\[(x * y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)\]
Test:
simple fma test
Bits:
128 bits
Bits error versus x
Bits error versus y
Bits error versus z
Time: 7.5 s
Input Error: 44.8
Output Error: 8.1
Log: ⚲
Profile: 🕒
\(\left((x * y + z)_* - \left(y \cdot x + z\right)\right) - 1\)
  1. Started with
    \[(x * y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)\]
    44.8
  2. Using strategy rm
    44.8
  3. Applied add-cube-cbrt to get
    \[(x * y + z)_* - \left(1 + \color{red}{\left(x \cdot y + z\right)}\right) \leadsto (x * y + z)_* - \left(1 + \color{blue}{{\left(\sqrt[3]{x \cdot y + z}\right)}^3}\right)\]
    45.5
  4. Using strategy rm
    45.5
  5. Applied add-cube-cbrt to get
    \[(x * y + z)_* - \left(1 + {\left(\sqrt[3]{\color{red}{x \cdot y + z}}\right)}^3\right) \leadsto (x * y + z)_* - \left(1 + {\left(\sqrt[3]{\color{blue}{{\left(\sqrt[3]{x \cdot y + z}\right)}^3}}\right)}^3\right)\]
    45.5
  6. Using strategy rm
    45.5
  7. Applied cube-mult to get
    \[(x * y + z)_* - \left(1 + {\left(\sqrt[3]{\color{red}{{\left(\sqrt[3]{x \cdot y + z}\right)}^3}}\right)}^3\right) \leadsto (x * y + z)_* - \left(1 + {\left(\sqrt[3]{\color{blue}{\sqrt[3]{x \cdot y + z} \cdot \left(\sqrt[3]{x \cdot y + z} \cdot \sqrt[3]{x \cdot y + z}\right)}}\right)}^3\right)\]
    45.5
  8. Applied cbrt-prod to get
    \[(x * y + z)_* - \left(1 + {\color{red}{\left(\sqrt[3]{\sqrt[3]{x \cdot y + z} \cdot \left(\sqrt[3]{x \cdot y + z} \cdot \sqrt[3]{x \cdot y + z}\right)}\right)}}^3\right) \leadsto (x * y + z)_* - \left(1 + {\color{blue}{\left(\sqrt[3]{\sqrt[3]{x \cdot y + z}} \cdot \sqrt[3]{\sqrt[3]{x \cdot y + z} \cdot \sqrt[3]{x \cdot y + z}}\right)}}^3\right)\]
    45.6
  9. Applied taylor to get
    \[(x * y + z)_* - \left(1 + {\left(\sqrt[3]{\sqrt[3]{x \cdot y + z}} \cdot \sqrt[3]{\sqrt[3]{x \cdot y + z} \cdot \sqrt[3]{x \cdot y + z}}\right)}^3\right) \leadsto (x * y + z)_* - \left({\left(\sqrt[3]{{\left(\sqrt[3]{y \cdot x + z}\right)}^2} \cdot \sqrt[3]{\sqrt[3]{y \cdot x + z}}\right)}^3 + 1\right)\]
    45.6
  10. Taylor expanded around 0 to get
    \[\color{red}{(x * y + z)_* - \left({\left(\sqrt[3]{{\left(\sqrt[3]{y \cdot x + z}\right)}^2} \cdot \sqrt[3]{\sqrt[3]{y \cdot x + z}}\right)}^3 + 1\right)} \leadsto \color{blue}{(x * y + z)_* - \left({\left(\sqrt[3]{{\left(\sqrt[3]{y \cdot x + z}\right)}^2} \cdot \sqrt[3]{\sqrt[3]{y \cdot x + z}}\right)}^3 + 1\right)}\]
    45.6
  11. Applied simplify to get
    \[(x * y + z)_* - \left({\left(\sqrt[3]{{\left(\sqrt[3]{y \cdot x + z}\right)}^2} \cdot \sqrt[3]{\sqrt[3]{y \cdot x + z}}\right)}^3 + 1\right) \leadsto \left((x * y + z)_* - {\left(\sqrt[3]{y \cdot x + z}\right)}^2 \cdot \sqrt[3]{y \cdot x + z}\right) - 1\]
    38.7
  12. Applied final simplification
  13. Applied simplify to get
    \[\color{red}{\left((x * y + z)_* - {\left(\sqrt[3]{y \cdot x + z}\right)}^2 \cdot \sqrt[3]{y \cdot x + z}\right) - 1} \leadsto \color{blue}{\left((x * y + z)_* - \left(y \cdot x + z\right)\right) - 1}\]
    8.1

Original test:


(lambda ((x default) (y default) (z default))
  #:name "simple fma test"
  (- (fma x y z) (+ 1 (+ (* x y) z)))
  #:target
  -1)