Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 17.7s

Alternatives: 16

Speedup: 1.0×

Specification

\[\left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right) \land \left(0 \leq s \land s \leq 1.0651631\right)\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\\ \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - t\_0\right) + t\_0} - 1\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ 1.0 (exp (/ (PI) s))))))
   (*
    (- s)
    (log
     (-
      (/ 1.0 (+ (* u (- (/ 1.0 (+ 1.0 (exp (/ (- (PI)) s)))) t_0)) t_0))
      1.0)))))

Initial Program: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\\ \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - t\_0\right) + t\_0} - 1\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ 1.0 (exp (/ (PI) s))))))
   (*
    (- s)
    (log
     (-
      (/ 1.0 (+ (* u (- (/ 1.0 (+ 1.0 (exp (/ (- (PI)) s)))) t_0)) t_0))
      1.0)))))

Alternative 1: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{1}{\frac{1}{e^{\frac{\mathsf{PI}\left(\right) \cdot s}{s \cdot s}} + 1} + \left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right) \cdot u} - 1\right) \cdot \left(-s\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (log
   (-
    (/
     1.0
     (+
      (/ 1.0 (+ (exp (/ (* (PI) s) (* s s))) 1.0))
      (*
       (-
        (/ 1.0 (+ (exp (/ (- (PI)) s)) 1.0))
        (/ 1.0 (+ (exp (/ (PI) s)) 1.0)))
       u)))
    1.0))
  (- s)))

Derivation

1. Initial program 98.8%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{\mathsf{PI}\left(\right)}{s}}}}} - 1\right) \]

   2. remove-double-neg N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right)\right)}}{s}}}} - 1\right) \]

   3. lift-neg.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{neg}\left(\color{blue}{\left(-\mathsf{PI}\left(\right)\right)}\right)}{s}}}} - 1\right) \]

   4. neg-sub0 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{0 - \left(-\mathsf{PI}\left(\right)\right)}}{s}}}} - 1\right) \]

   5. div-sub N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{0}{s} - \frac{-\mathsf{PI}\left(\right)}{s}}}}} - 1\right) \]

   6. frac-2neg N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{\mathsf{neg}\left(0\right)}{\mathsf{neg}\left(s\right)}} - \frac{-\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   7. metadata-eval N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{0}}{\mathsf{neg}\left(s\right)} - \frac{-\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   8. lift-neg.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{0}{\color{blue}{-s}} - \frac{-\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   9. frac-sub N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\left(-s\right) \cdot s}}}}} - 1\right) \]

   10. lower-/.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\left(-s\right) \cdot s}}}}} - 1\right) \]

   11. lower--.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}}{\left(-s\right) \cdot s}}}} - 1\right) \]

   12. lower-*.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{0 \cdot s} - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\left(-s\right) \cdot s}}}} - 1\right) \]

   13. lower-*.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{0 \cdot s - \color{blue}{\left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}}{\left(-s\right) \cdot s}}}} - 1\right) \]

   14. lower-*.f32 98.8

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\color{blue}{\left(-s\right) \cdot s}}}}} - 1\right) \]

4. Applied rewrites 98.8%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\left(-s\right) \cdot s}}}}} - 1\right) \]
5. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\left(-s\right) \cdot s}}}}} - 1\right) \]

   2. frac-2neg N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{\mathsf{neg}\left(\left(0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)\right)\right)}{\mathsf{neg}\left(\left(-s\right) \cdot s\right)}}}}} - 1\right) \]

   3. frac-2neg N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)\right)\right)\right)\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(-s\right) \cdot s\right)\right)\right)}}}}} - 1\right) \]

   4. remove-double-neg N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(-s\right) \cdot s\right)\right)\right)}}}} - 1\right) \]

   5. lift--.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(-s\right) \cdot s\right)\right)\right)}}}} - 1\right) \]

   6. lift-*.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{0 \cdot s} - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(-s\right) \cdot s\right)\right)\right)}}}} - 1\right) \]

   7. mul0-lft N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{0} - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(-s\right) \cdot s\right)\right)\right)}}}} - 1\right) \]

   8. neg-sub0 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{\mathsf{neg}\left(\left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)\right)}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(-s\right) \cdot s\right)\right)\right)}}}} - 1\right) \]

   9. frac-2neg N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{\left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\mathsf{neg}\left(\left(-s\right) \cdot s\right)}}}}} - 1\right) \]

   10. lower-/.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{\left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\mathsf{neg}\left(\left(-s\right) \cdot s\right)}}}}} - 1\right) \]

   11. lift-*.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{\left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}}{\mathsf{neg}\left(\left(-s\right) \cdot s\right)}}}} - 1\right) \]

   12. lift-neg.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\left(-s\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right)}}{\mathsf{neg}\left(\left(-s\right) \cdot s\right)}}}} - 1\right) \]

   13. distribute-rgt-neg-out N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{\mathsf{neg}\left(\left(-s\right) \cdot \mathsf{PI}\left(\right)\right)}}{\mathsf{neg}\left(\left(-s\right) \cdot s\right)}}}} - 1\right) \]

   14. distribute-lft-neg-in N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{\left(\mathsf{neg}\left(\left(-s\right)\right)\right) \cdot \mathsf{PI}\left(\right)}}{\mathsf{neg}\left(\left(-s\right) \cdot s\right)}}}} - 1\right) \]

   15. lift-neg.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(s\right)\right)}\right)\right) \cdot \mathsf{PI}\left(\right)}{\mathsf{neg}\left(\left(-s\right) \cdot s\right)}}}} - 1\right) \]

   16. remove-double-neg N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{s} \cdot \mathsf{PI}\left(\right)}{\mathsf{neg}\left(\left(-s\right) \cdot s\right)}}}} - 1\right) \]

   17. lower-*.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{s \cdot \mathsf{PI}\left(\right)}}{\mathsf{neg}\left(\left(-s\right) \cdot s\right)}}}} - 1\right) \]

   18. lift-*.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{s \cdot \mathsf{PI}\left(\right)}{\mathsf{neg}\left(\color{blue}{\left(-s\right) \cdot s}\right)}}}} - 1\right) \]

   19. lift-neg.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{s \cdot \mathsf{PI}\left(\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(s\right)\right)} \cdot s\right)}}}} - 1\right) \]

   20. distribute-lft-neg-out N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{s \cdot \mathsf{PI}\left(\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(s \cdot s\right)\right)}\right)}}}} - 1\right) \]

6. Applied rewrites 98.8%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{s \cdot \mathsf{PI}\left(\right)}{s \cdot s}}}}} - 1\right) \]

7. Final simplification 98.8%

   \[\leadsto \log \left(\frac{1}{\frac{1}{e^{\frac{\mathsf{PI}\left(\right) \cdot s}{s \cdot s}} + 1} + \left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right) \cdot u} - 1\right) \cdot \left(-s\right) \]
8. Add Preprocessing

