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12.1.3.3 Rule of Float Substitutability

When the arguments to an irrational mathematical *function* are all *rational* and the true mathematical result is also (mathematically) rational, then unless otherwise noted an implementation is free to return either an accurate *rational* result or a *single float* approximation. If the arguments are all *rational* but the result cannot be expressed as a *rational* number, then a *single float* approximation is always returned.

If the arguments to an irrational mathematical *function* are all of type `(or rational (complex rational))` and the true mathematical result is (mathematically) a complex number with rational real and imaginary parts, then unless otherwise noted an implementation is free to return either an accurate result of type `(or rational (complex rational))` or a *single float* (permissible only if the imaginary part of the true mathematical result is zero) or `(complex single-float)`. If the arguments are all of type `(or rational (complex rational))` but the result cannot be expressed as a *rational* or *complex rational*, then the returned value will be of *type* **single-float** (permissible only if the imaginary part of the true mathematical result is zero) or `(complex single-float)`.

Float substitutability applies neither to the rational *functions* **+**, **-**, *****, and **/** nor to the related *operators* **1+**, **1-**, **incf**, **decf**, and **conjugate**. For rational *functions*, if all arguments are *rational*, then the result is *rational*; if all arguments are of type `(or rational (complex rational))`, then the result is of type `(or rational (complex rational))`.

Function Sample Results
abs (abs #c(3 4)) => 5 or 5.0
acos (acos 1) => 0 or 0.0
acosh (acosh 1) => 0 or 0.0
asin (asin 0) => 0 or 0.0
asinh (asinh 0) => 0 or 0.0
atan (atan 0) => 0 or 0.0
atanh (atanh 0) => 0 or 0.0
cis (cis 0) => 1 or #c(1.0 0.0)
cos (cos 0) => 1 or 1.0
cosh (cosh 0) => 1 or 1.0
exp (exp 0) => 1 or 1.0
expt (expt 8 1/3) => 2 or 2.0
log (log 1) => 0 or 0.0
(log 8 2) => 3 or 3.0
phase (phase 7) => 0 or 0.0
signum (signum #c(3 4)) => #c(3/5 4/5) or #c(0.6 0.8)
sin (sin 0) => 0 or 0.0
sinh (sinh 0) => 0 or 0.0
sqrt (sqrt 4) => 2 or 2.0
(sqrt 9/16) => 3/4 or 0.75
tan (tan 0) => 0 or 0.0
tanh (tanh 0) => 0 or 0.0

**Figure 12-8. Functions Affected by Rule of Float Substitutability**

The following X3J13 cleanup issue, *not part of the specification*, applies to this section:

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