
        double r2452247 = B;
        double r2452248 = tan(r2452247);
        double r2452249 = r2452246 / r2452248;
        double r2452250 = r2452245 * r2452249;
        double r2452251 = -r2452250;
        double r2452252 = F;
        double r2452253 = sin(r2452247);
        double r2452254 = r2452252 / r2452253;
        double r2452255 = r2452252 * r2452252;
        double r2452256 = 2.0;
        double r2452257 = r2452255 + r2452256;
        double r2452258 = r2452256 * r2452245;
        double r2452259 = r2452257 + r2452258;
        double r2452260 = r2452246 / r2452256;
        double r2452261 = -r2452260;
        double r2452262 = pow(r2452259, r2452261);
        double r2452263 = r2452254 * r2452262;
        double r2452264 = r2452251 + r2452263;
        return r2452264;
}

↓

double f(double F, double B, double x) {
        double r2452265 = F;
        double r2452266 = -1.1125333120695832e+25;
        bool r2452267 = r2452265 <= r2452266;
        double r2452268 = 1.0;
        double r2452269 = r2452265 * r2452265;
        double r2452270 = r2452268 / r2452269;
        double r2452271 = 1.0;
        double r2452272 = r2452270 - r2452271;
        double r2452273 = B;
        double r2452274 = sin(r2452273);
        double r2452275 = r2452272 / r2452274;
        double r2452276 = x;
        double r2452277 = r2452268 * r2452276;
        double r2452278 = tan(r2452273);
        double r2452279 = r2452277 / r2452278;
        double r2452280 = r2452275 - r2452279;
        double r2452281 = 117877.59134790412;
        bool r2452282 = r2452265 <= r2452281;
        double r2452283 = 2.0;
        double r2452284 = r2452276 * r2452283;
        double r2452285 = r2452284 + r2452283;
        double r2452286 = r2452285 + r2452269;
        double r2452287 = r2452268 / r2452283;
        double r2452288 = pow(r2452286, r2452287);
        double r2452289 = r2452288 * r2452274;
        double r2452290 = r2452265 / r2452289;
        double r2452291 = r2452290 - r2452279;
        double r2452292 = r2452271 - r2452270;
        double r2452293 = r2452292 / r2452274;
        double r2452294 = r2452293 - r2452279;
        double r2452295 = r2452282 ? r2452291 : r2452294;
        double r2452296 = r2452267 ? r2452280 : r2452295;
        return r2452296;
}

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if F < -1.1125333120695832e+25

   1. Initial program 26.2

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]

   2. Simplified 20.6

      \[\leadsto \color{blue}{\frac{F}{\frac{\sin B}{{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{\left(-\frac{1}{2}\right)}}} - \frac{x \cdot 1}{\tan B}}\]
   3. Using strategy rm

   4. Applied pow-neg 20.6

      \[\leadsto \frac{F}{\frac{\sin B}{\color{blue}{\frac{1}{{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{\left(\frac{1}{2}\right)}}}}} - \frac{x \cdot 1}{\tan B}\]

   5. Applied associate-/r/ 20.5

      \[\leadsto \frac{F}{\color{blue}{\frac{\sin B}{1} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{\left(\frac{1}{2}\right)}}} - \frac{x \cdot 1}{\tan B}\]

   6. Applied *-un-lft-identity 20.5

      \[\leadsto \frac{\color{blue}{1 \cdot F}}{\frac{\sin B}{1} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{\left(\frac{1}{2}\right)}} - \frac{x \cdot 1}{\tan B}\]

   7. Applied times-frac 20.5

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin B}{1}} \cdot \frac{F}{{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{\left(\frac{1}{2}\right)}}} - \frac{x \cdot 1}{\tan B}\]

   8. Simplified 20.5

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \cdot \frac{F}{{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{\left(\frac{1}{2}\right)}} - \frac{x \cdot 1}{\tan B}\]
   9. Using strategy rm

   10. Applied associate-*l/ 20.5

       \[\leadsto \color{blue}{\frac{1 \cdot \frac{F}{{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{\left(\frac{1}{2}\right)}}}{\sin B}} - \frac{x \cdot 1}{\tan B}\]

   11. Simplified 20.5

       \[\leadsto \frac{\color{blue}{\frac{F}{{\left(F \cdot F + \left(2 + 2 \cdot x\right)\right)}^{\left(\frac{1}{2}\right)}}}}{\sin B} - \frac{x \cdot 1}{\tan B}\]

   12. Taylor expanded around -inf 0.2

       \[\leadsto \frac{\color{blue}{1 \cdot \frac{1}{{F}^{2}} - 1}}{\sin B} - \frac{x \cdot 1}{\tan B}\]

   13. Simplified 0.2

       \[\leadsto \frac{\color{blue}{\frac{1}{F \cdot F} - 1}}{\sin B} - \frac{x \cdot 1}{\tan B}\]

   if -1.1125333120695832e+25 < F < 117877.59134790412

   1. Initial program 0.4

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]

   2. Simplified 0.3

      \[\leadsto \color{blue}{\frac{F}{\frac{\sin B}{{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{\left(-\frac{1}{2}\right)}}} - \frac{x \cdot 1}{\tan B}}\]
   3. Using strategy rm

