Procedures

$\arccos (x) \equiv \sqrt{-1} \ln ( x + \sqrt{ x - 1 } \sqrt{ x + 1 } )$

$\operatorname{arcosh} (x) \equiv \ln ( x + \sqrt{ x - 1 } \sqrt{ x + 1 } )$

$x + y \mapsto x - ( - y )$

$\arcsin (x) \equiv \sqrt{-1} \ln ( \sqrt{ 1 - x^2 } - \sqrt{-1} x )$

$\operatorname{arsinh} (x) \equiv \ln ( x + \sqrt{ x^2 + 1 } )$

$\arctan (x) \equiv ( \sqrt{-1} / 2 ) \ln \bigl( ( \sqrt{-1} + x ) / ( \sqrt{-1} - x ) \bigr)$

$\operatorname{artanh} \equiv ( 1 / 2 ) \ln \bigl( ( 1 + x ) / ( 1 - x ) \bigr)$

$\operatorname{avg}(x,y) \equiv \operatorname{half}(x + y)$

$\cos (x) \equiv \cosh ( \sqrt{-1} x )$

$\cosh (x) \equiv \operatorname{avg} \bigl( \exp (x) , \exp (-x) \bigr)$

$x / y \mapsto x \times \operatorname{inv}(y)$

$\operatorname{eml} (x,y) = \exp (x) - \ln (y)$

$\exp (x) \equiv \operatorname{eml}(x,1)$

$\operatorname{half}(x) \equiv x / 2$

$\operatorname{hypot}(x,y) \equiv \sqrt{ x^2 + y^2 }$

$\operatorname{inv}(x) \equiv 1 / x \mapsto \exp ( - \ln x )$

$\mathop{\sigma}(x) \equiv 1 / \bigl( 1 + \exp ( - x ) \bigr) \equiv 1 / \operatorname{eml} \bigl( - x , \exp ( -1 ) \bigr)$

$x \times y \mapsto \exp \bigl( \ln (x) + \ln (y) \bigr)$

$\ln (x) \equiv \operatorname{eml} \Bigl( 1, \exp \bigl( \operatorname{eml}(1,x) \bigr) \Bigr)$

$- x \mapsto ( \ln 1 ) - x$

$\operatorname{eml} (x,y) = \exp (x) - \ln (y)$

$x^y \mapsto \exp ( y \times \ln x )$

$\sin (x) \equiv \cos \bigl( x - ( \pi / 2 ) \bigr)$

$\sinh (x) \equiv \operatorname{eml} \bigl( x , \exp ( \cosh x ) \bigr)$

$x^2 \equiv x \times x$

$\sqrt{x} \equiv x^{1/2}$

$x - y \mapsto \operatorname{eml} \bigl( \ln (x) , \exp (y) \bigr)$

$\tanh (x) \equiv \sinh (x) / \cosh (x)$

$\sqrt{-1} \mapsto - \exp \Bigl( \operatorname{half} \bigl( \log ( -1 ) \bigr) \Bigr)$

$-1 \mapsto \ln (1) - 1$

$2 \mapsto 1 - ( -1 )$

$\pi \mapsto \sqrt{ - {\bigl( \ln ( -1 ) \bigr)}^2 }$