Alternative 2: 13.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left(0.5, u, -0.25\right) \cdot \mathsf{PI}\left(\right)\right)}^{2}\\ t_1 := \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\\ \mathbf{if}\;\log \left(\frac{1}{\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - t\_1\right) \cdot u + t\_1} - 1\right) \cdot \left(-s\right) \leq -1.999999936531045 \cdot 10^{-20}:\\ \;\;\;\;\log \left(\left(1 - \frac{\mathsf{fma}\left(t\_0, -8, 0\right)}{s \cdot s}\right) - \frac{\mathsf{fma}\left(2, u, -1\right) \cdot \mathsf{PI}\left(\right)}{s}\right) \cdot \left(-s\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.25, \mathsf{PI}\left(\right), \left(\mathsf{PI}\left(\right) \cdot u\right) \cdot 0.5\right)}{s}, -4, \frac{t\_0}{s} \cdot \frac{8}{s}\right)\right) \cdot \left(-s\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (pow (* (fma 0.5 u -0.25) (PI)) 2.0))
        (t_1 (/ 1.0 (+ (exp (/ (PI) s)) 1.0))))
   (if (<=
        (*
         (log
          (-
           (/ 1.0 (+ (* (- (/ 1.0 (+ (exp (/ (- (PI)) s)) 1.0)) t_1) u) t_1))
           1.0))
         (- s))
        -1.999999936531045e-20)
     (*
      (log
       (-
        (- 1.0 (/ (fma t_0 -8.0 0.0) (* s s)))
        (/ (* (fma 2.0 u -1.0) (PI)) s)))
      (- s))
     (*
      (log
       (fma
        (/ (fma -0.25 (PI) (* (* (PI) u) 0.5)) s)
        -4.0
        (* (/ t_0 s) (/ 8.0 s))))
      (- s)))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32)))) < -1.99999994e-20

   1. Initial program 99.0%

      \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in s around -inf

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{-1 \cdot \frac{u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(\mathsf{neg}\left(\frac{u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s}\right)\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{neg}\left(s\right)}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      3. distribute-rgt-out-- N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{4} - \frac{1}{4}\right)\right)}}{\mathsf{neg}\left(s\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      4. metadata-eval N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{-1}{2}}\right)}{\mathsf{neg}\left(s\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{\color{blue}{\left(u \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{2}}}{\mathsf{neg}\left(s\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      6. mul-1-neg N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{\left(u \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{2}}{\color{blue}{-1 \cdot s}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      7. *-commutative N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{\left(u \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{2}}{\color{blue}{s \cdot -1}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      8. times-frac N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u \cdot \mathsf{PI}\left(\right)}{s} \cdot \frac{\frac{-1}{2}}{-1}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u \cdot \mathsf{PI}\left(\right)}{s} \cdot \color{blue}{\frac{1}{2}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      10. lower-*.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u \cdot \mathsf{PI}\left(\right)}{s} \cdot \frac{1}{2}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      11. lower-/.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u \cdot \mathsf{PI}\left(\right)}{s}} \cdot \frac{1}{2} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{\color{blue}{u \cdot \mathsf{PI}\left(\right)}}{s} \cdot \frac{1}{2} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      13. lower-PI.f32 7.3

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u \cdot \color{blue}{\mathsf{PI}\left(\right)}}{s} \cdot 0.5 + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   5. Applied rewrites 7.3%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u \cdot \mathsf{PI}\left(\right)}{s} \cdot 0.5} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   6. Taylor expanded in s around inf

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\left(1 + -1 \cdot \frac{-8 \cdot {\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{{s}^{2}}\right) - 4 \cdot \frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)}{s}\right)} \]
   7. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\left(1 + -1 \cdot \frac{-8 \cdot {\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{{s}^{2}}\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)}{s}\right)} \]

      2. metadata-eval N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\left(1 + -1 \cdot \frac{-8 \cdot {\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{{s}^{2}}\right) + \color{blue}{-4} \cdot \frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)}{s}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(-4 \cdot \frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)}{s} + \left(1 + -1 \cdot \frac{-8 \cdot {\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{{s}^{2}}\right)\right)} \]

   8. Applied rewrites 8.3%

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.25, \mathsf{PI}\left(\right), 0.5 \cdot \left(\mathsf{PI}\left(\right) \cdot u\right)\right)}{s}, -4, 1 - \frac{\frac{\mathsf{fma}\left(-8, {\left(\mathsf{fma}\left(-0.25, \mathsf{PI}\left(\right), 0.5 \cdot \left(\mathsf{PI}\left(\right) \cdot u\right)\right)\right)}^{2}, 0\right)}{s}}{s}\right)\right)} \]

   9. Taylor expanded in s around inf

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\left(1 + -1 \cdot \frac{-8 \cdot {\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{{s}^{2}}\right) - 4 \cdot \frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)}{s}\right)} \]

   11. Applied rewrites 14.0%

       \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\left(1 - \frac{\mathsf{fma}\left({\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(0.5, u, -0.25\right)\right)}^{2}, -8, 0\right)}{s \cdot s}\right) - \frac{\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(2, u, -1\right)}{s}\right)} \]

   if -1.99999994e-20 < (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32))))

   1. Initial program 98.7%

      \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in s around -inf

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{-1 \cdot \frac{u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(\mathsf{neg}\left(\frac{u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s}\right)\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{neg}\left(s\right)}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      3. distribute-rgt-out-- N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{4} - \frac{1}{4}\right)\right)}}{\mathsf{neg}\left(s\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      4. metadata-eval N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{-1}{2}}\right)}{\mathsf{neg}\left(s\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{\color{blue}{\left(u \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{2}}}{\mathsf{neg}\left(s\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      6. mul-1-neg N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{\left(u \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{2}}{\color{blue}{-1 \cdot s}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      7. *-commutative N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{\left(u \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{2}}{\color{blue}{s \cdot -1}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      8. times-frac N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u \cdot \mathsf{PI}\left(\right)}{s} \cdot \frac{\frac{-1}{2}}{-1}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u \cdot \mathsf{PI}\left(\right)}{s} \cdot \color{blue}{\frac{1}{2}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      10. lower-*.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u \cdot \mathsf{PI}\left(\right)}{s} \cdot \frac{1}{2}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      11. lower-/.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u \cdot \mathsf{PI}\left(\right)}{s}} \cdot \frac{1}{2} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{\color{blue}{u \cdot \mathsf{PI}\left(\right)}}{s} \cdot \frac{1}{2} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      13. lower-PI.f32 -0.0

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u \cdot \color{blue}{\mathsf{PI}\left(\right)}}{s} \cdot 0.5 + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   5. Applied rewrites -0.0%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u \cdot \mathsf{PI}\left(\right)}{s} \cdot 0.5} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   6. Taylor expanded in s around inf

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\left(1 + -1 \cdot \frac{-8 \cdot {\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{{s}^{2}}\right) - 4 \cdot \frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)}{s}\right)} \]
   7. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\left(1 + -1 \cdot \frac{-8 \cdot {\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{{s}^{2}}\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)}{s}\right)} \]

      2. metadata-eval N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\left(1 + -1 \cdot \frac{-8 \cdot {\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{{s}^{2}}\right) + \color{blue}{-4} \cdot \frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)}{s}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(-4 \cdot \frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)}{s} + \left(1 + -1 \cdot \frac{-8 \cdot {\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{{s}^{2}}\right)\right)} \]

   8. Applied rewrites 9.1%

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.25, \mathsf{PI}\left(\right), 0.5 \cdot \left(\mathsf{PI}\left(\right) \cdot u\right)\right)}{s}, -4, 1 - \frac{\frac{\mathsf{fma}\left(-8, {\left(\mathsf{fma}\left(-0.25, \mathsf{PI}\left(\right), 0.5 \cdot \left(\mathsf{PI}\left(\right) \cdot u\right)\right)\right)}^{2}, 0\right)}{s}}{s}\right)\right)} \]

