Destination given bearing on a great circle

Average Error: 0.2 → 0.2

Time: 50.8s

Precision: 64

• unused variable phi2

Math

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

↓

\[\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\frac{{\left(\cos delta\right)}^{3} - {\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \sin delta \cdot \cos \phi_1, \sin \phi_1 \cdot \cos delta\right)\right)\right)\right)\right)\right)}^{3}}{\left(\cos delta \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \sin delta \cdot \cos \phi_1, \sin \phi_1 \cdot \cos delta\right)\right)\right)\right)\right) + \left(\sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \sin delta \cdot \cos \phi_1, \sin \phi_1 \cdot \cos delta\right)\right)\right)\right)\right)} \cdot \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \sin delta \cdot \cos \phi_1, \sin \phi_1 \cdot \cos delta\right)\right)\right)\right)\right)}\right) \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \sin delta \cdot \cos \phi_1, \sin \phi_1 \cdot \cos delta\right)\right)\right)\right)\right) \cdot \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \sin delta \cdot \cos \phi_1, \sin \phi_1 \cdot \cos delta\right)\right)\right)\right)\right)}\right)\right) + \cos delta \cdot \cos delta}}\]

C

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r3723402 = lambda1;
        double r3723403 = theta;
        double r3723404 = sin(r3723403);
        double r3723405 = delta;
        double r3723406 = sin(r3723405);
        double r3723407 = r3723404 * r3723406;
        double r3723408 = phi1;
        double r3723409 = cos(r3723408);
        double r3723410 = r3723407 * r3723409;
        double r3723411 = cos(r3723405);
        double r3723412 = sin(r3723408);
        double r3723413 = r3723412 * r3723411;
        double r3723414 = r3723409 * r3723406;
        double r3723415 = cos(r3723403);
        double r3723416 = r3723414 * r3723415;
        double r3723417 = r3723413 + r3723416;
        double r3723418 = asin(r3723417);
        double r3723419 = sin(r3723418);
        double r3723420 = r3723412 * r3723419;
        double r3723421 = r3723411 - r3723420;
        double r3723422 = atan2(r3723410, r3723421);
        double r3723423 = r3723402 + r3723422;
        return r3723423;
}

↓

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r3723424 = lambda1;
        double r3723425 = theta;
        double r3723426 = sin(r3723425);
        double r3723427 = delta;
        double r3723428 = sin(r3723427);
        double r3723429 = phi1;
        double r3723430 = cos(r3723429);
        double r3723431 = r3723428 * r3723430;
        double r3723432 = r3723426 * r3723431;
        double r3723433 = cos(r3723427);
        double r3723434 = 3.0;
        double r3723435 = pow(r3723433, r3723434);
        double r3723436 = sin(r3723429);
        double r3723437 = cos(r3723425);
        double r3723438 = r3723436 * r3723433;
        double r3723439 = fma(r3723437, r3723431, r3723438);
        double r3723440 = asin(r3723439);
        double r3723441 = sin(r3723440);
        double r3723442 = r3723436 * r3723441;
        double r3723443 = log1p(r3723442);
        double r3723444 = expm1(r3723443);
        double r3723445 = pow(r3723444, r3723434);
        double r3723446 = r3723435 - r3723445;
        double r3723447 = r3723433 * r3723444;
        double r3723448 = cbrt(r3723444);
        double r3723449 = r3723448 * r3723448;
        double r3723450 = r3723444 * r3723448;
        double r3723451 = r3723449 * r3723450;
        double r3723452 = r3723447 + r3723451;
        double r3723453 = r3723433 * r3723433;
        double r3723454 = r3723452 + r3723453;
        double r3723455 = r3723446 / r3723454;
        double r3723456 = atan2(r3723432, r3723455);
        double r3723457 = r3723424 + r3723456;
        return r3723457;
}

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 0.2

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

2. Simplified 0.2

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

4. Applied associate-*r* 0.2

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

5. Simplified 0.2

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

7. Applied expm1-log1p-u 0.2

   \[\leadsto \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin delta\right) \cdot \sin theta}{\cos delta - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)\right)\right)}} + \lambda_1\]
8. Using strategy rm

9. Applied flip3-- 0.2

   \[\leadsto \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin delta\right) \cdot \sin theta}{\color{blue}{\frac{{\left(\cos delta\right)}^{3} - {\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)\right)\right)\right)}^{3}}{\cos delta \cdot \cos delta + \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)\right)\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)\right)\right) + \cos delta \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)\right)\right)\right)}}} + \lambda_1\]
10. Using strategy rm

11. Applied add-cube-cbrt 0.2

    \[\leadsto \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin delta\right) \cdot \sin theta}{\frac{{\left(\cos delta\right)}^{3} - {\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)\right)\right)\right)}^{3}}{\cos delta \cdot \cos delta + \left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)\right)\right)} \cdot \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)\right)\right)}\right) \cdot \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)\right)\right)}\right)} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)\right)\right) + \cos delta \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)\right)\right)\right)}} + \lambda_1\]

12. Applied associate-*l* 0.2

    \[\leadsto \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin delta\right) \cdot \sin theta}{\frac{{\left(\cos delta\right)}^{3} - {\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)\right)\right)\right)}^{3}}{\cos delta \cdot \cos delta + \left(\color{blue}{\left(\sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)\right)\right)} \cdot \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)\right)\right)}\right) \cdot \left(\sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)\right)\right)} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)\right)\right)\right)} + \cos delta \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)\right)\right)\right)}} + \lambda_1\]

13. Final simplification 0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\frac{{\left(\cos delta\right)}^{3} - {\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \sin delta \cdot \cos \phi_1, \sin \phi_1 \cdot \cos delta\right)\right)\right)\right)\right)\right)}^{3}}{\left(\cos delta \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \sin delta \cdot \cos \phi_1, \sin \phi_1 \cdot \cos delta\right)\right)\right)\right)\right) + \left(\sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \sin delta \cdot \cos \phi_1, \sin \phi_1 \cdot \cos delta\right)\right)\right)\right)\right)} \cdot \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \sin delta \cdot \cos \phi_1, \sin \phi_1 \cdot \cos delta\right)\right)\right)\right)\right)}\right) \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \sin delta \cdot \cos \phi_1, \sin \phi_1 \cdot \cos delta\right)\right)\right)\right)\right) \cdot \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \sin delta \cdot \cos \phi_1, \sin \phi_1 \cdot \cos delta\right)\right)\right)\right)\right)}\right)\right) + \cos delta \cdot \cos delta}}\]

Reproduce

herbie shell --seed 2019171 +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))))))))))