Average Error: 4.0 → 2.8
Time: 2.7m
Precision: 64
Internal Precision: 2368
\[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n}}\]
↓
\[\frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c_p} \cdot {\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n} \cdot (c_p \cdot \left((\frac{1}{2} \cdot t + \left(\log \frac{1}{2}\right))_*\right) + 1)_*}\]

Error

Bits error versus c_p

Bits error versus c_n

Bits error versus t

Bits error versus s

Target

Original4.0
Target2.2
Herbie2.8
\[{\left(\frac{1 + e^{-t}}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(\frac{1 + e^{t}}{1 + e^{s}}\right)}^{c_n}\]

Derivation

  1. 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 2.8

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

  3. Simplified2.8

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

  4. Final simplification2.8

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

Runtime

Time bar (total: 2.7m)Debug logProfile

herbie shell --seed 2018273 +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))))