Average Error: 3.9 → 0.3
Time: 2.8m
Precision: 64
Internal Precision: 1856
\[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;s \le -734371095.2098387:\\ \;\;\;\;e^{(\left(e^{\log \left(\log_* (1 + \frac{-1}{e^{-s} + 1}) - \log_* (1 + \frac{-1}{e^{-t} + 1})\right)}\right) \cdot c_n + \left(\left((\left((\frac{1}{8} \cdot t + \frac{-1}{2})_*\right) \cdot t + \left(\log 2\right))_* - (e^{\log_* (1 + \log_* (1 + e^{-s}))} - 1)^*\right) \cdot c_p\right))_*}\\ \mathbf{else}:\\ \;\;\;\;e^{(\left(\log_* (1 + \frac{-1}{e^{-s} + 1}) - \log_* (1 + \frac{-1}{e^{-t} + 1})\right) \cdot c_n + \left(\left((\left((\frac{1}{8} \cdot t + \frac{-1}{2})_*\right) \cdot t + \left(\log 2\right))_* - (\left((\frac{1}{8} \cdot s + \frac{-1}{2})_*\right) \cdot s + \left(\log 2\right))_*\right) \cdot c_p\right))_*}\\ \end{array}\]

Error

Bits error versus c_p

Bits error versus c_n

Bits error versus t

Bits error versus s

Target

Original3.9
Target1.8
Herbie0.3
\[{\left(\frac{1 + e^{-t}}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(\frac{1 + e^{t}}{1 + e^{s}}\right)}^{c_n}\]

Derivation

  1. Split input into 2 regimes

  2. if s < -734371095.2098387

    1. Initial program 3.4

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n}}\]

    2. Initial simplification3.4

      \[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}} \cdot \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c_p}}{{\left(\frac{1}{e^{-t} + 1}\right)}^{c_p}}\]
    3. Using strategy rm

    4. Applied add-exp-log3.4

      \[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}} \cdot \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c_p}}{{\color{blue}{\left(e^{\log \left(\frac{1}{e^{-t} + 1}\right)}\right)}}^{c_p}}\]

    5. Applied pow-exp3.4

      \[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}} \cdot \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c_p}}{\color{blue}{e^{\log \left(\frac{1}{e^{-t} + 1}\right) \cdot c_p}}}\]

    6. Applied add-exp-log3.4

      \[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}} \cdot \frac{{\left(\frac{1}{\color{blue}{e^{\log \left(e^{-s} + 1\right)}}}\right)}^{c_p}}{e^{\log \left(\frac{1}{e^{-t} + 1}\right) \cdot c_p}}\]

    7. Applied rec-exp3.4

      \[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}} \cdot \frac{{\color{blue}{\left(e^{-\log \left(e^{-s} + 1\right)}\right)}}^{c_p}}{e^{\log \left(\frac{1}{e^{-t} + 1}\right) \cdot c_p}}\]

    8. Applied pow-exp3.4

      \[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}} \cdot \frac{\color{blue}{e^{\left(-\log \left(e^{-s} + 1\right)\right) \cdot c_p}}}{e^{\log \left(\frac{1}{e^{-t} + 1}\right) \cdot c_p}}\]

    9. Applied div-exp2.2

      \[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}} \cdot \color{blue}{e^{\left(-\log \left(e^{-s} + 1\right)\right) \cdot c_p - \log \left(\frac{1}{e^{-t} + 1}\right) \cdot c_p}}\]

    10. Applied pow-to-exp2.2

      \[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{\color{blue}{e^{\log \left(1 - \frac{1}{e^{-t} + 1}\right) \cdot c_n}}} \cdot e^{\left(-\log \left(e^{-s} + 1\right)\right) \cdot c_p - \log \left(\frac{1}{e^{-t} + 1}\right) \cdot c_p}\]

