Error

Target

Derivation

1. Split input into 2 regimes

2. if (- (- (- (fma x y z) (* y x)) z) 1) < -1.0000000021634312 or -1.0 < (- (- (- (fma x y z) (* y x)) z) 1)

   1. Initial program 62.6

      \[(x \cdot y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)\]
   2. Using strategy rm

   3. Applied add-log-exp63.2

      \[\leadsto (x \cdot y + z)_* - \color{blue}{\log \left(e^{1 + \left(x \cdot y + z\right)}\right)}\]

   4. Applied add-log-exp63.6

      \[\leadsto \color{blue}{\log \left(e^{(x \cdot y + z)_*}\right)} - \log \left(e^{1 + \left(x \cdot y + z\right)}\right)\]

   5. Applied diff-log63.6

      \[\leadsto \color{blue}{\log \left(\frac{e^{(x \cdot y + z)_*}}{e^{1 + \left(x \cdot y + z\right)}}\right)}\]

   6. Applied simplify61.9

      \[\leadsto \log \color{blue}{\left(e^{\left((x \cdot y + z)_* - y \cdot x\right) - \left(z + 1\right)}\right)}\]
   7. Using strategy rm

   8. Applied add-log-exp62.0

      \[\leadsto \log \left(e^{\left((x \cdot y + z)_* - y \cdot x\right) - \color{blue}{\log \left(e^{z + 1}\right)}}\right)\]

   9. Applied add-log-exp63.4

      \[\leadsto \log \left(e^{\left((x \cdot y + z)_* - \color{blue}{\log \left(e^{y \cdot x}\right)}\right) - \log \left(e^{z + 1}\right)}\right)\]

   10. Applied add-log-exp63.6

       \[\leadsto \log \left(e^{\left(\color{blue}{\log \left(e^{(x \cdot y + z)_*}\right)} - \log \left(e^{y \cdot x}\right)\right) - \log \left(e^{z + 1}\right)}\right)\]

   11. Applied diff-log63.6

       \[\leadsto \log \left(e^{\color{blue}{\log \left(\frac{e^{(x \cdot y + z)_*}}{e^{y \cdot x}}\right)} - \log \left(e^{z + 1}\right)}\right)\]

   12. Applied diff-log63.6

       \[\leadsto \log \left(e^{\color{blue}{\log \left(\frac{\frac{e^{(x \cdot y + z)_*}}{e^{y \cdot x}}}{e^{z + 1}}\right)}}\right)\]

   13. Applied simplify62.7

       \[\leadsto \log \left(e^{\log \color{blue}{\left(e^{\left((x \cdot y + z)_* - z\right) - \left(y \cdot x + 1\right)}\right)}}\right)\]
   14. Using strategy rm

   15. Applied associate--r+37.5

       \[\leadsto \log \left(e^{\log \left(e^{\color{blue}{\left(\left((x \cdot y + z)_* - z\right) - y \cdot x\right) - 1}}\right)}\right)\]

   if -1.0000000021634312 < (- (- (- (fma x y z) (* y x)) z) 1) < -1.0

   1. Initial program 39.8

      \[(x \cdot y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)\]
   2. Using strategy rm

   3. Applied add-log-exp42.0

      \[\leadsto (x \cdot y + z)_* - \color{blue}{\log \left(e^{1 + \left(x \cdot y + z\right)}\right)}\]

   4. Applied add-log-exp42.5

      \[\leadsto \color{blue}{\log \left(e^{(x \cdot y + z)_*}\right)} - \log \left(e^{1 + \left(x \cdot y + z\right)}\right)\]

   5. Applied diff-log42.5

      \[\leadsto \color{blue}{\log \left(\frac{e^{(x \cdot y + z)_*}}{e^{1 + \left(x \cdot y + z\right)}}\right)}\]

   6. Applied simplify27.0

      \[\leadsto \log \color{blue}{\left(e^{\left((x \cdot y + z)_* - y \cdot x\right) - \left(z + 1\right)}\right)}\]
   7. Using strategy rm

   8. Applied associate--r+0.0

      \[\leadsto \log \left(e^{\color{blue}{\left(\left((x \cdot y + z)_* - y \cdot x\right) - z\right) - 1}}\right)\]
3. Recombined 2 regimes into one program.

4. Applied simplify8.0

   \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\left(\left((x \cdot y + z)_* - y \cdot x\right) - z\right) - 1 \le -1.0000000021634312 \lor \neg \left(\left(\left((x \cdot y + z)_* - y \cdot x\right) - z\right) - 1 \le -1.0\right):\\ \;\;\;\;\log \left(e^{\log \left(e^{\left(\left((x \cdot y + z)_* - z\right) - y \cdot x\right) - 1}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\left(\left((x \cdot y + z)_* - y \cdot x\right) - z\right) - 1}\right)\\ \end{array}}\]

Runtime

Time bar (total: 45.7s)Debug logProfile

herbie shell --seed '#(1071948828 1180510430 2986424009 997076509 406109801 420189285)' 
(FPCore (x y z)
  :name "simple fma test"

  :herbie-target
  -1

  (- (fma x y z) (+ 1 (+ (* x y) z))))