Average Error: 3.9 → 0.7
Time: 4.3m
Precision: 64
Internal Precision: 2112
\[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{1 + e^{-t}} \le 3.4772576107803896 \cdot 10^{-295}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}} \cdot \sqrt[3]{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}}{{\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n}} \cdot \frac{\sqrt[3]{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}}{\frac{(c_p \cdot \left((\frac{1}{2} \cdot t + \left(\log \frac{1}{2}\right))_*\right) + 1)_*}{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p}}}\\ \mathbf{else}:\\ \;\;\;\;e^{\log_* (1 + \frac{-1}{1 + e^{-s}}) \cdot c_n - (\left(\sqrt[3]{{\left(\log_* (1 + \frac{-1}{e^{-t} + 1})\right)}^{3}}\right) \cdot c_n + \left(c_p \cdot \left(\log_* (1 + e^{-s}) - \log_* (1 + e^{-t})\right)\right))_*}\\ \end{array}\]

Error

Bits error versus c_p

Bits error versus c_n

Bits error versus t

Bits error versus s

Target

Original3.9
Target2.2
Herbie0.7
\[{\left(\frac{1 + e^{-t}}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(\frac{1 + e^{t}}{1 + e^{s}}\right)}^{c_n}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ 1 (+ 1 (exp (- t)))) < 3.4772576107803896e-295

    1. Initial program 62.4

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

    2. Taylor expanded around 0 0.3

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

    3. Applied simplify0.3

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

    5. Applied add-cube-cbrt0.3

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}} \cdot \sqrt[3]{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}\right) \cdot \sqrt[3]{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}}}{{\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n} \cdot \frac{(c_p \cdot \left((\frac{1}{2} \cdot t + \left(\log \frac{1}{2}\right))_*\right) + 1)_*}{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p}}}\]

    6. Applied times-frac0.3

      \[\leadsto \color{blue}{\frac{\sqrt[3]{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}} \cdot \sqrt[3]{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}}{{\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n}} \cdot \frac{\sqrt[3]{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}}{\frac{(c_p \cdot \left((\frac{1}{2} \cdot t + \left(\log \frac{1}{2}\right))_*\right) + 1)_*}{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p}}}}\]

    if 3.4772576107803896e-295 < (/ 1 (+ 1 (exp (- t))))

    1. Initial program 2.7

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

    3. Applied add-exp-log2.8

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

    4. Applied add-exp-log2.8

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

    5. Applied pow-exp2.8

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

    6. Applied add-exp-log2.8

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

    7. Applied rec-exp2.8

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

    8. Applied pow-exp2.7

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

    9. Applied prod-exp2.7

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

    10. Applied div-exp2.7

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

    11. Applied simplify0.7

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

    13. Applied add-cbrt-cube0.7

      \[\leadsto e^{\log_* (1 + \frac{-1}{1 + e^{-s}}) \cdot c_n - (\color{blue}{\left(\sqrt[3]{\left(\log_* (1 + \frac{-1}{e^{-t} + 1}) \cdot \log_* (1 + \frac{-1}{e^{-t} + 1})\right) \cdot \log_* (1 + \frac{-1}{e^{-t} + 1})}\right)} \cdot c_n + \left(c_p \cdot \left(\log_* (1 + e^{-s}) - \log_* (1 + e^{-t})\right)\right))_*}\]

    14. Applied simplify0.7

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

Runtime

Time bar (total: 4.3m)Debug logProfile

herbie shell --seed '#(1072840222 1305617769 1692503039 1353360431 4178980589 1488672652)' +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))))