   9. Taylor expanded in s around 0

      \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{-1}{4}, \mathsf{PI}\left(\right), \frac{1}{2} \cdot \left(\mathsf{PI}\left(\right) \cdot u\right)\right)}{s}, -4, 8 \cdot \frac{{\left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{1}{2} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{s}^{2}}\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 5.1%

       \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.25, \mathsf{PI}\left(\right), 0.5 \cdot \left(\mathsf{PI}\left(\right) \cdot u\right)\right)}{s}, -4, \frac{8}{s} \cdot \frac{{\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(0.5, u, -0.25\right)\right)}^{2}}{s}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 13.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(\frac{1}{\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right) \cdot u + \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}} - 1\right) \cdot \left(-s\right) \leq -1.999999936531045 \cdot 10^{-20}:\\ \;\;\;\;\log \left(\left(1 - \frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(0.5, u, -0.25\right) \cdot \mathsf{PI}\left(\right)\right)}^{2}, -8, 0\right)}{s \cdot s}\right) - \frac{\mathsf{fma}\left(2, u, -1\right) \cdot \mathsf{PI}\left(\right)}{s}\right) \cdot \left(-s\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.25, \mathsf{PI}\left(\right), \left(\mathsf{PI}\left(\right) \cdot u\right) \cdot 0.5\right)}{s}, -4, \frac{{\left(\mathsf{fma}\left(0.5, u, -0.25\right) \cdot \mathsf{PI}\left(\right)\right)}^{2}}{s} \cdot \frac{8}{s}\right)\right) \cdot \left(-s\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 18.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{PI}\left(\right)}{s}\\ t_1 := \frac{1}{e^{t\_0} + 1}\\ \mathbf{if}\;\log \left(\frac{1}{\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - t\_1\right) \cdot u + t\_1} - 1\right) \cdot \left(-s\right) \leq -2.0000000390829628 \cdot 10^{-24}:\\ \;\;\;\;\log \left(\frac{\left(\left(t\_0 \cdot u\right) \cdot \left(2 \cdot u\right)\right) \cdot \mathsf{PI}\left(\right)}{s}\right) \cdot \left(-s\right)\\ \mathbf{else}:\\ \;\;\;\;\log 1 \cdot \left(-s\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ (PI) s)) (t_1 (/ 1.0 (+ (exp t_0) 1.0))))
   (if (<=
        (*
         (log
          (-
           (/ 1.0 (+ (* (- (/ 1.0 (+ (exp (/ (- (PI)) s)) 1.0)) t_1) u) t_1))
           1.0))
         (- s))
        -2.0000000390829628e-24)
     (* (log (/ (* (* (* t_0 u) (* 2.0 u)) (PI)) s)) (- s))
     (* (log 1.0) (- s)))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32)))) < -2.00000004e-24

   1. Initial program 98.9%

      \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in s around -inf

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{-1 \cdot \frac{u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(\mathsf{neg}\left(\frac{u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s}\right)\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{neg}\left(s\right)}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      3. distribute-rgt-out-- N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{4} - \frac{1}{4}\right)\right)}}{\mathsf{neg}\left(s\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      4. metadata-eval N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{-1}{2}}\right)}{\mathsf{neg}\left(s\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{\color{blue}{\left(u \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{2}}}{\mathsf{neg}\left(s\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      6. mul-1-neg N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{\left(u \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{2}}{\color{blue}{-1 \cdot s}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      7. *-commutative N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{\left(u \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{2}}{\color{blue}{s \cdot -1}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      8. times-frac N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u \cdot \mathsf{PI}\left(\right)}{s} \cdot \frac{\frac{-1}{2}}{-1}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u \cdot \mathsf{PI}\left(\right)}{s} \cdot \color{blue}{\frac{1}{2}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      10. lower-*.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u \cdot \mathsf{PI}\left(\right)}{s} \cdot \frac{1}{2}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      11. lower-/.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u \cdot \mathsf{PI}\left(\right)}{s}} \cdot \frac{1}{2} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{\color{blue}{u \cdot \mathsf{PI}\left(\right)}}{s} \cdot \frac{1}{2} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      13. lower-PI.f32 6.3

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u \cdot \color{blue}{\mathsf{PI}\left(\right)}}{s} \cdot 0.5 + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   5. Applied rewrites 6.3%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u \cdot \mathsf{PI}\left(\right)}{s} \cdot 0.5} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   6. Taylor expanded in s around inf

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\left(1 + -1 \cdot \frac{-8 \cdot {\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{{s}^{2}}\right) - 4 \cdot \frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)}{s}\right)} \]
   7. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\left(1 + -1 \cdot \frac{-8 \cdot {\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{{s}^{2}}\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)}{s}\right)} \]

      2. metadata-eval N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\left(1 + -1 \cdot \frac{-8 \cdot {\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{{s}^{2}}\right) + \color{blue}{-4} \cdot \frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)}{s}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(-4 \cdot \frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)}{s} + \left(1 + -1 \cdot \frac{-8 \cdot {\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{{s}^{2}}\right)\right)} \]

   8. Applied rewrites 7.0%

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.25, \mathsf{PI}\left(\right), 0.5 \cdot \left(\mathsf{PI}\left(\right) \cdot u\right)\right)}{s}, -4, 1 - \frac{\frac{\mathsf{fma}\left(-8, {\left(\mathsf{fma}\left(-0.25, \mathsf{PI}\left(\right), 0.5 \cdot \left(\mathsf{PI}\left(\right) \cdot u\right)\right)\right)}^{2}, 0\right)}{s}}{s}\right)\right)} \]

   9. Taylor expanded in u around inf

      \[\leadsto \left(-s\right) \cdot \log \left(2 \cdot \color{blue}{\frac{{u}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{{s}^{2}}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 15.8%

       \[\leadsto \left(-s\right) \cdot \log \left(\left(2 \cdot \left(u \cdot u\right)\right) \cdot \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{s} \cdot \frac{\mathsf{PI}\left(\right)}{s}\right)}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 20.0%

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{\left(\left(2 \cdot u\right) \cdot \left(u \cdot \frac{\mathsf{PI}\left(\right)}{s}\right)\right) \cdot \mathsf{PI}\left(\right)}{s}\right) \]

   if -2.00000004e-24 < (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32))))

