\[(x * y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)\]
Test:
simple fma test
Bits:
128 bits
Bits error versus x
Bits error versus y
Bits error versus z
Time: 14.6 s
Input Error: 45.4
Output Error: 10.1
Log: ⚲
Profile: 🕒
\(\begin{cases} \log \left(e^{\left(\left((x * y + z)_* - z\right) - y \cdot x\right) - 1}\right) & \text{when } x \le -9.644871466934235 \cdot 10^{+231} \\ (\left(\frac{1}{x}\right) * \left(\frac{1}{y}\right) + \left(\frac{1}{z}\right))_* - 1 & \text{when } x \le -2.9335963728403624 \cdot 10^{+168} \\ \log \left(e^{\left(\left((x * y + z)_* - z\right) - y \cdot x\right) - 1}\right) & \text{when } x \le 1.65537590089569 \cdot 10^{+195} \\ (\left(\frac{1}{x}\right) * \left(\frac{1}{y}\right) + \left(\frac{1}{z}\right))_* - 1 & \text{otherwise} \end{cases}\)

    if x < -9.644871466934235e+231 or -2.9335963728403624e+168 < x < 1.65537590089569e+195

    1. Started with
      \[(x * y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)\]
      42.7
    2. Using strategy rm
      42.7
    3. Applied add-cbrt-cube to get
      \[(x * y + z)_* - \left(1 + \color{red}{\left(x \cdot y + z\right)}\right) \leadsto (x * y + z)_* - \left(1 + \color{blue}{\sqrt[3]{{\left(x \cdot y + z\right)}^3}}\right)\]
      43.3
    4. Using strategy rm
      43.3
    5. Applied add-log-exp to get
      \[\color{red}{(x * y + z)_* - \left(1 + \sqrt[3]{{\left(x \cdot y + z\right)}^3}\right)} \leadsto \color{blue}{\log \left(e^{(x * y + z)_* - \left(1 + \sqrt[3]{{\left(x \cdot y + z\right)}^3}\right)}\right)}\]
      43.4
    6. Applied simplify to get
      \[\log \color{red}{\left(e^{(x * y + z)_* - \left(1 + \sqrt[3]{{\left(x \cdot y + z\right)}^3}\right)}\right)} \leadsto \log \color{blue}{\left(e^{\left((x * y + z)_* - z\right) - \left(y \cdot x + 1\right)}\right)}\]
      26.2
    7. Using strategy rm
      26.2
    8. Applied associate--r+ to get
      \[\log \left(e^{\color{red}{\left((x * y + z)_* - z\right) - \left(y \cdot x + 1\right)}}\right) \leadsto \log \left(e^{\color{blue}{\left(\left((x * y + z)_* - z\right) - y \cdot x\right) - 1}}\right)\]
      11.7

    if -9.644871466934235e+231 < x < -2.9335963728403624e+168 or 1.65537590089569e+195 < x

    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-cbrt-cube to get
      \[(x * y + z)_* - \left(1 + \color{red}{\left(x \cdot y + z\right)}\right) \leadsto (x * y + z)_* - \left(1 + \color{blue}{\sqrt[3]{{\left(x \cdot y + z\right)}^3}}\right)\]
      62.5
    4. Applied taylor to get
      \[(x * y + z)_* - \left(1 + \sqrt[3]{{\left(x \cdot y + z\right)}^3}\right) \leadsto (\left(\frac{1}{x}\right) * \left(\frac{1}{y}\right) + \left(\frac{1}{z}\right))_* - 1\]
      0.0
    5. Taylor expanded around inf to get
      \[\color{red}{(\left(\frac{1}{x}\right) * \left(\frac{1}{y}\right) + \left(\frac{1}{z}\right))_* - 1} \leadsto \color{blue}{(\left(\frac{1}{x}\right) * \left(\frac{1}{y}\right) + \left(\frac{1}{z}\right))_* - 1}\]
      0.0
  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)