Average Error: 4.0 → 0.7
Time: 8.4m
Precision: 64
Internal Precision: 2368
\[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{\left(\log \left(\sqrt{1 - \frac{1}{1 + e^{-s}}}\right) + \left(\log \left(\sqrt{1 - \frac{1}{1 + e^{-s}}}\right) - \log \left(1 - \frac{1}{e^{-t} + 1}\right)\right)\right) \cdot c_n - \left(\log \left(1 + e^{-s}\right) - \log \left(e^{-t} + 1\right)\right) \cdot c_p} \le 1.0000189574180463:\\ \;\;\;\;e^{\left(\log \left(\sqrt{1 - \frac{1}{1 + e^{-s}}}\right) + \left(\log \left(\sqrt{1 - \frac{1}{1 + e^{-s}}}\right) - \log \left(1 - \frac{1}{e^{-t} + 1}\right)\right)\right) \cdot c_n - \left(\log \left(1 + e^{-s}\right) - \log \left(e^{-t} + 1\right)\right) \cdot c_p}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\sqrt{1 - \frac{1}{1 + e^{-s}}}\right)}^{c_n}}{{1}^{c_n}} \cdot \left(\frac{{\left(\sqrt{1 - \frac{1}{1 + e^{-s}}}\right)}^{c_n}}{{\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n}} \cdot \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p}}{c_p \cdot \left(\log \frac{1}{2} + t \cdot \frac{1}{2}\right) + 1}\right)\\ \end{array}\]

Error

Bits error versus c_p

Bits error versus c_n

Bits error versus t

Bits error versus s

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original4.0
Target2.0
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 (exp (- (* (+ (log (sqrt (- 1 (/ 1 (+ 1 (exp (- s))))))) (- (log (sqrt (- 1 (/ 1 (+ 1 (exp (- s))))))) (log (- 1 (/ 1 (+ (exp (- t)) 1)))))) c_n) (* (- (log (+ 1 (exp (- s)))) (log (+ (exp (- t)) 1))) c_p))) < 1.0000189574180463

    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-log2.7

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

    4. Applied pow-exp2.7

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

    5. Applied add-exp-log2.7

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

    6. Applied rec-exp2.7

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

    7. Applied pow-exp2.7

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

    8. Applied prod-exp2.7

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

    9. Applied add-exp-log2.7

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

    10. Applied pow-exp2.7

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

    11. Applied add-exp-log2.7

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

    12. Applied rec-exp2.7

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

    13. Applied pow-exp2.7

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

    14. Applied prod-exp2.7

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

    15. Applied div-exp0.3

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

    16. Applied simplify0.3

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

    18. Applied add-sqr-sqrt0.6

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

    19. Applied log-prod0.6

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

    20. Applied associate--l+0.6

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

    if 1.0000189574180463 < (exp (- (* (+ (log (sqrt (- 1 (/ 1 (+ 1 (exp (- s))))))) (- (log (sqrt (- 1 (/ 1 (+ 1 (exp (- s))))))) (log (- 1 (/ 1 (+ (exp (- t)) 1)))))) c_n) (* (- (log (+ 1 (exp (- s)))) (log (+ (exp (- t)) 1))) c_p)))

    1. Initial program 62.2

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

    2. Taylor expanded around 0 9.1

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

    3. Applied simplify9.1

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

    5. Applied *-un-lft-identity9.1

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

    6. Applied unpow-prod-down9.1

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

    7. Applied add-sqr-sqrt9.1

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

    8. Applied unpow-prod-down9.1

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

    9. Applied times-frac9.1

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

    10. Applied associate-*l*9.1

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

Runtime

Time bar (total: 8.4m)Debug logProfile

herbie shell --seed 2018207 
(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))))