    11. Applied add-exp-log2.2

      \[\leadsto \frac{\color{blue}{e^{\log \left({\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}\right)}}}{e^{\log \left(1 - \frac{1}{e^{-t} + 1}\right) \cdot c_n}} \cdot e^{\left(-\log \left(e^{-s} + 1\right)\right) \cdot c_p - \log \left(\frac{1}{e^{-t} + 1}\right) \cdot c_p}\]

    12. Applied div-exp2.2

      \[\leadsto \color{blue}{e^{\log \left({\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}\right) - \log \left(1 - \frac{1}{e^{-t} + 1}\right) \cdot c_n}} \cdot e^{\left(-\log \left(e^{-s} + 1\right)\right) \cdot c_p - \log \left(\frac{1}{e^{-t} + 1}\right) \cdot c_p}\]

    13. Applied prod-exp1.2

      \[\leadsto \color{blue}{e^{\left(\log \left({\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}\right) - \log \left(1 - \frac{1}{e^{-t} + 1}\right) \cdot c_n\right) + \left(\left(-\log \left(e^{-s} + 1\right)\right) \cdot c_p - \log \left(\frac{1}{e^{-t} + 1}\right) \cdot c_p\right)}}\]

    14. Simplified1.2

      \[\leadsto e^{\color{blue}{(\left(\log_* (1 + \frac{-1}{e^{-s} + 1}) - \log_* (1 + \frac{-1}{e^{-t} + 1})\right) \cdot c_n + \left(c_p \cdot \left(\log_* (1 + e^{-t}) - \log_* (1 + e^{-s})\right)\right))_*}}\]

    15. Taylor expanded around 0 0.1

      \[\leadsto e^{(\left(\log_* (1 + \frac{-1}{e^{-s} + 1}) - \log_* (1 + \frac{-1}{e^{-t} + 1})\right) \cdot c_n + \left(c_p \cdot \left(\color{blue}{\left(\left(\log 2 + \frac{1}{8} \cdot {t}^{2}\right) - \frac{1}{2} \cdot t\right)} - \log_* (1 + e^{-s})\right)\right))_*}\]

    16. Simplified0.1

      \[\leadsto e^{(\left(\log_* (1 + \frac{-1}{e^{-s} + 1}) - \log_* (1 + \frac{-1}{e^{-t} + 1})\right) \cdot c_n + \left(c_p \cdot \left(\color{blue}{(\left((\frac{1}{8} \cdot t + \frac{-1}{2})_*\right) \cdot t + \left(\log 2\right))_*} - \log_* (1 + e^{-s})\right)\right))_*}\]
    17. Using strategy rm

    18. Applied expm1-log1p-u0.1

      \[\leadsto e^{(\left(\log_* (1 + \frac{-1}{e^{-s} + 1}) - \log_* (1 + \frac{-1}{e^{-t} + 1})\right) \cdot c_n + \left(c_p \cdot \left((\left((\frac{1}{8} \cdot t + \frac{-1}{2})_*\right) \cdot t + \left(\log 2\right))_* - \color{blue}{(e^{\log_* (1 + \log_* (1 + e^{-s}))} - 1)^*}\right)\right))_*}\]
    19. Using strategy rm

    20. Applied add-exp-log0.1

      \[\leadsto e^{(\color{blue}{\left(e^{\log \left(\log_* (1 + \frac{-1}{e^{-s} + 1}) - \log_* (1 + \frac{-1}{e^{-t} + 1})\right)}\right)} \cdot c_n + \left(c_p \cdot \left((\left((\frac{1}{8} \cdot t + \frac{-1}{2})_*\right) \cdot t + \left(\log 2\right))_* - (e^{\log_* (1 + \log_* (1 + e^{-s}))} - 1)^*\right)\right))_*}\]

    if -734371095.2098387 < s

    1. Initial program 4.1

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n}}\]

