\[(x * y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)\]
Test:
simple fma test
Bits:
128 bits
Bits error versus x
Bits error versus y
Bits error versus z
Time: 10.3 s
Input Error: 44.1
Output Error: 16.4
Log: ⚲
Profile: 🕒
\(\begin{cases} {\left(\sqrt[3]{(\left(\frac{-1}{x}\right) * \left(\frac{-1}{y}\right) + \left(\frac{-1}{z}\right))_* - 1}\right)}^3 & \text{when } z \le -1.0070763208517064 \cdot 10^{+21} \\ (x * y + z)_* - \left(1 + \left(x \cdot y + z\right)\right) & \text{when } z \le 3491808544826742.5 \\ {\left(\sqrt[3]{(\left(\frac{-1}{x}\right) * \left(\frac{-1}{y}\right) + \left(\frac{-1}{z}\right))_* - 1}\right)}^3 & \text{otherwise} \end{cases}\)

    if z < -1.0070763208517064e+21 or 3491808544826742.5 < z

    1. Started with
      \[(x * y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)\]
      62.4
    2. Using strategy rm
      62.4
    3. Applied add-cube-cbrt to get
      \[(x * y + z)_* - \left(1 + \color{red}{\left(x \cdot y + z\right)}\right) \leadsto (x * y + z)_* - \left(1 + \color{blue}{{\left(\sqrt[3]{x \cdot y + z}\right)}^3}\right)\]
      62.2
    4. Applied taylor to get
      \[(x * y + z)_* - \left(1 + {\left(\sqrt[3]{x \cdot y + z}\right)}^3\right) \leadsto (\left(\frac{-1}{x}\right) * \left(\frac{-1}{y}\right) + \left(\frac{-1}{z}\right))_* - 1\]
      0.2
    5. Taylor expanded around -inf to get
      \[\color{red}{(\left(\frac{-1}{x}\right) * \left(\frac{-1}{y}\right) + \left(\frac{-1}{z}\right))_* - 1} \leadsto \color{blue}{(\left(\frac{-1}{x}\right) * \left(\frac{-1}{y}\right) + \left(\frac{-1}{z}\right))_* - 1}\]
      0.2
    6. Using strategy rm
      0.2
    7. Applied add-cube-cbrt to get
      \[\color{red}{(\left(\frac{-1}{x}\right) * \left(\frac{-1}{y}\right) + \left(\frac{-1}{z}\right))_* - 1} \leadsto \color{blue}{{\left(\sqrt[3]{(\left(\frac{-1}{x}\right) * \left(\frac{-1}{y}\right) + \left(\frac{-1}{z}\right))_* - 1}\right)}^3}\]
      0.6

    if -1.0070763208517064e+21 < z < 3491808544826742.5

    1. Started with
      \[(x * y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)\]
      29.3
  1. Removed slow pow expressions

Original test:


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