VandenBroeck and Keller, Equation (23)

Average Error: 14.4 → 0.2

Time: 42.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;F \le -61781114441641127683218472587214714361090000:\\ \;\;\;\;\left(\frac{-1}{\sin B} + \frac{\frac{1}{F \cdot F}}{\sin B}\right) - \frac{x \cdot 1}{\tan B}\\ \mathbf{elif}\;F \le 662194.586172068375162780284881591796875:\\ \;\;\;\;\frac{F}{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{\left(\frac{1}{2}\right)} \cdot \sin B} - \frac{x \cdot 1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \frac{\frac{1}{\sin B}}{F \cdot F}\right) - \frac{x \cdot 1}{\tan B}\\ \end{array}\]

C

double f(double F, double B, double x) {
        double r2072995 = x;
        double r2072996 = 1.0;
        double r2072997 = B;
        double r2072998 = tan(r2072997);
        double r2072999 = r2072996 / r2072998;
        double r2073000 = r2072995 * r2072999;
        double r2073001 = -r2073000;
        double r2073002 = F;
        double r2073003 = sin(r2072997);
        double r2073004 = r2073002 / r2073003;
        double r2073005 = r2073002 * r2073002;
        double r2073006 = 2.0;
        double r2073007 = r2073005 + r2073006;
        double r2073008 = r2073006 * r2072995;
        double r2073009 = r2073007 + r2073008;
        double r2073010 = r2072996 / r2073006;
        double r2073011 = -r2073010;
        double r2073012 = pow(r2073009, r2073011);
        double r2073013 = r2073004 * r2073012;
        double r2073014 = r2073001 + r2073013;
        return r2073014;
}

↓

double f(double F, double B, double x) {
        double r2073015 = F;
        double r2073016 = -6.178111444164113e+43;
        bool r2073017 = r2073015 <= r2073016;
        double r2073018 = -1.0;
        double r2073019 = B;
        double r2073020 = sin(r2073019);
        double r2073021 = r2073018 / r2073020;
        double r2073022 = 1.0;
        double r2073023 = r2073015 * r2073015;
        double r2073024 = r2073022 / r2073023;
        double r2073025 = r2073024 / r2073020;
        double r2073026 = r2073021 + r2073025;
        double r2073027 = x;
        double r2073028 = r2073027 * r2073022;
        double r2073029 = tan(r2073019);
        double r2073030 = r2073028 / r2073029;
        double r2073031 = r2073026 - r2073030;
        double r2073032 = 662194.5861720684;
        bool r2073033 = r2073015 <= r2073032;
        double r2073034 = 2.0;
        double r2073035 = fma(r2073034, r2073027, r2073034);
        double r2073036 = fma(r2073015, r2073015, r2073035);
        double r2073037 = r2073022 / r2073034;
        double r2073038 = pow(r2073036, r2073037);
        double r2073039 = r2073038 * r2073020;
        double r2073040 = r2073015 / r2073039;
        double r2073041 = r2073040 - r2073030;
        double r2073042 = 1.0;
        double r2073043 = r2073042 / r2073020;
        double r2073044 = r2073022 / r2073020;
        double r2073045 = r2073044 / r2073023;
        double r2073046 = r2073043 - r2073045;
        double r2073047 = r2073046 - r2073030;
        double r2073048 = r2073033 ? r2073041 : r2073047;
        double r2073049 = r2073017 ? r2073031 : r2073048;
        return r2073049;
}

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if F < -6.178111444164113e+43

   1. Initial program 29.0

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

   2. Simplified 22.3

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

   4. Applied associate-*l/ 22.3

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

   6. Applied div-inv 22.3

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

   7. Applied *-un-lft-identity 22.3

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

   8. Applied times-frac 22.2

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

   9. Simplified 22.2

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

   10. Taylor expanded around -inf 0.2

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

   11. Simplified 0.2

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

   if -6.178111444164113e+43 < F < 662194.5861720684

   1. Initial program 0.5

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

   2. Simplified 0.4

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

   4. Applied associate-*l/ 0.3

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

   6. Applied div-inv 0.3

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

   7. Applied *-un-lft-identity 0.3

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

   8. Applied times-frac 0.3

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

   9. Simplified 0.3

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

   11. Applied distribute-frac-neg 0.3

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

   12. Applied pow-neg 0.3

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

   13. Applied un-div-inv 0.3

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

   14. Applied frac-times 0.3

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

   15. Simplified 0.3

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

   if 662194.5861720684 < F

   1. Initial program 25.7

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

   2. Simplified 20.1

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

   4. Applied associate-*l/ 20.0

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

   6. Applied div-inv 20.0

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

   7. Applied *-un-lft-identity 20.0

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

   8. Applied times-frac 20.0

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

   9. Simplified 20.0

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

   11. Applied distribute-frac-neg 20.0

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

   12. Applied pow-neg 20.0

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

   13. Applied un-div-inv 19.9

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

   14. Applied frac-times 20.0

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

   15. Simplified 20.0

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

   16. Taylor expanded around inf 0.2

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

   17. Simplified 0.2

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -61781114441641127683218472587214714361090000:\\ \;\;\;\;\left(\frac{-1}{\sin B} + \frac{\frac{1}{F \cdot F}}{\sin B}\right) - \frac{x \cdot 1}{\tan B}\\ \mathbf{elif}\;F \le 662194.586172068375162780284881591796875:\\ \;\;\;\;\frac{F}{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{\left(\frac{1}{2}\right)} \cdot \sin B} - \frac{x \cdot 1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \frac{\frac{1}{\sin B}}{F \cdot F}\right) - \frac{x \cdot 1}{\tan B}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(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))))))