Average Error: 3.9 → 1.3
Time: 2.5m
Precision: 64
Internal Precision: 2176
\[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;-s \le -189.33005877633454:\\ \;\;\;\;\frac{\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p}}{(c_p \cdot \left((\frac{1}{2} \cdot t + \left(\log \frac{1}{2}\right))_*\right) + 1)_*}}{{\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n}} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}\\ \mathbf{if}\;-s \le 25182448.00233434:\\ \;\;\;\;(\left(s \cdot \frac{1}{2}\right) \cdot \left(c_p - c_n\right) + 1)_*\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p}}{(c_p \cdot \left((\frac{1}{2} \cdot t + \left(\log \frac{1}{2}\right))_*\right) + 1)_*}}{{\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n}} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}\\ \end{array}\]

Error

Bits error versus c_p

Bits error versus c_n

Bits error versus t

Bits error versus s

Target

Original3.9
Target1.9
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) < -189.33005877633454

    1. Initial program 3.7

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

    2. Taylor expanded around 0 1.8

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

    3. Applied simplify1.8

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

    if -189.33005877633454 < (- s) < 25182448.00233434

    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. Taylor expanded around 0 1.0

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

    3. Applied simplify1.0

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

    if 25182448.00233434 < (- s)

    1. Initial program 3.7

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

    2. Taylor expanded around 0 1.6

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

    3. Applied simplify1.6

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

Runtime

Time bar (total: 2.5m)Debug logProfile

herbie shell --seed '#(1064173506 2580572819 2847706409 4129882574 1125180799 1845288547)' +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))))