Destination given bearing on a great circle

Average Error: 0.2 → 0.2

Time: 43.7s

Precision: 64

• unused variable phi2

Math

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

↓

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

C

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r1737091 = lambda1;
        double r1737092 = theta;
        double r1737093 = sin(r1737092);
        double r1737094 = delta;
        double r1737095 = sin(r1737094);
        double r1737096 = r1737093 * r1737095;
        double r1737097 = phi1;
        double r1737098 = cos(r1737097);
        double r1737099 = r1737096 * r1737098;
        double r1737100 = cos(r1737094);
        double r1737101 = sin(r1737097);
        double r1737102 = r1737101 * r1737100;
        double r1737103 = r1737098 * r1737095;
        double r1737104 = cos(r1737092);
        double r1737105 = r1737103 * r1737104;
        double r1737106 = r1737102 + r1737105;
        double r1737107 = asin(r1737106);
        double r1737108 = sin(r1737107);
        double r1737109 = r1737101 * r1737108;
        double r1737110 = r1737100 - r1737109;
        double r1737111 = atan2(r1737099, r1737110);
        double r1737112 = r1737091 + r1737111;
        return r1737112;
}

↓

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r1737113 = phi1;
        double r1737114 = cos(r1737113);
        double r1737115 = theta;
        double r1737116 = sin(r1737115);
        double r1737117 = r1737114 * r1737116;
        double r1737118 = delta;
        double r1737119 = sin(r1737118);
        double r1737120 = r1737117 * r1737119;
        double r1737121 = cos(r1737118);
        double r1737122 = r1737121 * r1737121;
        double r1737123 = r1737122 * r1737121;
        double r1737124 = cos(r1737115);
        double r1737125 = r1737114 * r1737124;
        double r1737126 = sin(r1737113);
        double r1737127 = r1737126 * r1737121;
        double r1737128 = fma(r1737125, r1737119, r1737127);
        double r1737129 = asin(r1737128);
        double r1737130 = sin(r1737129);
        double r1737131 = r1737130 * r1737126;
        double r1737132 = r1737131 * r1737131;
        double r1737133 = cbrt(r1737132);
        double r1737134 = r1737133 * r1737133;
        double r1737135 = r1737133 * r1737134;
        double r1737136 = r1737135 * r1737131;
        double r1737137 = r1737123 - r1737136;
        double r1737138 = r1737121 + r1737131;
        double r1737139 = r1737138 * r1737131;
        double r1737140 = fma(r1737121, r1737121, r1737139);
        double r1737141 = r1737137 / r1737140;
        double r1737142 = atan2(r1737120, r1737141);
        double r1737143 = lambda1;
        double r1737144 = r1737142 + r1737143;
        return r1737144;
}

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 flip3-- 0.2

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

5. Simplified 0.2

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

6. Simplified 0.2

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

8. Applied add-cube-cbrt 0.2

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

9. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019156 +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))))))))))