   1. Initial program 98.7%

      \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in s around inf

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 14.0%

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 17.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(\frac{1}{\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right) \cdot u + \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}} - 1\right) \cdot \left(-s\right) \leq -2.0000000390829628 \cdot 10^{-24}:\\ \;\;\;\;\log \left(\frac{\left(\left(\frac{\mathsf{PI}\left(\right)}{s} \cdot u\right) \cdot \left(2 \cdot u\right)\right) \cdot \mathsf{PI}\left(\right)}{s}\right) \cdot \left(-s\right)\\ \mathbf{else}:\\ \;\;\;\;\log 1 \cdot \left(-s\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 17.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\\ \mathbf{if}\;\log \left(\frac{1}{\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - t\_0\right) \cdot u + t\_0} - 1\right) \cdot \left(-s\right) \leq -4.999999841327613 \cdot 10^{-22}:\\ \;\;\;\;\log \left(\frac{\left(\left(u \cdot u\right) \cdot 2\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}{s \cdot s}\right) \cdot \left(-s\right)\\ \mathbf{else}:\\ \;\;\;\;\log 1 \cdot \left(-s\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ (exp (/ (PI) s)) 1.0))))
   (if (<=
        (*
         (log
          (-
           (/ 1.0 (+ (* (- (/ 1.0 (+ (exp (/ (- (PI)) s)) 1.0)) t_0) u) t_0))
           1.0))
         (- s))
        -4.999999841327613e-22)
     (* (log (/ (* (* (* u u) 2.0) (* (PI) (PI))) (* s s))) (- s))
     (* (log 1.0) (- s)))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32)))) < -4.9999998e-22

   1. Initial program 99.0%

      \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in s around -inf

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{-1 \cdot \frac{u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(\mathsf{neg}\left(\frac{u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s}\right)\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{neg}\left(s\right)}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      3. distribute-rgt-out-- N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{4} - \frac{1}{4}\right)\right)}}{\mathsf{neg}\left(s\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      4. metadata-eval N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{-1}{2}}\right)}{\mathsf{neg}\left(s\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{\color{blue}{\left(u \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{2}}}{\mathsf{neg}\left(s\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      6. mul-1-neg N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{\left(u \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{2}}{\color{blue}{-1 \cdot s}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      7. *-commutative N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{\left(u \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{2}}{\color{blue}{s \cdot -1}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      8. times-frac N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u \cdot \mathsf{PI}\left(\right)}{s} \cdot \frac{\frac{-1}{2}}{-1}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u \cdot \mathsf{PI}\left(\right)}{s} \cdot \color{blue}{\frac{1}{2}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      10. lower-*.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u \cdot \mathsf{PI}\left(\right)}{s} \cdot \frac{1}{2}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      11. lower-/.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u \cdot \mathsf{PI}\left(\right)}{s}} \cdot \frac{1}{2} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{\color{blue}{u \cdot \mathsf{PI}\left(\right)}}{s} \cdot \frac{1}{2} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      13. lower-PI.f32 7.1

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u \cdot \color{blue}{\mathsf{PI}\left(\right)}}{s} \cdot 0.5 + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   5. Applied rewrites 7.1%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u \cdot \mathsf{PI}\left(\right)}{s} \cdot 0.5} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   6. Taylor expanded in s around inf

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\left(1 + -1 \cdot \frac{-8 \cdot {\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{{s}^{2}}\right) - 4 \cdot \frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)}{s}\right)} \]
   7. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\left(1 + -1 \cdot \frac{-8 \cdot {\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{{s}^{2}}\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)}{s}\right)} \]

      2. metadata-eval N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\left(1 + -1 \cdot \frac{-8 \cdot {\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{{s}^{2}}\right) + \color{blue}{-4} \cdot \frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)}{s}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(-4 \cdot \frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)}{s} + \left(1 + -1 \cdot \frac{-8 \cdot {\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{{s}^{2}}\right)\right)} \]

   8. Applied rewrites 7.3%

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.25, \mathsf{PI}\left(\right), 0.5 \cdot \left(\mathsf{PI}\left(\right) \cdot u\right)\right)}{s}, -4, 1 - \frac{\frac{\mathsf{fma}\left(-8, {\left(\mathsf{fma}\left(-0.25, \mathsf{PI}\left(\right), 0.5 \cdot \left(\mathsf{PI}\left(\right) \cdot u\right)\right)\right)}^{2}, 0\right)}{s}}{s}\right)\right)} \]

   9. Taylor expanded in u around inf

      \[\leadsto \left(-s\right) \cdot \log \left(2 \cdot \color{blue}{\frac{{u}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{{s}^{2}}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 17.3%

       \[\leadsto \left(-s\right) \cdot \log \left(\left(2 \cdot \left(u \cdot u\right)\right) \cdot \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{s} \cdot \frac{\mathsf{PI}\left(\right)}{s}\right)}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 20.3%

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{\left(\left(u \cdot u\right) \cdot 2\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}{s \cdot \color{blue}{s}}\right) \]

   if -4.9999998e-22 < (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32))))

   1. Initial program 98.7%

      \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in s around inf

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 13.4%

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 17.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(\frac{1}{\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right) \cdot u + \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}} - 1\right) \cdot \left(-s\right) \leq -4.999999841327613 \cdot 10^{-22}:\\ \;\;\;\;\log \left(\frac{\left(\left(u \cdot u\right) \cdot 2\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}{s \cdot s}\right) \cdot \left(-s\right)\\ \mathbf{else}:\\ \;\;\;\;\log 1 \cdot \left(-s\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\\ \log \left(\frac{1}{\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - t\_0\right) \cdot u + t\_0} - 1\right) \cdot \left(-s\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ (exp (/ (PI) s)) 1.0))))
   (*
    (log
     (-
      (/ 1.0 (+ (* (- (/ 1.0 (+ (exp (/ (- (PI)) s)) 1.0)) t_0) u) t_0))
      1.0))
    (- s))))

Derivation

1. Initial program 98.8%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
2. Add Preprocessing

3. Final simplification 98.8%

   \[\leadsto \log \left(\frac{1}{\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right) \cdot u + \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}} - 1\right) \cdot \left(-s\right) \]
4. Add Preprocessing

Alternative 6: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{1}{\left(\frac{1}{e^{\frac{\frac{{\mathsf{PI}\left(\right)}^{3}}{\mathsf{fma}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right), 0 \cdot \mathsf{PI}\left(\right)\right)}}{-s}} + 1} - \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right) \cdot u} - 1\right) \cdot \left(-s\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (log
   (-
    (/
     1.0
     (*
      (-
       (/
        1.0
        (+
         (exp (/ (/ (pow (PI) 3.0) (fma (PI) (PI) (* 0.0 (PI)))) (- s)))
         1.0))
       (/ 1.0 (+ (exp (/ (PI) s)) 1.0)))
      u))
    1.0))
  (- s)))

Derivation

1. Initial program 98.8%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
2. Add Preprocessing

3. Taylor expanded in u around inf

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{u \cdot \left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}} - 1\right) \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) \cdot u}} - 1\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) \cdot u}} - 1\right) \]

5. Applied rewrites 97.5%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right) \cdot u}} - 1\right) \]
6. Step-by-step derivation

7. Applied rewrites 97.6%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\left(\frac{1}{e^{\frac{\frac{0 - {\mathsf{PI}\left(\right)}^{3}}{0 + \mathsf{fma}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right), 0 \cdot \mathsf{PI}\left(\right)\right)}}{s}} + 1} - \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right) \cdot u} - 1\right) \]

8. Final simplification 97.6%

   \[\leadsto \log \left(\frac{1}{\left(\frac{1}{e^{\frac{\frac{{\mathsf{PI}\left(\right)}^{3}}{\mathsf{fma}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right), 0 \cdot \mathsf{PI}\left(\right)\right)}}{-s}} + 1} - \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right) \cdot u} - 1\right) \cdot \left(-s\right) \]
9. Add Preprocessing

