\[(x * y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)\]
Test:
simple fma test
Bits:
128 bits
Bits error versus x
Bits error versus y
Bits error versus z
Time: 6.3 s
Input Error: 45.1
Output Error: 30.8
Log: ⚲
Profile: 🕒
\(\left(\frac{1}{z} + (\left(\frac{-1}{x}\right) * \left(\frac{-1}{y}\right) + \left(\frac{-1}{z}\right))_*\right) - \left(\frac{1}{y \cdot x} + 1\right)\)
  1. Started with
    \[(x * y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)\]
    45.1
  2. Using strategy rm
    45.1
  3. Applied add-cube-cbrt to get
    \[(x * y + z)_* - \left(1 + \color{red}{\left(x \cdot y + z\right)}\right) \leadsto (x * y + z)_* - \left(1 + \color{blue}{{\left(\sqrt[3]{x \cdot y + z}\right)}^3}\right)\]
    45.7
  4. Using strategy rm
    45.7
  5. Applied pow3 to get
    \[(x * y + z)_* - \left(1 + \color{red}{{\left(\sqrt[3]{x \cdot y + z}\right)}^3}\right) \leadsto (x * y + z)_* - \left(1 + \color{blue}{{\left(\sqrt[3]{x \cdot y + z}\right)}^{3}}\right)\]
    45.7
  6. Applied taylor to get
    \[(x * y + z)_* - \left(1 + {\left(\sqrt[3]{x \cdot y + z}\right)}^{3}\right) \leadsto (\left(\frac{-1}{x}\right) * \left(\frac{-1}{y}\right) + \left(\frac{-1}{z}\right))_* - \left({\left(\sqrt[3]{\frac{1}{y \cdot x} - \frac{1}{z}}\right)}^{3} + 1\right)\]
    45.4
  7. Taylor expanded around -inf to get
    \[\color{red}{(\left(\frac{-1}{x}\right) * \left(\frac{-1}{y}\right) + \left(\frac{-1}{z}\right))_* - \left({\left(\sqrt[3]{\frac{1}{y \cdot x} - \frac{1}{z}}\right)}^{3} + 1\right)} \leadsto \color{blue}{(\left(\frac{-1}{x}\right) * \left(\frac{-1}{y}\right) + \left(\frac{-1}{z}\right))_* - \left({\left(\sqrt[3]{\frac{1}{y \cdot x} - \frac{1}{z}}\right)}^{3} + 1\right)}\]
    45.4
  8. Applied simplify to get
    \[\color{red}{(\left(\frac{-1}{x}\right) * \left(\frac{-1}{y}\right) + \left(\frac{-1}{z}\right))_* - \left({\left(\sqrt[3]{\frac{1}{y \cdot x} - \frac{1}{z}}\right)}^{3} + 1\right)} \leadsto \color{blue}{\left(\frac{1}{z} + (\left(\frac{-1}{x}\right) * \left(\frac{-1}{y}\right) + \left(\frac{-1}{z}\right))_*\right) - \left(\frac{1}{y \cdot x} + 1\right)}\]
    30.8

Original test:


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