\[(x * y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)\]
Test:
simple fma test
Bits:
128 bits
Bits error versus x
Bits error versus y
Bits error versus z
Time: 9.9 s
Input Error: 44.4
Output Error: 17.6
Log: ⚲
Profile: 🕒
\(\begin{cases} (\left(\frac{-1}{x}\right) * \left(\frac{-1}{y}\right) + \left(\frac{-1}{z}\right))_* - 1 & \text{when } z \le -1.0261015636691071 \cdot 10^{+29} \\ (x * y + z)_* - \left(1 + {\left(\sqrt[3]{x \cdot y + z}\right)}^{3}\right) & \text{when } z \le 8618.608349889819 \\ (\left(\frac{-1}{x}\right) * \left(\frac{-1}{y}\right) + \left(\frac{-1}{z}\right))_* - 1 & \text{otherwise} \end{cases}\)

    if z < -1.0261015636691071e+29 or 8618.608349889819 < 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

    if -1.0261015636691071e+29 < z < 8618.608349889819

    1. Started with
      \[(x * y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)\]
      30.4
    2. Using strategy rm
      30.4
    3. Applied add-cube-cbrt to get
      \[(x * y + z)_* - \left(1 + \color{red}{\left(x \cdot y + z\right)}\right) \leadsto (x * y + z)_* - \left(1 + \color{blue}{{\left(\sqrt[3]{x \cdot y + z}\right)}^3}\right)\]
      31.2
    4. Using strategy rm
      31.2
    5. Applied pow3 to get
      \[(x * y + z)_* - \left(1 + \color{red}{{\left(\sqrt[3]{x \cdot y + z}\right)}^3}\right) \leadsto (x * y + z)_* - \left(1 + \color{blue}{{\left(\sqrt[3]{x \cdot y + z}\right)}^{3}}\right)\]
      31.2
  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)