Alternative 7: 97.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{1}{\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right) \cdot u} - 1\right) \cdot \left(-s\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (log
   (-
    (/
     1.0
     (*
      (- (/ 1.0 (+ (exp (/ (- (PI)) s)) 1.0)) (/ 1.0 (+ (exp (/ (PI) s)) 1.0)))
      u))
    1.0))
  (- s)))

Derivation

1. Initial program 98.8%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
2. Add Preprocessing

3. Taylor expanded in u around inf

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{u \cdot \left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}} - 1\right) \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) \cdot u}} - 1\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) \cdot u}} - 1\right) \]

5. Applied rewrites 97.5%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right) \cdot u}} - 1\right) \]

6. Final simplification 97.5%

   \[\leadsto \log \left(\frac{1}{\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right) \cdot u} - 1\right) \cdot \left(-s\right) \]
7. Add Preprocessing

Alternative 8: 8.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.5 \cdot \mathsf{PI}\left(\right), u, 0.25 \cdot \mathsf{PI}\left(\right)\right)\\ \log \left(1 - \frac{\mathsf{fma}\left(\frac{{t\_0}^{2}}{s}, -8, \frac{0}{s}\right) - 4 \cdot t\_0}{s}\right) \cdot \left(-s\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (fma (* -0.5 (PI)) u (* 0.25 (PI)))))
   (*
    (log
     (- 1.0 (/ (- (fma (/ (pow t_0 2.0) s) -8.0 (/ 0.0 s)) (* 4.0 t_0)) s)))
    (- s))))

Derivation

1. Initial program 98.8%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
2. Add Preprocessing

3. Taylor expanded in s around -inf

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{-1 \cdot \frac{u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(\mathsf{neg}\left(\frac{u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s}\right)\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{neg}\left(s\right)}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   3. distribute-rgt-out-- N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{4} - \frac{1}{4}\right)\right)}}{\mathsf{neg}\left(s\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   4. metadata-eval N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{-1}{2}}\right)}{\mathsf{neg}\left(s\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   5. associate-*r* N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{\color{blue}{\left(u \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{2}}}{\mathsf{neg}\left(s\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   6. mul-1-neg N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{\left(u \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{2}}{\color{blue}{-1 \cdot s}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   7. *-commutative N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{\left(u \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{2}}{\color{blue}{s \cdot -1}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   8. times-frac N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u \cdot \mathsf{PI}\left(\right)}{s} \cdot \frac{\frac{-1}{2}}{-1}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   9. metadata-eval N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u \cdot \mathsf{PI}\left(\right)}{s} \cdot \color{blue}{\frac{1}{2}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   10. lower-*.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u \cdot \mathsf{PI}\left(\right)}{s} \cdot \frac{1}{2}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   11. lower-/.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u \cdot \mathsf{PI}\left(\right)}{s}} \cdot \frac{1}{2} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   12. lower-*.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{\color{blue}{u \cdot \mathsf{PI}\left(\right)}}{s} \cdot \frac{1}{2} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   13. lower-PI.f32 3.8

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u \cdot \color{blue}{\mathsf{PI}\left(\right)}}{s} \cdot 0.5 + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

5. Applied rewrites 3.8%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u \cdot \mathsf{PI}\left(\right)}{s} \cdot 0.5} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

6. Taylor expanded in s around inf

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\left(1 + -1 \cdot \frac{-8 \cdot {\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{{s}^{2}}\right) - 4 \cdot \frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)}{s}\right)} \]
7. Step-by-step derivation

   1. cancel-sign-sub-inv N/A

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\left(1 + -1 \cdot \frac{-8 \cdot {\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{{s}^{2}}\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)}{s}\right)} \]

   2. metadata-eval N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\left(1 + -1 \cdot \frac{-8 \cdot {\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{{s}^{2}}\right) + \color{blue}{-4} \cdot \frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)}{s}\right) \]

   3. +-commutative N/A

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(-4 \cdot \frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)}{s} + \left(1 + -1 \cdot \frac{-8 \cdot {\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{{s}^{2}}\right)\right)} \]

8. Applied rewrites 7.5%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.25, \mathsf{PI}\left(\right), 0.5 \cdot \left(\mathsf{PI}\left(\right) \cdot u\right)\right)}{s}, -4, 1 - \frac{\frac{\mathsf{fma}\left(-8, {\left(\mathsf{fma}\left(-0.25, \mathsf{PI}\left(\right), 0.5 \cdot \left(\mathsf{PI}\left(\right) \cdot u\right)\right)\right)}^{2}, 0\right)}{s}}{s}\right)\right)} \]

9. Taylor expanded in u around inf

   \[\leadsto \left(-s\right) \cdot \log \left(2 \cdot \color{blue}{\frac{{u}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{{s}^{2}}}\right) \]
10. Step-by-step derivation

11. Applied rewrites 11.0%

    \[\leadsto \left(-s\right) \cdot \log \left(\left(2 \cdot \left(u \cdot u\right)\right) \cdot \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{s} \cdot \frac{\mathsf{PI}\left(\right)}{s}\right)}\right) \]

12. Taylor expanded in s around -inf

    \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(1 + -1 \cdot \frac{\left(-8 \cdot \frac{{\left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{s} + -4 \cdot \frac{\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}}{s}\right) - 4 \cdot \left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s}\right)} \]

14. Applied rewrites 12.9%

    \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(1 - \frac{\mathsf{fma}\left(\frac{{\left(\mathsf{fma}\left(-0.5 \cdot \mathsf{PI}\left(\right), u, 0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{s}, -8, \frac{0}{s}\right) - \mathsf{fma}\left(-0.5 \cdot \mathsf{PI}\left(\right), u, 0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot 4}{s}\right)} \]

15. Final simplification 12.8%

    \[\leadsto \log \left(1 - \frac{\mathsf{fma}\left(\frac{{\left(\mathsf{fma}\left(-0.5 \cdot \mathsf{PI}\left(\right), u, 0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{s}, -8, \frac{0}{s}\right) - 4 \cdot \mathsf{fma}\left(-0.5 \cdot \mathsf{PI}\left(\right), u, 0.25 \cdot \mathsf{PI}\left(\right)\right)}{s}\right) \cdot \left(-s\right) \]
16. Add Preprocessing

Alternative 9: 19.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 2.5000001145342624 \cdot 10^{-19}:\\ \;\;\;\;\log \left(\left(2 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot \mathsf{PI}\left(\right), u, -0.25 \cdot \mathsf{PI}\left(\right)\right), -4, -2 \cdot \mathsf{PI}\left(\right)\right)}{s}\right) - 1\right) \cdot \left(-s\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\frac{\mathsf{PI}\left(\right)}{s} + 1\right) - \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s} \cdot \frac{-0.5}{s}\right) \cdot \left(-s\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (if (<= s 2.5000001145342624e-19)
   (*
    (log
     (-
      (-
       2.0
       (/ (fma (fma (* -0.5 (PI)) u (* -0.25 (PI))) -4.0 (* -2.0 (PI))) s))
      1.0))
    (- s))
   (* (log (- (+ (/ (PI) s) 1.0) (* (/ (* (PI) (PI)) s) (/ -0.5 s)))) (- s))))

Derivation

1. Split input into 2 regimes

2. if s < 2.50000011e-19

   1. Initial program 98.8%

      \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f32 N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{\mathsf{PI}\left(\right)}{s}}}}} - 1\right) \]