   4. Applied pow-neg 0.3

      \[\leadsto \frac{F}{\frac{\sin B}{\color{blue}{\frac{1}{{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{\left(\frac{1}{2}\right)}}}}} - \frac{x \cdot 1}{\tan B}\]

   5. Applied associate-/r/ 0.3

      \[\leadsto \frac{F}{\color{blue}{\frac{\sin B}{1} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{\left(\frac{1}{2}\right)}}} - \frac{x \cdot 1}{\tan B}\]

   6. Applied *-un-lft-identity 0.3

      \[\leadsto \frac{\color{blue}{1 \cdot F}}{\frac{\sin B}{1} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{\left(\frac{1}{2}\right)}} - \frac{x \cdot 1}{\tan B}\]

   7. Applied times-frac 0.3

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin B}{1}} \cdot \frac{F}{{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{\left(\frac{1}{2}\right)}}} - \frac{x \cdot 1}{\tan B}\]

   8. Simplified 0.3

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \cdot \frac{F}{{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{\left(\frac{1}{2}\right)}} - \frac{x \cdot 1}{\tan B}\]
   9. Using strategy rm

   10. Applied associate-*l/ 0.3

       \[\leadsto \color{blue}{\frac{1 \cdot \frac{F}{{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{\left(\frac{1}{2}\right)}}}{\sin B}} - \frac{x \cdot 1}{\tan B}\]

   11. Simplified 0.3

       \[\leadsto \frac{\color{blue}{\frac{F}{{\left(F \cdot F + \left(2 + 2 \cdot x\right)\right)}^{\left(\frac{1}{2}\right)}}}}{\sin B} - \frac{x \cdot 1}{\tan B}\]
   12. Using strategy rm

   13. Applied associate-/l/ 0.3

       \[\leadsto \color{blue}{\frac{F}{\sin B \cdot {\left(F \cdot F + \left(2 + 2 \cdot x\right)\right)}^{\left(\frac{1}{2}\right)}}} - \frac{x \cdot 1}{\tan B}\]

   if 117877.59134790412 < F

   1. Initial program 25.2

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]

   2. Simplified 19.2

      \[\leadsto \color{blue}{\frac{F}{\frac{\sin B}{{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{\left(-\frac{1}{2}\right)}}} - \frac{x \cdot 1}{\tan B}}\]
   3. Using strategy rm

   4. Applied pow-neg 19.2

      \[\leadsto \frac{F}{\frac{\sin B}{\color{blue}{\frac{1}{{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{\left(\frac{1}{2}\right)}}}}} - \frac{x \cdot 1}{\tan B}\]

   5. Applied associate-/r/ 19.2

      \[\leadsto \frac{F}{\color{blue}{\frac{\sin B}{1} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{\left(\frac{1}{2}\right)}}} - \frac{x \cdot 1}{\tan B}\]

   6. Applied *-un-lft-identity 19.2

      \[\leadsto \frac{\color{blue}{1 \cdot F}}{\frac{\sin B}{1} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{\left(\frac{1}{2}\right)}} - \frac{x \cdot 1}{\tan B}\]

   7. Applied times-frac 19.1

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin B}{1}} \cdot \frac{F}{{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{\left(\frac{1}{2}\right)}}} - \frac{x \cdot 1}{\tan B}\]

   8. Simplified 19.1

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \cdot \frac{F}{{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{\left(\frac{1}{2}\right)}} - \frac{x \cdot 1}{\tan B}\]
   9. Using strategy rm

   10. Applied associate-*l/ 19.1

       \[\leadsto \color{blue}{\frac{1 \cdot \frac{F}{{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{\left(\frac{1}{2}\right)}}}{\sin B}} - \frac{x \cdot 1}{\tan B}\]

   11. Simplified 19.1

       \[\leadsto \frac{\color{blue}{\frac{F}{{\left(F \cdot F + \left(2 + 2 \cdot x\right)\right)}^{\left(\frac{1}{2}\right)}}}}{\sin B} - \frac{x \cdot 1}{\tan B}\]

   12. Taylor expanded around inf 0.1

       \[\leadsto \frac{\color{blue}{1 - 1 \cdot \frac{1}{{F}^{2}}}}{\sin B} - \frac{x \cdot 1}{\tan B}\]

   13. Simplified 0.1

       \[\leadsto \frac{\color{blue}{1 - \frac{1}{F \cdot F}}}{\sin B} - \frac{x \cdot 1}{\tan B}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -11125333120695831522967552:\\ \;\;\;\;\frac{\frac{1}{F \cdot F} - 1}{\sin B} - \frac{1 \cdot x}{\tan B}\\ \mathbf{elif}\;F \le 117877.5913479041191749274730682373046875:\\ \;\;\;\;\frac{F}{{\left(\left(x \cdot 2 + 2\right) + F \cdot F\right)}^{\left(\frac{1}{2}\right)} \cdot \sin B} - \frac{1 \cdot x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{1}{F \cdot F}}{\sin B} - \frac{1 \cdot x}{\tan B}\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 
(FPCore (F B x)
  :name "VandenBroeck and Keller, Equation (23)"
  (+ (- (* x (/ 1.0 (tan B)))) (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0))))))