VandenBroeck and Keller, Equation (23)

Average Error: 13.9 → 0.3

Time: 1.1m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;F \le -5.807335808175188240909392334287986159325:\\ \;\;\;\;\left(\frac{\frac{1}{F \cdot F}}{\sin B} + \frac{-1}{\sin B}\right) - 1 \cdot \frac{x}{\tan B}\\ \mathbf{elif}\;F \le 86.69108830173802004992467118427157402039:\\ \;\;\;\;\frac{F}{\sin B \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{\left(\frac{1}{2}\right)}} - \left(\cos B \cdot \frac{x}{\sin B}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \frac{\frac{1}{F \cdot F}}{\sin B}\right) - 1 \cdot \frac{x}{\tan B}\\ \end{array}\]

C

double f(double F, double B, double x) {
        double r2581169 = x;
        double r2581170 = 1.0;
        double r2581171 = B;
        double r2581172 = tan(r2581171);
        double r2581173 = r2581170 / r2581172;
        double r2581174 = r2581169 * r2581173;
        double r2581175 = -r2581174;
        double r2581176 = F;
        double r2581177 = sin(r2581171);
        double r2581178 = r2581176 / r2581177;
        double r2581179 = r2581176 * r2581176;
        double r2581180 = 2.0;
        double r2581181 = r2581179 + r2581180;
        double r2581182 = r2581180 * r2581169;
        double r2581183 = r2581181 + r2581182;
        double r2581184 = r2581170 / r2581180;
        double r2581185 = -r2581184;
        double r2581186 = pow(r2581183, r2581185);
        double r2581187 = r2581178 * r2581186;
        double r2581188 = r2581175 + r2581187;
        return r2581188;
}

↓

double f(double F, double B, double x) {
        double r2581189 = F;
        double r2581190 = -5.807335808175188;
        bool r2581191 = r2581189 <= r2581190;
        double r2581192 = 1.0;
        double r2581193 = r2581189 * r2581189;
        double r2581194 = r2581192 / r2581193;
        double r2581195 = B;
        double r2581196 = sin(r2581195);
        double r2581197 = r2581194 / r2581196;
        double r2581198 = -1.0;
        double r2581199 = r2581198 / r2581196;
        double r2581200 = r2581197 + r2581199;
        double r2581201 = x;
        double r2581202 = tan(r2581195);
        double r2581203 = r2581201 / r2581202;
        double r2581204 = r2581192 * r2581203;
        double r2581205 = r2581200 - r2581204;
        double r2581206 = 86.69108830173802;
        bool r2581207 = r2581189 <= r2581206;
        double r2581208 = 2.0;
        double r2581209 = r2581208 + r2581193;
        double r2581210 = r2581201 * r2581208;
        double r2581211 = r2581209 + r2581210;
        double r2581212 = r2581192 / r2581208;
        double r2581213 = pow(r2581211, r2581212);
        double r2581214 = r2581196 * r2581213;
        double r2581215 = r2581189 / r2581214;
        double r2581216 = cos(r2581195);
        double r2581217 = r2581201 / r2581196;
        double r2581218 = r2581216 * r2581217;
        double r2581219 = r2581218 * r2581192;
        double r2581220 = r2581215 - r2581219;
        double r2581221 = 1.0;
        double r2581222 = r2581221 / r2581196;
        double r2581223 = r2581222 - r2581197;
        double r2581224 = r2581223 - r2581204;
        double r2581225 = r2581207 ? r2581220 : r2581224;
        double r2581226 = r2581191 ? r2581205 : r2581225;
        return r2581226;
}

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if F < -5.807335808175188

   1. Initial program 25.5

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

   2. Simplified 25.5

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

   4. Applied distribute-frac-neg 25.5

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

   5. Applied pow-neg 25.5

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

   6. Applied frac-times 19.9

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

   7. Simplified 19.9

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

   9. Applied div-inv 19.9

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

   10. Applied associate-*l* 19.9

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

   11. Simplified 19.8

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

   13. Applied div-inv 19.8

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

   14. Taylor expanded around -inf 0.3

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

   15. Simplified 0.3

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

   if -5.807335808175188 < F < 86.69108830173802

   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 0.4

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

   4. Applied distribute-frac-neg 0.4

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

   5. Applied pow-neg 0.4

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

   6. Applied frac-times 0.4

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

   7. Simplified 0.4

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

   9. Applied div-inv 0.4

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

   10. Applied associate-*l* 0.4

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

   11. Simplified 0.3

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

   13. Applied tan-quot 0.3

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

   14. Applied associate-/r/ 0.3

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

   if 86.69108830173802 < F

   1. Initial program 24.5

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

   2. Simplified 24.5

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

   4. Applied distribute-frac-neg 24.5

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

   5. Applied pow-neg 24.5

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

   6. Applied frac-times 18.5

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

   7. Simplified 18.5

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

   9. Applied div-inv 18.5

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

   10. Applied associate-*l* 18.5

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

   11. Simplified 18.4

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

   12. Taylor expanded around inf 0.2

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

   13. Simplified 0.2

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -5.807335808175188240909392334287986159325:\\ \;\;\;\;\left(\frac{\frac{1}{F \cdot F}}{\sin B} + \frac{-1}{\sin B}\right) - 1 \cdot \frac{x}{\tan B}\\ \mathbf{elif}\;F \le 86.69108830173802004992467118427157402039:\\ \;\;\;\;\frac{F}{\sin B \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{\left(\frac{1}{2}\right)}} - \left(\cos B \cdot \frac{x}{\sin B}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \frac{\frac{1}{F \cdot F}}{\sin B}\right) - 1 \cdot \frac{x}{\tan B}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(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))))))