Destination given bearing on a great circle

Average Error: 0.1 → 0.2

Time: 1.1m

Precision: 64

• unused variable phi2

Math

\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]

↓

\[\lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \sin delta}{\log \left(e^{\mathsf{fma}\left(-\sin \phi_1, \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \cos delta \cdot \sin \phi_1\right)\right)\right), \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \cos delta \cdot \sin \phi_1\right)\right)\right) \cdot \sin \phi_1\right) + \sqrt[3]{\mathsf{fma}\left(1, \cos delta, \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \cos delta \cdot \sin \phi_1\right)\right)\right) \cdot \left(-\sin \phi_1\right)\right) \cdot \left(\mathsf{fma}\left(1, \cos delta, \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \cos delta \cdot \sin \phi_1\right)\right)\right) \cdot \left(-\sin \phi_1\right)\right) \cdot \mathsf{fma}\left(1, \cos delta, \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \cos delta \cdot \sin \phi_1\right)\right)\right) \cdot \left(-\sin \phi_1\right)\right)\right)}}\right)}\]

C

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r3053202 = lambda1;
        double r3053203 = theta;
        double r3053204 = sin(r3053203);
        double r3053205 = delta;
        double r3053206 = sin(r3053205);
        double r3053207 = r3053204 * r3053206;
        double r3053208 = phi1;
        double r3053209 = cos(r3053208);
        double r3053210 = r3053207 * r3053209;
        double r3053211 = cos(r3053205);
        double r3053212 = sin(r3053208);
        double r3053213 = r3053212 * r3053211;
        double r3053214 = r3053209 * r3053206;
        double r3053215 = cos(r3053203);
        double r3053216 = r3053214 * r3053215;
        double r3053217 = r3053213 + r3053216;
        double r3053218 = asin(r3053217);
        double r3053219 = sin(r3053218);
        double r3053220 = r3053212 * r3053219;
        double r3053221 = r3053211 - r3053220;
        double r3053222 = atan2(r3053210, r3053221);
        double r3053223 = r3053202 + r3053222;
        return r3053223;
}

↓

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r3053224 = lambda1;
        double r3053225 = phi1;
        double r3053226 = cos(r3053225);
        double r3053227 = theta;
        double r3053228 = sin(r3053227);
        double r3053229 = r3053226 * r3053228;
        double r3053230 = delta;
        double r3053231 = sin(r3053230);
        double r3053232 = r3053229 * r3053231;
        double r3053233 = sin(r3053225);
        double r3053234 = -r3053233;
        double r3053235 = cos(r3053227);
        double r3053236 = r3053226 * r3053235;
        double r3053237 = cos(r3053230);
        double r3053238 = r3053237 * r3053233;
        double r3053239 = fma(r3053231, r3053236, r3053238);
        double r3053240 = asin(r3053239);
        double r3053241 = sin(r3053240);
        double r3053242 = r3053241 * r3053233;
        double r3053243 = fma(r3053234, r3053241, r3053242);
        double r3053244 = 1.0;
        double r3053245 = r3053241 * r3053234;
        double r3053246 = fma(r3053244, r3053237, r3053245);
        double r3053247 = r3053246 * r3053246;
        double r3053248 = r3053246 * r3053247;
        double r3053249 = cbrt(r3053248);
        double r3053250 = r3053243 + r3053249;
        double r3053251 = exp(r3053250);
        double r3053252 = log(r3053251);
        double r3053253 = atan2(r3053232, r3053252);
        double r3053254 = r3053224 + r3053253;
        return r3053254;
}

Error

Bits error versus lambda1

Bits error versus phi1

Bits error versus phi2

Bits error versus delta

Bits error versus theta

Derivation

1. Initial program 0.1

   \[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]

2. Simplified 0.1

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)} + \lambda_1}\]
3. Using strategy rm

4. Applied add-log-exp 0.2

   \[\leadsto \tan^{-1}_* \frac{\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)}{\cos delta - \color{blue}{\log \left(e^{\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)}\right)}} + \lambda_1\]