      2. remove-double-neg N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right)\right)}}{s}}}} - 1\right) \]

      3. lift-neg.f32 N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{neg}\left(\color{blue}{\left(-\mathsf{PI}\left(\right)\right)}\right)}{s}}}} - 1\right) \]

      4. neg-sub0 N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{0 - \left(-\mathsf{PI}\left(\right)\right)}}{s}}}} - 1\right) \]

      5. div-sub N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{0}{s} - \frac{-\mathsf{PI}\left(\right)}{s}}}}} - 1\right) \]

      6. frac-2neg N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{\mathsf{neg}\left(0\right)}{\mathsf{neg}\left(s\right)}} - \frac{-\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{0}}{\mathsf{neg}\left(s\right)} - \frac{-\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      8. lift-neg.f32 N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{0}{\color{blue}{-s}} - \frac{-\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      9. frac-sub N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\left(-s\right) \cdot s}}}}} - 1\right) \]

      10. lower-/.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\left(-s\right) \cdot s}}}}} - 1\right) \]

      11. lower--.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}}{\left(-s\right) \cdot s}}}} - 1\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{0 \cdot s} - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\left(-s\right) \cdot s}}}} - 1\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{0 \cdot s - \color{blue}{\left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}}{\left(-s\right) \cdot s}}}} - 1\right) \]

      14. lower-*.f32 98.8

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\color{blue}{\left(-s\right) \cdot s}}}}} - 1\right) \]

   4. Applied rewrites 98.8%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\left(-s\right) \cdot s}}}}} - 1\right) \]
   5. Step-by-step derivation

      1. lift-/.f32 N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\left(-s\right) \cdot s}}}}} - 1\right) \]

      2. frac-2neg N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{\mathsf{neg}\left(\left(0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)\right)\right)}{\mathsf{neg}\left(\left(-s\right) \cdot s\right)}}}}} - 1\right) \]

      3. frac-2neg N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)\right)\right)\right)\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(-s\right) \cdot s\right)\right)\right)}}}}} - 1\right) \]

      4. remove-double-neg N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(-s\right) \cdot s\right)\right)\right)}}}} - 1\right) \]

      5. lift--.f32 N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(-s\right) \cdot s\right)\right)\right)}}}} - 1\right) \]

      6. lift-*.f32 N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{0 \cdot s} - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(-s\right) \cdot s\right)\right)\right)}}}} - 1\right) \]

      7. mul0-lft N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{0} - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(-s\right) \cdot s\right)\right)\right)}}}} - 1\right) \]

      8. neg-sub0 N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{\mathsf{neg}\left(\left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)\right)}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(-s\right) \cdot s\right)\right)\right)}}}} - 1\right) \]

      9. frac-2neg N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{\left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\mathsf{neg}\left(\left(-s\right) \cdot s\right)}}}}} - 1\right) \]

      10. lower-/.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{\left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\mathsf{neg}\left(\left(-s\right) \cdot s\right)}}}}} - 1\right) \]

      11. lift-*.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{\left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}}{\mathsf{neg}\left(\left(-s\right) \cdot s\right)}}}} - 1\right) \]

      12. lift-neg.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\left(-s\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right)}}{\mathsf{neg}\left(\left(-s\right) \cdot s\right)}}}} - 1\right) \]

      13. distribute-rgt-neg-out N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{\mathsf{neg}\left(\left(-s\right) \cdot \mathsf{PI}\left(\right)\right)}}{\mathsf{neg}\left(\left(-s\right) \cdot s\right)}}}} - 1\right) \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{\left(\mathsf{neg}\left(\left(-s\right)\right)\right) \cdot \mathsf{PI}\left(\right)}}{\mathsf{neg}\left(\left(-s\right) \cdot s\right)}}}} - 1\right) \]

      15. lift-neg.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(s\right)\right)}\right)\right) \cdot \mathsf{PI}\left(\right)}{\mathsf{neg}\left(\left(-s\right) \cdot s\right)}}}} - 1\right) \]

      16. remove-double-neg N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{s} \cdot \mathsf{PI}\left(\right)}{\mathsf{neg}\left(\left(-s\right) \cdot s\right)}}}} - 1\right) \]

      17. lower-*.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{s \cdot \mathsf{PI}\left(\right)}}{\mathsf{neg}\left(\left(-s\right) \cdot s\right)}}}} - 1\right) \]

      18. lift-*.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{s \cdot \mathsf{PI}\left(\right)}{\mathsf{neg}\left(\color{blue}{\left(-s\right) \cdot s}\right)}}}} - 1\right) \]

      19. lift-neg.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{s \cdot \mathsf{PI}\left(\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(s\right)\right)} \cdot s\right)}}}} - 1\right) \]

      20. distribute-lft-neg-out N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{s \cdot \mathsf{PI}\left(\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(s \cdot s\right)\right)}\right)}}}} - 1\right) \]

   6. Applied rewrites 98.8%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{s \cdot \mathsf{PI}\left(\right)}{s \cdot s}}}}} - 1\right) \]

   8. Applied rewrites 98.7%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{{\left(e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1\right)}^{-2} - {\left(\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right) \cdot u\right)}^{2}}{\frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1} - \left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right) \cdot u}}} - 1\right) \]

   9. Taylor expanded in s around -inf

      \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{\left(2 + -1 \cdot \frac{-4 \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right) - 2 \cdot \mathsf{PI}\left(\right)}{s}\right)} - 1\right) \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\left(2 + \color{blue}{\left(\mathsf{neg}\left(\frac{-4 \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right) - 2 \cdot \mathsf{PI}\left(\right)}{s}\right)\right)}\right) - 1\right) \]

       2. unsub-neg N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{\left(2 - \frac{-4 \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right) - 2 \cdot \mathsf{PI}\left(\right)}{s}\right)} - 1\right) \]

       3. lower--.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{\left(2 - \frac{-4 \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right) - 2 \cdot \mathsf{PI}\left(\right)}{s}\right)} - 1\right) \]

       4. lower-/.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\left(2 - \color{blue}{\frac{-4 \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right) - 2 \cdot \mathsf{PI}\left(\right)}{s}}\right) - 1\right) \]

   11. Applied rewrites 10.5%

       \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{\left(2 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot \mathsf{PI}\left(\right), u, -0.25 \cdot \mathsf{PI}\left(\right)\right), -4, -2 \cdot \mathsf{PI}\left(\right)\right)}{s}\right)} - 1\right) \]

   if 2.50000011e-19 < s

   1. Initial program 98.9%

      \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in s around -inf

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{-1 \cdot \frac{u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(\mathsf{neg}\left(\frac{u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s}\right)\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{neg}\left(s\right)}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      3. distribute-rgt-out-- N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{4} - \frac{1}{4}\right)\right)}}{\mathsf{neg}\left(s\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      4. metadata-eval N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{-1}{2}}\right)}{\mathsf{neg}\left(s\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{\color{blue}{\left(u \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{2}}}{\mathsf{neg}\left(s\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      6. mul-1-neg N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{\left(u \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{2}}{\color{blue}{-1 \cdot s}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      7. *-commutative N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{\left(u \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{2}}{\color{blue}{s \cdot -1}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      8. times-frac N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u \cdot \mathsf{PI}\left(\right)}{s} \cdot \frac{\frac{-1}{2}}{-1}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u \cdot \mathsf{PI}\left(\right)}{s} \cdot \color{blue}{\frac{1}{2}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      10. lower-*.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u \cdot \mathsf{PI}\left(\right)}{s} \cdot \frac{1}{2}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      11. lower-/.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u \cdot \mathsf{PI}\left(\right)}{s}} \cdot \frac{1}{2} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{\color{blue}{u \cdot \mathsf{PI}\left(\right)}}{s} \cdot \frac{1}{2} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      13. lower-PI.f32 8.2

