Average Error: 44.7 → 8.1
Time: 44.4s
Precision: 64
Internal Precision: 2368
\[(x \cdot y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\log \left(e^{\log \left(e^{\left(\left((x \cdot y + z)_* - z\right) - y \cdot x\right) - 1}\right)}\right) \le -1.0000000000293188:\\ \;\;\;\;\log \left(e^{\left(\left((x \cdot y + z)_* - y \cdot x\right) - z\right) - 1}\right)\\ \mathbf{if}\;\log \left(e^{\log \left(e^{\left(\left((x \cdot y + z)_* - z\right) - y \cdot x\right) - 1}\right)}\right) \le -1.0:\\ \;\;\;\;\log \left(e^{\log \left(e^{\left(\left((x \cdot y + z)_* - z\right) - y \cdot x\right) - 1}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\left(\left((x \cdot y + z)_* - y \cdot x\right) - z\right) - 1}\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original44.7
Target0
Herbie8.1
\[-1\]

Derivation

  1. Split input into 2 regimes

  2. if (log (exp (log (exp (- (- (- (fma x y z) z) (* y x)) 1))))) < -1.0000000000293188 or -1.0 < (log (exp (log (exp (- (- (- (fma x y z) z) (* y x)) 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 simplify62.6

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

    8. Applied associate--r+33.3

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

    if -1.0000000000293188 < (log (exp (log (exp (- (- (- (fma x y z) z) (* y x)) 1))))) < -1.0

    1. Initial program 39.1

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

    3. Applied add-log-exp41.2

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

    4. Applied add-log-exp41.8

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

    5. Applied diff-log41.8

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

    6. Applied simplify25.5

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

    8. Applied add-log-exp26.6

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

    9. Applied add-log-exp41.3

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

    10. Applied add-log-exp41.7

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

    11. Applied diff-log41.7

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

    12. Applied diff-log41.8

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

    13. Applied simplify20.8

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

    15. Applied associate--r+0.0

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

Runtime

Time bar (total: 44.4s)Debug logProfile

herbie shell --seed '#(1071501266 3581234924 1086666455 2685055582 1243441566 1802958749)' 
(FPCore (x y z)
  :name "simple fma test"

  :herbie-target
  -1

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