5. Applied add-log-exp 0.2

   \[\leadsto \tan^{-1}_* \frac{\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)}{\color{blue}{\log \left(e^{\cos delta}\right)} - \log \left(e^{\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)}\right)} + \lambda_1\]

6. Applied diff-log 0.2

   \[\leadsto \tan^{-1}_* \frac{\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)}{\color{blue}{\log \left(\frac{e^{\cos delta}}{e^{\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)}}\right)}} + \lambda_1\]

7. Simplified 0.2

   \[\leadsto \tan^{-1}_* \frac{\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)}{\log \color{blue}{\left(e^{\cos delta - \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \sin \phi_1 \cdot \cos delta\right)\right)\right) \cdot \sin \phi_1}\right)}} + \lambda_1\]
8. Using strategy rm

9. Applied *-un-lft-identity 0.2

   \[\leadsto \tan^{-1}_* \frac{\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)}{\log \left(e^{\color{blue}{1 \cdot \cos delta} - \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \sin \phi_1 \cdot \cos delta\right)\right)\right) \cdot \sin \phi_1}\right)} + \lambda_1\]

10. Applied prod-diff 0.2

    \[\leadsto \tan^{-1}_* \frac{\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)}{\log \left(e^{\color{blue}{\mathsf{fma}\left(1, \cos delta, -\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \sin \phi_1 \cdot \cos delta\right)\right)\right)\right) + \mathsf{fma}\left(-\sin \phi_1, \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \sin \phi_1 \cdot \cos delta\right)\right)\right), \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \sin \phi_1 \cdot \cos delta\right)\right)\right)\right)}}\right)} + \lambda_1\]
11. Using strategy rm

12. Applied add-cbrt-cube 0.2

    \[\leadsto \tan^{-1}_* \frac{\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)}{\log \left(e^{\color{blue}{\sqrt[3]{\left(\mathsf{fma}\left(1, \cos delta, -\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \sin \phi_1 \cdot \cos delta\right)\right)\right)\right) \cdot \mathsf{fma}\left(1, \cos delta, -\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \sin \phi_1 \cdot \cos delta\right)\right)\right)\right)\right) \cdot \mathsf{fma}\left(1, \cos delta, -\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \sin \phi_1 \cdot \cos delta\right)\right)\right)\right)}} + \mathsf{fma}\left(-\sin \phi_1, \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \sin \phi_1 \cdot \cos delta\right)\right)\right), \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \sin \phi_1 \cdot \cos delta\right)\right)\right)\right)}\right)} + \lambda_1\]

13. Final simplification 0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \sin delta}{\log \left(e^{\mathsf{fma}\left(-\sin \phi_1, \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \cos delta \cdot \sin \phi_1\right)\right)\right), \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \cos delta \cdot \sin \phi_1\right)\right)\right) \cdot \sin \phi_1\right) + \sqrt[3]{\mathsf{fma}\left(1, \cos delta, \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \cos delta \cdot \sin \phi_1\right)\right)\right) \cdot \left(-\sin \phi_1\right)\right) \cdot \left(\mathsf{fma}\left(1, \cos delta, \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \cos delta \cdot \sin \phi_1\right)\right)\right) \cdot \left(-\sin \phi_1\right)\right) \cdot \mathsf{fma}\left(1, \cos delta, \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \cos delta \cdot \sin \phi_1\right)\right)\right) \cdot \left(-\sin \phi_1\right)\right)\right)}}\right)}\]

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(FPCore (lambda1 phi1 phi2 delta theta)
  :name "Destination given bearing on a great circle"
  (+ lambda1 (atan2 (* (* (sin theta) (sin delta)) (cos phi1)) (- (cos delta) (* (sin phi1) (sin (asin (+ (* (sin phi1) (cos delta)) (* (* (cos phi1) (sin delta)) (cos theta))))))))))