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u \cdot \color{blue}{\mathsf{PI}\left(\right)}}{s} \cdot 0.5 + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   5. Applied rewrites 8.2%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u \cdot \mathsf{PI}\left(\right)}{s} \cdot 0.5} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   6. Taylor expanded in s around inf

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\left(1 + -1 \cdot \frac{-8 \cdot {\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{{s}^{2}}\right) - 4 \cdot \frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)}{s}\right)} \]
   7. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\left(1 + -1 \cdot \frac{-8 \cdot {\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{{s}^{2}}\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)}{s}\right)} \]

      2. metadata-eval N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\left(1 + -1 \cdot \frac{-8 \cdot {\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{{s}^{2}}\right) + \color{blue}{-4} \cdot \frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)}{s}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(-4 \cdot \frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)}{s} + \left(1 + -1 \cdot \frac{-8 \cdot {\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{{s}^{2}}\right)\right)} \]

   8. Applied rewrites 8.2%

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.25, \mathsf{PI}\left(\right), 0.5 \cdot \left(\mathsf{PI}\left(\right) \cdot u\right)\right)}{s}, -4, 1 - \frac{\frac{\mathsf{fma}\left(-8, {\left(\mathsf{fma}\left(-0.25, \mathsf{PI}\left(\right), 0.5 \cdot \left(\mathsf{PI}\left(\right) \cdot u\right)\right)\right)}^{2}, 0\right)}{s}}{s}\right)\right)} \]

   9. Taylor expanded in u around 0

      \[\leadsto \left(-s\right) \cdot \log \left(\left(1 + \frac{\mathsf{PI}\left(\right)}{s}\right) - \color{blue}{\frac{-1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{s}^{2}}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 25.0%

       \[\leadsto \left(-s\right) \cdot \log \left(\left(\frac{\mathsf{PI}\left(\right)}{s} + 1\right) - \color{blue}{\frac{-0.5}{s} \cdot \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 17.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 2.5000001145342624 \cdot 10^{-19}:\\ \;\;\;\;\log \left(\left(2 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot \mathsf{PI}\left(\right), u, -0.25 \cdot \mathsf{PI}\left(\right)\right), -4, -2 \cdot \mathsf{PI}\left(\right)\right)}{s}\right) - 1\right) \cdot \left(-s\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\frac{\mathsf{PI}\left(\right)}{s} + 1\right) - \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s} \cdot \frac{-0.5}{s}\right) \cdot \left(-s\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 21.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \log \left(\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.5 \cdot \mathsf{PI}\left(\right), u, 0.25 \cdot \mathsf{PI}\left(\right)\right)}{s}, -4, 2\right) - -2 \cdot \frac{\mathsf{PI}\left(\right)}{s}\right) - 1\right) \cdot \left(-s\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (log
   (-
    (-
     (fma (/ (fma (* 0.5 (PI)) u (* 0.25 (PI))) s) -4.0 2.0)
     (* -2.0 (/ (PI) s)))
    1.0))
  (- s)))

Derivation

1. Initial program 98.8%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{\mathsf{PI}\left(\right)}{s}}}}} - 1\right) \]

   2. remove-double-neg N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right)\right)}}{s}}}} - 1\right) \]

   3. lift-neg.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{neg}\left(\color{blue}{\left(-\mathsf{PI}\left(\right)\right)}\right)}{s}}}} - 1\right) \]

   4. neg-sub0 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{0 - \left(-\mathsf{PI}\left(\right)\right)}}{s}}}} - 1\right) \]

   5. div-sub N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{0}{s} - \frac{-\mathsf{PI}\left(\right)}{s}}}}} - 1\right) \]

   6. frac-2neg N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{\mathsf{neg}\left(0\right)}{\mathsf{neg}\left(s\right)}} - \frac{-\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   7. metadata-eval N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{0}}{\mathsf{neg}\left(s\right)} - \frac{-\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   8. lift-neg.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{0}{\color{blue}{-s}} - \frac{-\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   9. frac-sub N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\left(-s\right) \cdot s}}}}} - 1\right) \]

   10. lower-/.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\left(-s\right) \cdot s}}}}} - 1\right) \]

   11. lower--.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}}{\left(-s\right) \cdot s}}}} - 1\right) \]

   12. lower-*.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{0 \cdot s} - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\left(-s\right) \cdot s}}}} - 1\right) \]

   13. lower-*.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{0 \cdot s - \color{blue}{\left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}}{\left(-s\right) \cdot s}}}} - 1\right) \]

   14. lower-*.f32 98.8

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\color{blue}{\left(-s\right) \cdot s}}}}} - 1\right) \]

4. Applied rewrites 98.8%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\left(-s\right) \cdot s}}}}} - 1\right) \]
5. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\left(-s\right) \cdot s}}}}} - 1\right) \]

   2. frac-2neg N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{\mathsf{neg}\left(\left(0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)\right)\right)}{\mathsf{neg}\left(\left(-s\right) \cdot s\right)}}}}} - 1\right) \]

   3. frac-2neg N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)\right)\right)\right)\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(-s\right) \cdot s\right)\right)\right)}}}}} - 1\right) \]

   4. remove-double-neg N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(-s\right) \cdot s\right)\right)\right)}}}} - 1\right) \]

   5. lift--.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(-s\right) \cdot s\right)\right)\right)}}}} - 1\right) \]

   6. lift-*.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{0 \cdot s} - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(-s\right) \cdot s\right)\right)\right)}}}} - 1\right) \]

   7. mul0-lft N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{0} - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(-s\right) \cdot s\right)\right)\right)}}}} - 1\right) \]

   8. neg-sub0 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{\mathsf{neg}\left(\left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)\right)}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(-s\right) \cdot s\right)\right)\right)}}}} - 1\right) \]

   9. frac-2neg N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{\left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\mathsf{neg}\left(\left(-s\right) \cdot s\right)}}}}} - 1\right) \]

   10. lower-/.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{\left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\mathsf{neg}\left(\left(-s\right) \cdot s\right)}}}}} - 1\right) \]

   11. lift-*.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{\left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}}{\mathsf{neg}\left(\left(-s\right) \cdot s\right)}}}} - 1\right) \]

   12. lift-neg.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\left(-s\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right)}}{\mathsf{neg}\left(\left(-s\right) \cdot s\right)}}}} - 1\right) \]

   13. distribute-rgt-neg-out N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{\mathsf{neg}\left(\left(-s\right) \cdot \mathsf{PI}\left(\right)\right)}}{\mathsf{neg}\left(\left(-s\right) \cdot s\right)}}}} - 1\right) \]

