Average Error: 3.9 → 0.9
Time: 3.7m
Precision: 64
Internal Precision: 2624
\[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{(\left(\log_* (1 + \frac{-1}{1 + e^{-s}}) - \log_* (1 + \frac{-1}{e^{-t} + 1})\right) \cdot c_n + \left(c_p \cdot \left(\left(\sqrt[3]{\log_* (1 + e^{-t}) - \log_* (1 + e^{-s})} \cdot \sqrt[3]{\frac{{\left(\log_* (1 + e^{-t})\right)}^{3} - {\left(\log_* (1 + e^{-s})\right)}^{3}}{(\left(\log_* (1 + e^{-s}) + \log_* (1 + e^{-t})\right) \cdot \left(\log_* (1 + e^{-s})\right) + \left(\log_* (1 + e^{-t}) \cdot \log_* (1 + e^{-t})\right))_*}}\right) \cdot \sqrt[3]{(\left(\sqrt{\log_* (1 + e^{-t})}\right) \cdot \left(\sqrt{\log_* (1 + e^{-t})}\right) + \left(-\log_* (1 + e^{-s})\right))_*}\right)\right))_*} \le 149.71600564685664:\\ \;\;\;\;e^{(\left(\log_* (1 + \frac{-1}{1 + e^{-s}}) - \log_* (1 + \frac{-1}{e^{-t} + 1})\right) \cdot c_n + \left(c_p \cdot \left(\left(\sqrt[3]{\log_* (1 + e^{-t}) - \log_* (1 + e^{-s})} \cdot \sqrt[3]{\frac{{\left(\log_* (1 + e^{-t})\right)}^{3} - {\left(\log_* (1 + e^{-s})\right)}^{3}}{(\left(\log_* (1 + e^{-s}) + \log_* (1 + e^{-t})\right) \cdot \left(\log_* (1 + e^{-s})\right) + \left(\log_* (1 + e^{-t}) \cdot \log_* (1 + e^{-t})\right))_*}}\right) \cdot \sqrt[3]{(\left(\sqrt{\log_* (1 + e^{-t})}\right) \cdot \left(\sqrt{\log_* (1 + e^{-t})}\right) + \left(-\log_* (1 + e^{-s})\right))_*}\right)\right))_*}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n} \cdot \frac{(c_p \cdot \left((\frac{1}{2} \cdot t + \left(\log \frac{1}{2}\right))_*\right) + 1)_*}{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p}}}\\ \end{array}\]

Error

Bits error versus c_p

Bits error versus c_n

Bits error versus t

Bits error versus s

Target

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

Derivation

  1. Split input into 2 regimes

  2. if (exp (fma (- (log1p (/ (- 1) (+ 1 (exp (- s))))) (log1p (/ (- 1) (+ (exp (- t)) 1)))) c_n (* c_p (* (* (cbrt (- (log1p (exp (- t))) (log1p (exp (- s))))) (cbrt (/ (- (pow (log1p (exp (- t))) 3) (pow (log1p (exp (- s))) 3)) (fma (+ (log1p (exp (- s))) (log1p (exp (- t)))) (log1p (exp (- s))) (* (log1p (exp (- t))) (log1p (exp (- t)))))))) (cbrt (fma (sqrt (log1p (exp (- t)))) (sqrt (log1p (exp (- t)))) (- (log1p (exp (- s)))))))))) < 149.71600564685664

    1. Initial program 2.4

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

    3. Applied add-exp-log2.4

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

    4. Applied pow-exp2.4

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

    5. Applied add-exp-log2.4

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

    6. Applied rec-exp2.4

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

    7. Applied pow-exp2.4

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

    8. Applied prod-exp2.4

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

    9. Applied add-exp-log2.4

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

    10. Applied pow-exp2.4

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

    11. Applied add-exp-log2.4

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

    12. Applied rec-exp2.4

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

    13. Applied pow-exp2.4

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

    14. Applied prod-exp2.4

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

    15. Applied div-exp0.1

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

    16. Applied simplify0.1

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

    18. Applied add-cube-cbrt0.1

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

    20. Applied add-sqr-sqrt0.1

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

    21. Applied fma-neg0.1

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

    23. Applied flip3--0.1

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

    24. Applied simplify0.1

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

    if 149.71600564685664 < (exp (fma (- (log1p (/ (- 1) (+ 1 (exp (- s))))) (log1p (/ (- 1) (+ (exp (- t)) 1)))) c_n (* c_p (* (* (cbrt (- (log1p (exp (- t))) (log1p (exp (- s))))) (cbrt (/ (- (pow (log1p (exp (- t))) 3) (pow (log1p (exp (- s))) 3)) (fma (+ (log1p (exp (- s))) (log1p (exp (- t)))) (log1p (exp (- s))) (* (log1p (exp (- t))) (log1p (exp (- t)))))))) (cbrt (fma (sqrt (log1p (exp (- t)))) (sqrt (log1p (exp (- t)))) (- (log1p (exp (- s))))))))))

    1. Initial program 8.2

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

    2. Taylor expanded around 0 3.0

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

    3. Applied simplify3.0

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

Runtime

Time bar (total: 3.7m)Debug logProfile

herbie shell --seed 2018208 +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))))