## 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: