Average Error: 4.1 → 1.3
Time: 5.7m
Precision: 64
Internal Precision: 1920
\[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;-s \le -659.1730621603887:\\ \;\;\;\;\frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n} \cdot \frac{(c_p \cdot \left((\frac{1}{2} \cdot t + \left(\log \frac{1}{2}\right))_*\right) + 1)_*}{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p}}}\\ \mathbf{if}\;-s \le 311579297.9590099:\\ \;\;\;\;(\left(s \cdot \frac{1}{2}\right) \cdot \left(c_p - c_n\right) + 1)_*\\ \mathbf{else}:\\ \;\;\;\;e^{(c_p \cdot \left(\log_* (1 + e^{-t}) - \log_* (1 + e^{-s})\right) + \left(c_n \cdot (\left(-s\right) \cdot \left((\frac{1}{8} \cdot s + \frac{1}{2})_*\right) + \left(t \cdot \frac{1}{2}\right))_*\right))_*}\\ \end{array}\]

Error

Bits error versus c_p

Bits error versus c_n

Bits error versus t

Bits error versus s

Target

Original4.1
Target2.2
Herbie1.3
\[{\left(\frac{1 + e^{-t}}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(\frac{1 + e^{t}}{1 + e^{s}}\right)}^{c_n}\]

Derivation

  1. Split input into 3 regimes

  2. if (- s) < -659.1730621603887

    1. Initial program 3.5

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

    2. Taylor expanded around 0 1.1

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

    3. Applied simplify1.1

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

    if -659.1730621603887 < (- s) < 311579297.9590099

    1. Initial program 4.4

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

    2. Taylor expanded around 0 1.2

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(s \cdot c_p\right) + 1\right) - \frac{1}{2} \cdot \left(c_n \cdot s\right)}\]

    3. Applied simplify1.2

      \[\leadsto \color{blue}{(\left(s \cdot \frac{1}{2}\right) \cdot \left(c_p - c_n\right) + 1)_*}\]

    if 311579297.9590099 < (- s)

    1. Initial program 4.0

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

    3. Applied add-exp-log4.0

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

    4. Applied pow-exp4.0

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

    5. Applied add-exp-log4.0

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

    6. Applied rec-exp4.0

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

    7. Applied pow-exp4.0

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

    8. Applied prod-exp4.0

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

    9. Applied add-exp-log4.0

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

    10. Applied pow-exp4.0

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

    11. Applied add-exp-log4.0

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

    12. Applied rec-exp4.0

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

    13. Applied pow-exp4.0

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

    14. Applied prod-exp4.0

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

    15. Applied div-exp1.7

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

    16. Applied simplify1.7

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

    17. Taylor expanded around 0 1.6

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

    18. Applied simplify1.6

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

Runtime

Time bar (total: 5.7m)Debug logProfile

herbie shell --seed '#(1070864556 424010669 783715395 1203517814 4070606583 4107618214)' +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))))