Destination given bearing on a great circle

Average Error: 0.2 → 0.2

Time: 57.1s

Precision: 64

• unused variable phi2

Math

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

↓

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

C

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r4178052 = lambda1;
        double r4178053 = theta;
        double r4178054 = sin(r4178053);
        double r4178055 = delta;
        double r4178056 = sin(r4178055);
        double r4178057 = r4178054 * r4178056;
        double r4178058 = phi1;
        double r4178059 = cos(r4178058);
        double r4178060 = r4178057 * r4178059;
        double r4178061 = cos(r4178055);
        double r4178062 = sin(r4178058);
        double r4178063 = r4178062 * r4178061;
        double r4178064 = r4178059 * r4178056;
        double r4178065 = cos(r4178053);
        double r4178066 = r4178064 * r4178065;
        double r4178067 = r4178063 + r4178066;
        double r4178068 = asin(r4178067);
        double r4178069 = sin(r4178068);
        double r4178070 = r4178062 * r4178069;
        double r4178071 = r4178061 - r4178070;
        double r4178072 = atan2(r4178060, r4178071);
        double r4178073 = r4178052 + r4178072;
        return r4178073;
}

↓

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r4178074 = lambda1;
        double r4178075 = phi1;
        double r4178076 = cos(r4178075);
        double r4178077 = delta;
        double r4178078 = sin(r4178077);
        double r4178079 = theta;
        double r4178080 = sin(r4178079);
        double r4178081 = r4178078 * r4178080;
        double r4178082 = r4178076 * r4178081;
        double r4178083 = cos(r4178077);
        double r4178084 = r4178083 * r4178083;
        double r4178085 = r4178083 * r4178084;
        double r4178086 = r4178076 * r4178078;
        double r4178087 = cos(r4178079);
        double r4178088 = sin(r4178075);
        double r4178089 = r4178083 * r4178088;
        double r4178090 = fma(r4178086, r4178087, r4178089);
        double r4178091 = asin(r4178090);
        double r4178092 = sin(r4178091);
        double r4178093 = r4178092 * r4178088;
        double r4178094 = log1p(r4178093);
        double r4178095 = expm1(r4178094);
        double r4178096 = r4178095 * r4178093;
        double r4178097 = sqrt(r4178096);
        double r4178098 = r4178097 * r4178097;
        double r4178099 = r4178098 * r4178093;
        double r4178100 = r4178085 - r4178099;
        double r4178101 = fma(r4178092, r4178089, r4178084);
        double r4178102 = fma(r4178093, r4178093, r4178101);
        double r4178103 = r4178100 / r4178102;
        double r4178104 = atan2(r4178082, r4178103);
        double r4178105 = r4178074 + r4178104;
        return r4178105;
}

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. Using strategy rm

3. Applied flip3-- 0.2

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

4. Simplified 0.2

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

5. Simplified 0.2

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

7. Applied expm1-log1p-u 0.2

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

9. Applied add-sqr-sqrt 0.2

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

10. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019163 +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))))))))))