By Donald E. Knuth

The is the fourth quantity in a five-volume sequence on pcs and Typesetting, all authored through Donald E. Knuth.

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**Extra resources for METAFONT the program **

**Sample text**

If there is more than one way to print the result with the optimum number of digits following the decimal point, the closest possible value is given. The invariant relation in the repeat loop is that a sequence of decimal digits yet to be printed will yield the original number if and only if they form a fraction f in the range s − δ ≤ 10 · 216 f < s. We can stop if and only if f = 0 satisfies this condition; the loop will terminate before s can possibly become zero. "); repeat if delta > unity then s ← s + ´100000 − (delta div 2); { round the final digit } print char ("0" + (s div unity )); s ← 10 ∗ (s mod unity ); delta ← delta ∗ 10; until s ≤ delta ; end; end; 104.

44 PART 7: ARITHMETIC WITH SCALED NUMBERS 116. METAFONT §116 Here is a typical example of how the routines above can be used. It computes the function √ 1 1 sin φ)(sin φ − 16 sin θ)(cos θ − cos φ) τ −1 2 + 2 (sin θ − 16 1 √ √ , f (θ, φ) = 1 1 3τ 3 1 + 2 ( 5 − 1) cos θ + 2 (3 − 5 ) cos φ where τ is a scaled “tension” parameter. This is METAFONT’s magic fudge factor for placing the first control point of a curve that starts at an angle θ and ends at an angle φ from the straight path. ) √ The trigonometric quantity to be multiplied by 2 is less than 2.

Then x and y are divided by r. procedure n sin cos (z : angle ); { computes a multiple of the sine and cosine } var k: small number ; { loop control variable } q: 0 . 7; { specifies the quadrant } r: fraction ; { magnitude of (x, y) } x, y, t: integer ; { temporary registers } begin while z < 0 do z ← z + three sixty deg ; z ← z mod three sixty deg ; { now 0 ≤ z < three sixty deg } q ← z div forty five deg ; z ← z mod forty five deg ; x ← fraction one ; y ← x; if ¬odd (q) then z ← forty five deg − z; Subtract angle z from (x, y) 147 ; Convert (x, y) to the octant determined by q 146 ; r ← pyth add (x, y); n cos ← make fraction (x, r); n sin ← make fraction (y, r); end; 146.