    2. Initial simplification4.1

      \[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}} \cdot \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c_p}}{{\left(\frac{1}{e^{-t} + 1}\right)}^{c_p}}\]
    3. Using strategy rm

    4. Applied add-exp-log4.1

      \[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}} \cdot \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c_p}}{{\color{blue}{\left(e^{\log \left(\frac{1}{e^{-t} + 1}\right)}\right)}}^{c_p}}\]

    5. Applied pow-exp4.1

      \[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}} \cdot \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c_p}}{\color{blue}{e^{\log \left(\frac{1}{e^{-t} + 1}\right) \cdot c_p}}}\]

    6. Applied add-exp-log4.1

      \[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}} \cdot \frac{{\left(\frac{1}{\color{blue}{e^{\log \left(e^{-s} + 1\right)}}}\right)}^{c_p}}{e^{\log \left(\frac{1}{e^{-t} + 1}\right) \cdot c_p}}\]

    7. Applied rec-exp4.1

      \[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}} \cdot \frac{{\color{blue}{\left(e^{-\log \left(e^{-s} + 1\right)}\right)}}^{c_p}}{e^{\log \left(\frac{1}{e^{-t} + 1}\right) \cdot c_p}}\]

    8. Applied pow-exp4.1

      \[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}} \cdot \frac{\color{blue}{e^{\left(-\log \left(e^{-s} + 1\right)\right) \cdot c_p}}}{e^{\log \left(\frac{1}{e^{-t} + 1}\right) \cdot c_p}}\]

    9. Applied div-exp3.3

      \[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}} \cdot \color{blue}{e^{\left(-\log \left(e^{-s} + 1\right)\right) \cdot c_p - \log \left(\frac{1}{e^{-t} + 1}\right) \cdot c_p}}\]

    10. Applied pow-to-exp3.3

      \[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{\color{blue}{e^{\log \left(1 - \frac{1}{e^{-t} + 1}\right) \cdot c_n}}} \cdot e^{\left(-\log \left(e^{-s} + 1\right)\right) \cdot c_p - \log \left(\frac{1}{e^{-t} + 1}\right) \cdot c_p}\]

    11. Applied add-exp-log3.3

      \[\leadsto \frac{\color{blue}{e^{\log \left({\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}\right)}}}{e^{\log \left(1 - \frac{1}{e^{-t} + 1}\right) \cdot c_n}} \cdot e^{\left(-\log \left(e^{-s} + 1\right)\right) \cdot c_p - \log \left(\frac{1}{e^{-t} + 1}\right) \cdot c_p}\]

    12. Applied div-exp2.8

      \[\leadsto \color{blue}{e^{\log \left({\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}\right) - \log \left(1 - \frac{1}{e^{-t} + 1}\right) \cdot c_n}} \cdot e^{\left(-\log \left(e^{-s} + 1\right)\right) \cdot c_p - \log \left(\frac{1}{e^{-t} + 1}\right) \cdot c_p}\]

    13. Applied prod-exp2.4

      \[\leadsto \color{blue}{e^{\left(\log \left({\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}\right) - \log \left(1 - \frac{1}{e^{-t} + 1}\right) \cdot c_n\right) + \left(\left(-\log \left(e^{-s} + 1\right)\right) \cdot c_p - \log \left(\frac{1}{e^{-t} + 1}\right) \cdot c_p\right)}}\]

    14. Simplified1.7

      \[\leadsto e^{\color{blue}{(\left(\log_* (1 + \frac{-1}{e^{-s} + 1}) - \log_* (1 + \frac{-1}{e^{-t} + 1})\right) \cdot c_n + \left(c_p \cdot \left(\log_* (1 + e^{-t}) - \log_* (1 + e^{-s})\right)\right))_*}}\]

