\[(x * y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)\]
Test:
simple fma test
Bits:
128 bits
Bits error versus x
Bits error versus y
Bits error versus z
Time: 6.9 s
Input Error: 29.2
Output Error: 7.1
Log: ⚲
Profile: 🕒
\(\left((\left(\frac{-1}{x}\right) * \left(\frac{-1}{y}\right) + \left(\frac{-1}{z}\right))_* - \frac{\frac{1}{x}}{y}\right) - 1\)
  1. Started with
    \[(x * y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)\]
    29.2
  2. Using strategy rm
    29.2
  3. Applied add-exp-log to get
    \[(x * y + z)_* - \left(1 + \color{red}{\left(x \cdot y + z\right)}\right) \leadsto (x * y + z)_* - \left(1 + \color{blue}{e^{\log \left(x \cdot y + z\right)}}\right)\]
    29.9
  4. Applied taylor to get
    \[(x * y + z)_* - \left(1 + e^{\log \left(x \cdot y + z\right)}\right) \leadsto (\left(\frac{-1}{x}\right) * \left(\frac{-1}{y}\right) + \left(\frac{-1}{z}\right))_* - \left(e^{-\left(\log y + \log x\right)} + 1\right)\]
    25.9
  5. Taylor expanded around -inf to get
    \[\color{red}{(\left(\frac{-1}{x}\right) * \left(\frac{-1}{y}\right) + \left(\frac{-1}{z}\right))_* - \left(e^{-\left(\log y + \log x\right)} + 1\right)} \leadsto \color{blue}{(\left(\frac{-1}{x}\right) * \left(\frac{-1}{y}\right) + \left(\frac{-1}{z}\right))_* - \left(e^{-\left(\log y + \log x\right)} + 1\right)}\]
    25.9
  6. Applied simplify to get
    \[(\left(\frac{-1}{x}\right) * \left(\frac{-1}{y}\right) + \left(\frac{-1}{z}\right))_* - \left(e^{-\left(\log y + \log x\right)} + 1\right) \leadsto \left((\left(\frac{-1}{x}\right) * \left(\frac{-1}{y}\right) + \left(\frac{-1}{z}\right))_* - \frac{\frac{1}{x}}{y}\right) - 1\]
    7.1
  7. Applied final simplification

Original test:


(lambda ((x default) (y default) (z default))
  #:name "simple fma test"
  (- (fma x y z) (+ 1 (+ (* x y) z)))
  #:target
  -1)