Function tan

src/compute-engine/interval/trigonometric.ts:90–118

(x: Interval | IntervalResult)

/**
 * Tangent of an interval.
 *
 * Has singularities at pi/2 + n*pi. Within a single branch,
 * tan is monotonically increasing.
 */
export function tan(x: Interval | IntervalResult): IntervalResult {
  const unwrapped = unwrapOrPropagate(x);
  if (!Array.isArray(unwrapped)) return unwrapped;
  const [xVal] = unwrapped;
  // Case 1: Interval spans a full period - certainly crosses a singularity
  if (xVal.hi - xVal.lo >= PI) {
    return { kind: 'singular' };
  }

  // Case 2: Check if interval contains a pole at pi/2 + n*pi
  if (containsExtremum(xVal, HALF_PI, PI)) {
    // Find the pole location for refinement hints
    const n = Math.ceil((xVal.lo - HALF_PI) / PI);
    const poleAt = HALF_PI + n * PI;
    return { kind: 'singular', at: poleAt };
  }

  // Case 3: Safe interval - tan is monotonic on this branch
  const tanLo = Math.tan(xVal.lo);
  const tanHi = Math.tan(xVal.hi);

  // Sanity check: if results have opposite signs with large magnitude,
  // we may have crossed a branch due to floating-point error
  if ((tanLo > 1e10 && tanHi < -1e10) || (tanLo < -1e10 && tanHi > 1e10)) {
    return { kind: 'singular' };
  }