\[(x * y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)\]
Test:
simple fma test
Bits:
128 bits
Bits error versus x
Bits error versus y
Bits error versus z
Time: 19.4 s
Input Error: 44.9
Output Error: 9.7
Log: ⚲
Profile: 🕒
\(\begin{cases} (\left(\frac{1}{x}\right) * \left(\frac{1}{y}\right) + \left(\frac{1}{z}\right))_* - 1 & \text{when } z \le -1.6846166027071804 \cdot 10^{+162} \\ \log \left(e^{\left(\left((x * y + z)_* - z\right) - y \cdot x\right) - 1}\right) & \text{when } z \le 2.8499054335704732 \cdot 10^{+197} \\ (\left(\frac{1}{x}\right) * \left(\frac{1}{y}\right) + \left(\frac{1}{z}\right))_* - 1 & \text{otherwise} \end{cases}\)

    if z < -1.6846166027071804e+162 or 2.8499054335704732e+197 < z

    1. Started with
      \[(x * y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)\]
      62.5
    2. Using strategy rm
      62.5
    3. Applied add-cbrt-cube to get
      \[(x * y + z)_* - \left(1 + \color{red}{\left(x \cdot y + z\right)}\right) \leadsto (x * y + z)_* - \left(1 + \color{blue}{\sqrt[3]{{\left(x \cdot y + z\right)}^3}}\right)\]
      62.7
    4. Applied taylor to get
      \[(x * y + z)_* - \left(1 + \sqrt[3]{{\left(x \cdot y + z\right)}^3}\right) \leadsto (\left(\frac{1}{x}\right) * \left(\frac{1}{y}\right) + \left(\frac{1}{z}\right))_* - 1\]
      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 -1.6846166027071804e+162 < z < 2.8499054335704732e+197

    1. Started with
      \[(x * y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)\]
      41.1
    2. Using strategy rm
      41.1
    3. Applied add-cbrt-cube to get
      \[(x * y + z)_* - \left(1 + \color{red}{\left(x \cdot y + z\right)}\right) \leadsto (x * y + z)_* - \left(1 + \color{blue}{\sqrt[3]{{\left(x \cdot y + z\right)}^3}}\right)\]
      41.6
    4. Using strategy rm
      41.6
    5. Applied add-log-exp to get
      \[\color{red}{(x * y + z)_* - \left(1 + \sqrt[3]{{\left(x \cdot y + z\right)}^3}\right)} \leadsto \color{blue}{\log \left(e^{(x * y + z)_* - \left(1 + \sqrt[3]{{\left(x \cdot y + z\right)}^3}\right)}\right)}\]
      41.8
    6. Applied simplify to get
      \[\log \color{red}{\left(e^{(x * y + z)_* - \left(1 + \sqrt[3]{{\left(x \cdot y + z\right)}^3}\right)}\right)} \leadsto \log \color{blue}{\left(e^{\left((x * y + z)_* - z\right) - \left(y \cdot x + 1\right)}\right)}\]
      30.7
    7. Using strategy rm
      30.7
    8. Applied associate--r+ to get
      \[\log \left(e^{\color{red}{\left((x * y + z)_* - z\right) - \left(y \cdot x + 1\right)}}\right) \leadsto \log \left(e^{\color{blue}{\left(\left((x * y + z)_* - z\right) - y \cdot x\right) - 1}}\right)\]
      11.8
  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)