\[(x * y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)\]
Test:
simple fma test
Bits:
128 bits
Bits error versus x
Bits error versus y
Bits error versus z
Time: 13.2 s
Input Error: 44.3
Output Error: 16.3
Log: ⚲
Profile: 🕒
\(\begin{cases} (\left(\frac{1}{x}\right) * \left(\frac{1}{y}\right) + \left(\frac{1}{z}\right))_* - 1 & \text{when } z \le -809543.3727046504 \\ {\left(\sqrt[3]{(x * y + z)_* - \left(1 + {\left(\sqrt[3]{{\left(\sqrt[3]{x \cdot y + z}\right)}^3}\right)}^3\right)}\right)}^3 & \text{when } z \le 18533971.955403153 \\ (\left(\frac{-1}{x}\right) * \left(\frac{-1}{y}\right) + \left(\frac{-1}{z}\right))_* - 1 & \text{otherwise} \end{cases}\)

    if z < -809543.3727046504

    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
      \[\color{red}{(x * y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)} \leadsto \color{blue}{{\left(\sqrt[3]{(x * y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)}\right)}^3}\]
      62.4
    4. Using strategy rm
      62.4
    5. Applied add-cube-cbrt to get
      \[{\left(\sqrt[3]{(x * y + z)_* - \left(1 + \color{red}{\left(x \cdot y + z\right)}\right)}\right)}^3 \leadsto {\left(\sqrt[3]{(x * y + z)_* - \left(1 + \color{blue}{{\left(\sqrt[3]{x \cdot y + z}\right)}^3}\right)}\right)}^3\]
      62.3
    6. Applied taylor to get
      \[{\left(\sqrt[3]{(x * y + z)_* - \left(1 + {\left(\sqrt[3]{x \cdot y + z}\right)}^3\right)}\right)}^3 \leadsto {\left(\sqrt[3]{(\left(\frac{1}{x}\right) * \left(\frac{1}{y}\right) + \left(\frac{1}{z}\right))_* - 1}\right)}^3\]
      0.6
    7. Taylor expanded around inf to get
      \[{\left(\sqrt[3]{\color{red}{(\left(\frac{1}{x}\right) * \left(\frac{1}{y}\right) + \left(\frac{1}{z}\right))_* - 1}}\right)}^3 \leadsto {\left(\sqrt[3]{\color{blue}{(\left(\frac{1}{x}\right) * \left(\frac{1}{y}\right) + \left(\frac{1}{z}\right))_* - 1}}\right)}^3\]
      0.6
    8. Applied simplify to get
      \[\color{red}{{\left(\sqrt[3]{(\left(\frac{1}{x}\right) * \left(\frac{1}{y}\right) + \left(\frac{1}{z}\right))_* - 1}\right)}^3} \leadsto \color{blue}{(\left(\frac{1}{x}\right) * \left(\frac{1}{y}\right) + \left(\frac{1}{z}\right))_* - 1}\]
      0.2

    if -809543.3727046504 < z < 18533971.955403153

    1. Started with
      \[(x * y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)\]
      29.1
    2. Using strategy rm
      29.1
    3. Applied add-cube-cbrt to get
      \[\color{red}{(x * y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)} \leadsto \color{blue}{{\left(\sqrt[3]{(x * y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)}\right)}^3}\]
      29.1
    4. Using strategy rm
      29.1
    5. Applied add-cube-cbrt to get
      \[{\left(\sqrt[3]{(x * y + z)_* - \left(1 + \color{red}{\left(x \cdot y + z\right)}\right)}\right)}^3 \leadsto {\left(\sqrt[3]{(x * y + z)_* - \left(1 + \color{blue}{{\left(\sqrt[3]{x \cdot y + z}\right)}^3}\right)}\right)}^3\]
      29.8
    6. Using strategy rm
      29.8
    7. Applied add-cube-cbrt to get
      \[{\left(\sqrt[3]{(x * y + z)_* - \left(1 + {\left(\sqrt[3]{\color{red}{x \cdot y + z}}\right)}^3\right)}\right)}^3 \leadsto {\left(\sqrt[3]{(x * y + z)_* - \left(1 + {\left(\sqrt[3]{\color{blue}{{\left(\sqrt[3]{x \cdot y + z}\right)}^3}}\right)}^3\right)}\right)}^3\]
      29.8

    if 18533971.955403153 < 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
      \[\color{red}{(x * y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)} \leadsto \color{blue}{{\left(\sqrt[3]{(x * y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)}\right)}^3}\]
      62.4
    4. Using strategy rm
      62.4
    5. Applied add-cube-cbrt to get
      \[{\left(\sqrt[3]{(x * y + z)_* - \left(1 + \color{red}{\left(x \cdot y + z\right)}\right)}\right)}^3 \leadsto {\left(\sqrt[3]{(x * y + z)_* - \left(1 + \color{blue}{{\left(\sqrt[3]{x \cdot y + z}\right)}^3}\right)}\right)}^3\]
      62.2
    6. Applied taylor to get
      \[{\left(\sqrt[3]{(x * y + z)_* - \left(1 + {\left(\sqrt[3]{x \cdot y + z}\right)}^3\right)}\right)}^3 \leadsto {\left(\sqrt[3]{(\left(\frac{-1}{x}\right) * \left(\frac{-1}{y}\right) + \left(\frac{-1}{z}\right))_* - 1}\right)}^3\]
      0.6
    7. Taylor expanded around -inf to get
      \[{\left(\sqrt[3]{\color{red}{(\left(\frac{-1}{x}\right) * \left(\frac{-1}{y}\right) + \left(\frac{-1}{z}\right))_* - 1}}\right)}^3 \leadsto {\left(\sqrt[3]{\color{blue}{(\left(\frac{-1}{x}\right) * \left(\frac{-1}{y}\right) + \left(\frac{-1}{z}\right))_* - 1}}\right)}^3\]
      0.6
    8. Applied simplify to get
      \[\color{red}{{\left(\sqrt[3]{(\left(\frac{-1}{x}\right) * \left(\frac{-1}{y}\right) + \left(\frac{-1}{z}\right))_* - 1}\right)}^3} \leadsto \color{blue}{(\left(\frac{-1}{x}\right) * \left(\frac{-1}{y}\right) + \left(\frac{-1}{z}\right))_* - 1}\]
      0.2
  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)