For values far from 1, log_2 will be dominated by the integer portion, which IEEE FP representation gives you for free, so it's not surprising that outputs even

For values far from 1, log_2 will be dominated by the integer portion, which IEEE FP representation gives you for free, so it's not surprising that outputs eventually look right. I think the problem is that the approximation works with (x - 1), but only the "significand >= 1.5" branch subtracts that offset. Really, if you need speed, I'd probably start with an approximation over [1, 2)... Re-implementation in SBCL: CL-USER> (defun approx (x) (let ((o (1- x))) (values (/ (+ (* o o 0.338953) (* o 2.198599)) (+ o 1.523692)) (log x 2.0)))) CL-USER> (loop for x from 0.70 upto 1.51 by 0.1 do (format t "~,3F: ~{~F ~F~}~%" x (multiple-value-list (approx x)))) 0.700: -0.51407874 -0.5145732 0.800: -0.32194924 -0.32192805 0.900: -0.15204856 -0.15200303 1.000: 0.0 0.0 1.100: 0.13749498 0.13750356 1.200: 0.26296926 0.26303446 1.300: 0.3784003 0.37851173 1.400: 0.48535433 0.48542693 1.500: 0.585088 0.5849626 That's in the right ballpark for [0,7, 1.5] CL-USER> (defun almost-log2 (x) (let* ((bits (sb-kernel:single-float-bits x)) (exp (ash bits -23))) (values (if (logbitp 22 bits) (+ (- exp 126) (approx (sb-kernel:make-single-float (dpb 126 (byte 9 23) bits)))) (+ (- exp 127) (approx (sb-kernel:make-single-float (dpb 127 (byte 9 23) bits))))) (log x 2.0)))) CL-USER> (loop for x from 0.1 upto 10.1 by 0.1 do (format t "~,3F: ~{~,8F ~,8F~}~%" x (multiple-value-list (almost-log2 x)))) 0.100: -3.32194920 -3.32192800 0.200: -2.32194920 -2.32192800 0.300: -1.73703070 -1.73696570 0.400: -1.32194920 -1.32192800 0.500: -1.00000000 -1.00000000 0.600: -0.73703074 -0.73696554 0.700: -0.51464570 -0.51457310 0.800: -0.32194915 -0.32192796 0.900: -0.15204845 -0.15200295 1.000: 0.00000017 0.00000017 1.100: 0.13749513 0.13750371 1.200: 0.26296943 0.26303460 1.300: 0.37840047 0.37851185 1.400: 0.48535448 0.48542705 1.500: 0.58509207 0.58496270 1.600: 0.67805094 0.67807215 1.700: 0.76547647 0.76553500 1.800: 0.84795165 0.84799720 1.900: 0.92598030 0.92599964 2.000: 1.00000010 1.00000010 2.100: 1.07039330 1.07038940 2.200: 1.13749500 1.13750360 [...] 2.800: 1.48535400 1.48542650 2.900: 1.53605470 1.53605260 3.000: 1.58508750 1.58496210 3.100: 1.63230250 1.63226800 3.200: 1.67805030 1.67807150 [...] 3.800: 1.92597950 1.92599880 3.900: 1.96346550 1.96347340 4.000: 1.99999940 1.99999930 4.100: 2.03562740 2.03562330 4.200: 2.07039260 2.07038880 [...] 5.000: 2.32183340 2.32192750 [...] 6.000: 2.58508730 2.58496170 [...] 9.900: 3.30734100 3.30742860 10.000: 3.32183430 3.32192850