    15. Taylor expanded around 0 0.7

      \[\leadsto e^{(\left(\log_* (1 + \frac{-1}{e^{-s} + 1}) - \log_* (1 + \frac{-1}{e^{-t} + 1})\right) \cdot c_n + \left(c_p \cdot \left(\color{blue}{\left(\left(\log 2 + \frac{1}{8} \cdot {t}^{2}\right) - \frac{1}{2} \cdot t\right)} - \log_* (1 + e^{-s})\right)\right))_*}\]

    16. Simplified0.7

      \[\leadsto e^{(\left(\log_* (1 + \frac{-1}{e^{-s} + 1}) - \log_* (1 + \frac{-1}{e^{-t} + 1})\right) \cdot c_n + \left(c_p \cdot \left(\color{blue}{(\left((\frac{1}{8} \cdot t + \frac{-1}{2})_*\right) \cdot t + \left(\log 2\right))_*} - \log_* (1 + e^{-s})\right)\right))_*}\]

    17. Taylor expanded around 0 0.4

      \[\leadsto e^{(\left(\log_* (1 + \frac{-1}{e^{-s} + 1}) - \log_* (1 + \frac{-1}{e^{-t} + 1})\right) \cdot c_n + \left(c_p \cdot \left((\left((\frac{1}{8} \cdot t + \frac{-1}{2})_*\right) \cdot t + \left(\log 2\right))_* - \color{blue}{\left(\left(\log 2 + \frac{1}{8} \cdot {s}^{2}\right) - \frac{1}{2} \cdot s\right)}\right)\right))_*}\]

    18. Simplified0.4

      \[\leadsto e^{(\left(\log_* (1 + \frac{-1}{e^{-s} + 1}) - \log_* (1 + \frac{-1}{e^{-t} + 1})\right) \cdot c_n + \left(c_p \cdot \left((\left((\frac{1}{8} \cdot t + \frac{-1}{2})_*\right) \cdot t + \left(\log 2\right))_* - \color{blue}{(\left((\frac{1}{8} \cdot s + \frac{-1}{2})_*\right) \cdot s + \left(\log 2\right))_*}\right)\right))_*}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;s \le -734371095.2098387:\\ \;\;\;\;e^{(\left(e^{\log \left(\log_* (1 + \frac{-1}{e^{-s} + 1}) - \log_* (1 + \frac{-1}{e^{-t} + 1})\right)}\right) \cdot c_n + \left(\left((\left((\frac{1}{8} \cdot t + \frac{-1}{2})_*\right) \cdot t + \left(\log 2\right))_* - (e^{\log_* (1 + \log_* (1 + e^{-s}))} - 1)^*\right) \cdot c_p\right))_*}\\ \mathbf{else}:\\ \;\;\;\;e^{(\left(\log_* (1 + \frac{-1}{e^{-s} + 1}) - \log_* (1 + \frac{-1}{e^{-t} + 1})\right) \cdot c_n + \left(\left((\left((\frac{1}{8} \cdot t + \frac{-1}{2})_*\right) \cdot t + \left(\log 2\right))_* - (\left((\frac{1}{8} \cdot s + \frac{-1}{2})_*\right) \cdot s + \left(\log 2\right))_*\right) \cdot c_p\right))_*}\\ \end{array}\]

Runtime

Time bar (total: 2.8m)Debug logProfile

herbie shell --seed 2018256 +o rules:numerics
(FPCore (c_p c_n t s)
  :name "Harley's example"
  :pre (and (< 0 c_p) (< 0 c_n))

  :herbie-target
  (* (pow (/ (+ 1 (exp (- t))) (+ 1 (exp (- s)))) c_p) (pow (/ (+ 1 (exp t)) (+ 1 (exp s))) c_n))

  (/ (* (pow (/ 1 (+ 1 (exp (- s)))) c_p) (pow (- 1 (/ 1 (+ 1 (exp (- s))))) c_n)) (* (pow (/ 1 (+ 1 (exp (- t)))) c_p) (pow (- 1 (/ 1 (+ 1 (exp (- t))))) c_n))))