\[(x * y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)\]
Test:
simple fma test
Bits:
128 bits
Bits error versus x
Bits error versus y
Bits error versus z
Time: 10.1 s
Input Error: 19.4
Output Error: 5.1
Log: ⚲
Profile: 🕒
\(\begin{cases} (\left(\frac{-1}{x}\right) * \left(\frac{-1}{y}\right) + \left(\frac{-1}{z}\right))_* - 1 & \text{when } z \le -1.7226329f+15 \\ \left((x * y + z)_* - 1\right) - \left(x \cdot y + z\right) & \text{when } z \le -4.1334874f-06 \\ \left((x * y + z)_* - y \cdot x\right) - \left(z + 1\right) & \text{when } z \le 583.3768f0 \\ \left((x * y + z)_* - 1\right) - \left(x \cdot y + z\right) & \text{when } z \le 1.5366934f+17 \\ (\left(\frac{-1}{x}\right) * \left(\frac{-1}{y}\right) + \left(\frac{-1}{z}\right))_* - 1 & \text{otherwise} \end{cases}\)

    if z < -1.7226329f+15 or 1.5366934f+17 < z

    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
      \[(x * y + z)_* - \left(1 + \color{red}{\left(x \cdot y + z\right)}\right) \leadsto (x * y + z)_* - \left(1 + \color{blue}{{\left(\sqrt[3]{x \cdot y + z}\right)}^3}\right)\]
      30.1
    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
    6. Applied simplify to get
      \[(\left(\frac{-1}{x}\right) * \left(\frac{-1}{y}\right) + \left(\frac{-1}{z}\right))_* - 1 \leadsto (\left(\frac{-1}{x}\right) * \left(\frac{-1}{y}\right) + \left(\frac{-1}{z}\right))_* - 1\]
      0.2
    7. Applied final simplification

    if -1.7226329f+15 < z < -4.1334874f-06 or 583.3768f0 < z < 1.5366934f+17

    1. Started with
      \[(x * y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)\]
      21.3
    2. Using strategy rm
      21.3
    3. Applied associate--r+ to get
      \[\color{red}{(x * y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)} \leadsto \color{blue}{\left((x * y + z)_* - 1\right) - \left(x \cdot y + z\right)}\]
      11.2

    if -4.1334874f-06 < z < 583.3768f0

    1. Started with
      \[(x * y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)\]
      12.6
    2. Using strategy rm
      12.6
    3. Applied associate--r+ to get
      \[\color{red}{(x * y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)} \leadsto \color{blue}{\left((x * y + z)_* - 1\right) - \left(x \cdot y + z\right)}\]
      9.6
    4. Using strategy rm
      9.6
    5. Applied add-log-exp to get
      \[\left((x * y + z)_* - 1\right) - \color{red}{\left(x \cdot y + z\right)} \leadsto \left((x * y + z)_* - 1\right) - \color{blue}{\log \left(e^{x \cdot y + z}\right)}\]
      15.0
    6. Applied add-log-exp to get
      \[\color{red}{\left((x * y + z)_* - 1\right)} - \log \left(e^{x \cdot y + z}\right) \leadsto \color{blue}{\log \left(e^{(x * y + z)_* - 1}\right)} - \log \left(e^{x \cdot y + z}\right)\]
      15.4
    7. Applied diff-log to get
      \[\color{red}{\log \left(e^{(x * y + z)_* - 1}\right) - \log \left(e^{x \cdot y + z}\right)} \leadsto \color{blue}{\log \left(\frac{e^{(x * y + z)_* - 1}}{e^{x \cdot y + z}}\right)}\]
      15.3
    8. Applied simplify to get
      \[\log \color{red}{\left(\frac{e^{(x * y + z)_* - 1}}{e^{x \cdot y + z}}\right)} \leadsto \log \color{blue}{\left(e^{\left((x * y + z)_* - \left(1 + z\right)\right) - y \cdot x}\right)}\]
      9.8
    9. Applied taylor to get
      \[\log \left(e^{\left((x * y + z)_* - \left(1 + z\right)\right) - y \cdot x}\right) \leadsto \log \left(e^{(x * y + z)_* - \left(y \cdot x + \left(1 + z\right)\right)}\right)\]
      12.6
    10. Taylor expanded around 0 to get
      \[\log \left(e^{\color{red}{(x * y + z)_* - \left(y \cdot x + \left(1 + z\right)\right)}}\right) \leadsto \log \left(e^{\color{blue}{(x * y + z)_* - \left(y \cdot x + \left(1 + z\right)\right)}}\right)\]
      12.6
    11. Applied simplify to get
      \[\log \left(e^{(x * y + z)_* - \left(y \cdot x + \left(1 + z\right)\right)}\right) \leadsto \left((x * y + z)_* - y \cdot x\right) - \left(z + 1\right)\]
      4.8
    12. Applied final simplification
  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)