\[(x * y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)\]
Test:
simple fma test
Bits:
128 bits
Bits error versus x
Bits error versus y
Bits error versus z
Time: 15.2 s
Input Error: 44.6
Output Error: 27.7
Log: ⚲
Profile: 🕒
\(\begin{cases} \left((\left(\frac{1}{x}\right) * \left(\frac{1}{y}\right) + \left(\frac{1}{z}\right))_* - 1\right) - \left(\frac{1}{y \cdot x} + \frac{1}{z}\right) & \text{when } z \le -4.020529270782478 \cdot 10^{-179} \\ (x * y + z)_* - \left(1 + \sqrt[3]{{\left(x \cdot y + z\right)}^3}\right) & \text{when } z \le 3491808544826742.5 \\ (\left(\frac{-1}{x}\right) * \left(\frac{-1}{y}\right) + \left(\frac{-1}{z}\right))_* - 1 & \text{otherwise} \end{cases}\)

    if z < -4.020529270782478e-179

    1. Started with
      \[(x * y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)\]
      49.0
    2. Using strategy rm
      49.0
    3. Applied add-cube-cbrt to get
      \[(x * y + z)_* - \left(1 + \color{red}{\left(x \cdot y + z\right)}\right) \leadsto (x * y + z)_* - \left(1 + \color{blue}{{\left(\sqrt[3]{x \cdot y + z}\right)}^3}\right)\]
      49.6
    4. Using strategy rm
      49.6
    5. Applied pow3 to get
      \[(x * y + z)_* - \left(1 + \color{red}{{\left(\sqrt[3]{x \cdot y + z}\right)}^3}\right) \leadsto (x * y + z)_* - \left(1 + \color{blue}{{\left(\sqrt[3]{x \cdot y + z}\right)}^{3}}\right)\]
      49.7
    6. Applied taylor to get
      \[(x * y + z)_* - \left(1 + {\left(\sqrt[3]{x \cdot y + z}\right)}^{3}\right) \leadsto (\left(\frac{1}{x}\right) * \left(\frac{1}{y}\right) + \left(\frac{1}{z}\right))_* - \left(1 + {\left(\sqrt[3]{\frac{1}{z} + \frac{1}{y \cdot x}}\right)}^{3}\right)\]
      40.9
    7. Taylor expanded around inf to get
      \[\color{red}{(\left(\frac{1}{x}\right) * \left(\frac{1}{y}\right) + \left(\frac{1}{z}\right))_* - \left(1 + {\left(\sqrt[3]{\frac{1}{z} + \frac{1}{y \cdot x}}\right)}^{3}\right)} \leadsto \color{blue}{(\left(\frac{1}{x}\right) * \left(\frac{1}{y}\right) + \left(\frac{1}{z}\right))_* - \left(1 + {\left(\sqrt[3]{\frac{1}{z} + \frac{1}{y \cdot x}}\right)}^{3}\right)}\]
      40.9
    8. Applied simplify to get
      \[(\left(\frac{1}{x}\right) * \left(\frac{1}{y}\right) + \left(\frac{1}{z}\right))_* - \left(1 + {\left(\sqrt[3]{\frac{1}{z} + \frac{1}{y \cdot x}}\right)}^{3}\right) \leadsto \left((\left(\frac{1}{x}\right) * \left(\frac{1}{y}\right) + \left(\frac{1}{z}\right))_* - 1\right) - \left(\frac{1}{y \cdot x} + \frac{1}{z}\right)\]
      40.5
    9. Applied final simplification

    if -4.020529270782478e-179 < z < 3491808544826742.5

    1. Started with
      \[(x * y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)\]
      29.0
    2. Using strategy rm
      29.0
    3. Applied add-cbrt-cube to get
      \[(x * y + z)_* - \left(1 + \color{red}{\left(x \cdot y + z\right)}\right) \leadsto (x * y + z)_* - \left(1 + \color{blue}{\sqrt[3]{{\left(x \cdot y + z\right)}^3}}\right)\]
      29.7

    if 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
      \[(x * y + z)_* - \left(1 + \color{red}{\left(x \cdot y + z\right)}\right) \leadsto (x * y + z)_* - \left(1 + \color{blue}{{\left(\sqrt[3]{x \cdot y + z}\right)}^3}\right)\]
      62.1
    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
  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)