\[(x * y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)\]
Test:
simple fma test
Bits:
128 bits
Bits error versus x
Bits error versus y
Bits error versus z
Time: 11.4 s
Input Error: 45.4
Output Error: 22.2
Log: ⚲
Profile: 🕒
\(\begin{cases} \left((\left(\frac{-1}{x}\right) * \left(\frac{-1}{y}\right) + \left(\frac{-1}{z}\right))_* - \frac{\frac{1}{x}}{y}\right) - 1 & \text{when } z \le -809543.3727046504 \\ {\left(\sqrt[3]{(x * y + z)_* - \left(1 + {\left(\sqrt[3]{x \cdot y + z}\right)}^3\right)}\right)}^3 & \text{when } z \le 3.932270195760501 \cdot 10^{-13} \\ \left((\left(\frac{-1}{x}\right) * \left(\frac{-1}{y}\right) + \left(\frac{-1}{z}\right))_* - \frac{\frac{1}{x}}{y}\right) - 1 & \text{otherwise} \end{cases}\)

    if z < -809543.3727046504 or 3.932270195760501e-13 < z

    1. Started with
      \[(x * y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)\]
      61.4
    2. Using strategy rm
      61.4
    3. Applied add-exp-log to get
      \[(x * y + z)_* - \left(1 + \color{red}{\left(x \cdot y + z\right)}\right) \leadsto (x * y + z)_* - \left(1 + \color{blue}{e^{\log \left(x \cdot y + z\right)}}\right)\]
      62.6
    4. Applied taylor to get
      \[(x * y + z)_* - \left(1 + e^{\log \left(x \cdot y + z\right)}\right) \leadsto (\left(\frac{-1}{x}\right) * \left(\frac{-1}{y}\right) + \left(\frac{-1}{z}\right))_* - \left(e^{-\left(\log y + \log x\right)} + 1\right)\]
      55.1
    5. Taylor expanded around -inf to get
      \[\color{red}{(\left(\frac{-1}{x}\right) * \left(\frac{-1}{y}\right) + \left(\frac{-1}{z}\right))_* - \left(e^{-\left(\log y + \log x\right)} + 1\right)} \leadsto \color{blue}{(\left(\frac{-1}{x}\right) * \left(\frac{-1}{y}\right) + \left(\frac{-1}{z}\right))_* - \left(e^{-\left(\log y + \log x\right)} + 1\right)}\]
      55.1
    6. Applied simplify to get
      \[(\left(\frac{-1}{x}\right) * \left(\frac{-1}{y}\right) + \left(\frac{-1}{z}\right))_* - \left(e^{-\left(\log y + \log x\right)} + 1\right) \leadsto \left((\left(\frac{-1}{x}\right) * \left(\frac{-1}{y}\right) + \left(\frac{-1}{z}\right))_* - \frac{\frac{1}{x}}{y}\right) - 1\]
      14.6
    7. Applied final simplification

    if -809543.3727046504 < z < 3.932270195760501e-13

    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-cube-cbrt to get
      \[\color{red}{(x * y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)} \leadsto \color{blue}{{\left(\sqrt[3]{(x * y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)}\right)}^3}\]
      29.2
    4. Using strategy rm
      29.2
    5. Applied add-cube-cbrt to get
      \[{\left(\sqrt[3]{(x * y + z)_* - \left(1 + \color{red}{\left(x \cdot y + z\right)}\right)}\right)}^3 \leadsto {\left(\sqrt[3]{(x * y + z)_* - \left(1 + \color{blue}{{\left(\sqrt[3]{x \cdot y + z}\right)}^3}\right)}\right)}^3\]
      29.9
  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)