Average Error: 0.2 → 0.2
Time: 51.9s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\sqrt[3]{\log \left(\frac{e^{\cos delta}}{e^{\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)\right)}}\right) \cdot \left(\left(\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)\right)\right) \cdot \left(\cos delta - \left(\left(\sin \phi_1 \cdot \sin \phi_1\right) \cdot \cos delta + \left(\left(\cos \phi_1 \cdot \sin \phi_1\right) \cdot \cos theta\right) \cdot \sin delta\right)\right)\right)}}\]
\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)}{\sqrt[3]{\log \left(\frac{e^{\cos delta}}{e^{\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)\right)}}\right) \cdot \left(\left(\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)\right)\right) \cdot \left(\cos delta - \left(\left(\sin \phi_1 \cdot \sin \phi_1\right) \cdot \cos delta + \left(\left(\cos \phi_1 \cdot \sin \phi_1\right) \cdot \cos theta\right) \cdot \sin delta\right)\right)\right)}}
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r1957519 = lambda1;
        double r1957520 = theta;
        double r1957521 = sin(r1957520);
        double r1957522 = delta;
        double r1957523 = sin(r1957522);
        double r1957524 = r1957521 * r1957523;
        double r1957525 = phi1;
        double r1957526 = cos(r1957525);
        double r1957527 = r1957524 * r1957526;
        double r1957528 = cos(r1957522);
        double r1957529 = sin(r1957525);
        double r1957530 = r1957529 * r1957528;
        double r1957531 = r1957526 * r1957523;
        double r1957532 = cos(r1957520);
        double r1957533 = r1957531 * r1957532;
        double r1957534 = r1957530 + r1957533;
        double r1957535 = asin(r1957534);
        double r1957536 = sin(r1957535);
        double r1957537 = r1957529 * r1957536;
        double r1957538 = r1957528 - r1957537;
        double r1957539 = atan2(r1957527, r1957538);
        double r1957540 = r1957519 + r1957539;
        return r1957540;
}
↓
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r1957541 = lambda1;
        double r1957542 = phi1;
        double r1957543 = cos(r1957542);
        double r1957544 = delta;
        double r1957545 = sin(r1957544);
        double r1957546 = theta;
        double r1957547 = sin(r1957546);
        double r1957548 = r1957545 * r1957547;
        double r1957549 = r1957543 * r1957548;
        double r1957550 = cos(r1957544);
        double r1957551 = exp(r1957550);
        double r1957552 = sin(r1957542);
        double r1957553 = r1957550 * r1957552;
        double r1957554 = cos(r1957546);
        double r1957555 = r1957543 * r1957545;
        double r1957556 = r1957554 * r1957555;
        double r1957557 = r1957553 + r1957556;
        double r1957558 = asin(r1957557);
        double r1957559 = sin(r1957558);
        double r1957560 = r1957552 * r1957559;
        double r1957561 = exp(r1957560);
        double r1957562 = r1957551 / r1957561;
        double r1957563 = log(r1957562);
        double r1957564 = r1957550 - r1957560;
        double r1957565 = r1957552 * r1957552;
        double r1957566 = r1957565 * r1957550;
        double r1957567 = r1957543 * r1957552;
        double r1957568 = r1957567 * r1957554;
        double r1957569 = r1957568 * r1957545;
        double r1957570 = r1957566 + r1957569;
        double r1957571 = r1957550 - r1957570;
        double r1957572 = r1957564 * r1957571;
        double r1957573 = r1957563 * r1957572;
        double r1957574 = cbrt(r1957573);
        double r1957575 = atan2(r1957549, r1957574);
        double r1957576 = r1957541 + r1957575;
        return r1957576;
}

Error

Bits error versus lambda1

Bits error versus phi1

Bits error versus phi2

Bits error versus delta

Bits error versus theta

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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 add-cbrt-cube0.2

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

  5. Applied add-log-exp0.2

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

  6. Applied add-log-exp0.2

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

  7. Applied diff-log0.2

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

  8. Taylor expanded around inf 0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\sqrt[3]{\left(\left(\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) \cdot \color{blue}{\left(\cos delta - \left({\left(\sin \phi_1\right)}^{2} \cdot \cos delta + \sin \phi_1 \cdot \left(\cos \phi_1 \cdot \left(\cos theta \cdot \sin delta\right)\right)\right)\right)}\right) \cdot \log \left(\frac{e^{\cos delta}}{e^{\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}}\right)}}\]

  9. Simplified0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2019155 
(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))))))))))