   14. distribute-lft-neg-in N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{\left(\mathsf{neg}\left(\left(-s\right)\right)\right) \cdot \mathsf{PI}\left(\right)}}{\mathsf{neg}\left(\left(-s\right) \cdot s\right)}}}} - 1\right) \]

   15. lift-neg.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(s\right)\right)}\right)\right) \cdot \mathsf{PI}\left(\right)}{\mathsf{neg}\left(\left(-s\right) \cdot s\right)}}}} - 1\right) \]

   16. remove-double-neg N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{s} \cdot \mathsf{PI}\left(\right)}{\mathsf{neg}\left(\left(-s\right) \cdot s\right)}}}} - 1\right) \]

   17. lower-*.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{s \cdot \mathsf{PI}\left(\right)}}{\mathsf{neg}\left(\left(-s\right) \cdot s\right)}}}} - 1\right) \]

   18. lift-*.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{s \cdot \mathsf{PI}\left(\right)}{\mathsf{neg}\left(\color{blue}{\left(-s\right) \cdot s}\right)}}}} - 1\right) \]

   19. lift-neg.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{s \cdot \mathsf{PI}\left(\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(s\right)\right)} \cdot s\right)}}}} - 1\right) \]

   20. distribute-lft-neg-out N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{s \cdot \mathsf{PI}\left(\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(s \cdot s\right)\right)}\right)}}}} - 1\right) \]

6. Applied rewrites 98.8%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{s \cdot \mathsf{PI}\left(\right)}{s \cdot s}}}}} - 1\right) \]

8. Applied rewrites 98.8%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{{\left(e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1\right)}^{-2} - {\left(\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right) \cdot u\right)}^{2}}{\frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1} - \left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right) \cdot u}}} - 1\right) \]

9. Taylor expanded in s around inf

   \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{\left(\left(2 + -4 \cdot \frac{\frac{1}{4} \cdot \mathsf{PI}\left(\right) + u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s}\right) - -2 \cdot \frac{\mathsf{PI}\left(\right)}{s}\right)} - 1\right) \]
10. Step-by-step derivation

    1. lower--.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{\left(\left(2 + -4 \cdot \frac{\frac{1}{4} \cdot \mathsf{PI}\left(\right) + u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s}\right) - -2 \cdot \frac{\mathsf{PI}\left(\right)}{s}\right)} - 1\right) \]

11. Applied rewrites 21.6%

    \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.5 \cdot \mathsf{PI}\left(\right), u, 0.25 \cdot \mathsf{PI}\left(\right)\right)}{s}, -4, 2\right) - -2 \cdot \frac{\mathsf{PI}\left(\right)}{s}\right)} - 1\right) \]

12. Final simplification 21.7%

    \[\leadsto \log \left(\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.5 \cdot \mathsf{PI}\left(\right), u, 0.25 \cdot \mathsf{PI}\left(\right)\right)}{s}, -4, 2\right) - -2 \cdot \frac{\mathsf{PI}\left(\right)}{s}\right) - 1\right) \cdot \left(-s\right) \]
13. Add Preprocessing

Alternative 11: 17.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 1.8000000105227632 \cdot 10^{-25}:\\ \;\;\;\;\log \left(\left(2 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot \mathsf{PI}\left(\right), u, -0.25 \cdot \mathsf{PI}\left(\right)\right), -4, -2 \cdot \mathsf{PI}\left(\right)\right)}{s}\right) - 1\right) \cdot \left(-s\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{\left(\left(\frac{\mathsf{PI}\left(\right)}{s} \cdot u\right) \cdot \left(2 \cdot u\right)\right) \cdot \mathsf{PI}\left(\right)}{s}\right) \cdot \left(-s\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (if (<= s 1.8000000105227632e-25)
   (*
    (log
     (-
      (-
       2.0
       (/ (fma (fma (* -0.5 (PI)) u (* -0.25 (PI))) -4.0 (* -2.0 (PI))) s))
      1.0))
    (- s))
   (* (log (/ (* (* (* (/ (PI) s) u) (* 2.0 u)) (PI)) s)) (- s))))

Derivation

1. Split input into 2 regimes

2. if s < 1.80000001e-25

   1. Initial program 99.0%

      \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f32 N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{\mathsf{PI}\left(\right)}{s}}}}} - 1\right) \]

      2. remove-double-neg N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right)\right)}}{s}}}} - 1\right) \]

      3. lift-neg.f32 N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{neg}\left(\color{blue}{\left(-\mathsf{PI}\left(\right)\right)}\right)}{s}}}} - 1\right) \]

      4. neg-sub0 N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{0 - \left(-\mathsf{PI}\left(\right)\right)}}{s}}}} - 1\right) \]

      5. div-sub N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{0}{s} - \frac{-\mathsf{PI}\left(\right)}{s}}}}} - 1\right) \]

      6. frac-2neg N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{\mathsf{neg}\left(0\right)}{\mathsf{neg}\left(s\right)}} - \frac{-\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{0}}{\mathsf{neg}\left(s\right)} - \frac{-\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      8. lift-neg.f32 N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{0}{\color{blue}{-s}} - \frac{-\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      9. frac-sub N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\left(-s\right) \cdot s}}}}} - 1\right) \]

      10. lower-/.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\left(-s\right) \cdot s}}}}} - 1\right) \]

      11. lower--.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}}{\left(-s\right) \cdot s}}}} - 1\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{0 \cdot s} - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\left(-s\right) \cdot s}}}} - 1\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{0 \cdot s - \color{blue}{\left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}}{\left(-s\right) \cdot s}}}} - 1\right) \]

      14. lower-*.f32 99.0

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\color{blue}{\left(-s\right) \cdot s}}}}} - 1\right) \]

   4. Applied rewrites 99.0%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\left(-s\right) \cdot s}}}}} - 1\right) \]
   5. Step-by-step derivation

      1. lift-/.f32 N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\left(-s\right) \cdot s}}}}} - 1\right) \]

      2. frac-2neg N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{\mathsf{neg}\left(\left(0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)\right)\right)}{\mathsf{neg}\left(\left(-s\right) \cdot s\right)}}}}} - 1\right) \]

      3. frac-2neg N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)\right)\right)\right)\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(-s\right) \cdot s\right)\right)\right)}}}}} - 1\right) \]

      4. remove-double-neg N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(-s\right) \cdot s\right)\right)\right)}}}} - 1\right) \]

      5. lift--.f32 N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(-s\right) \cdot s\right)\right)\right)}}}} - 1\right) \]

      6. lift-*.f32 N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{0 \cdot s} - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(-s\right) \cdot s\right)\right)\right)}}}} - 1\right) \]

      7. mul0-lft N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{0} - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(-s\right) \cdot s\right)\right)\right)}}}} - 1\right) \]

      8. neg-sub0 N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{\mathsf{neg}\left(\left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)\right)}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(-s\right) \cdot s\right)\right)\right)}}}} - 1\right) \]

      9. frac-2neg N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{\left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\mathsf{neg}\left(\left(-s\right) \cdot s\right)}}}}} - 1\right) \]

      10. lower-/.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{\left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\mathsf{neg}\left(\left(-s\right) \cdot s\right)}}}}} - 1\right) \]

      11. lift-*.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{\left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}}{\mathsf{neg}\left(\left(-s\right) \cdot s\right)}}}} - 1\right) \]

      12. lift-neg.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\left(-s\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right)}}{\mathsf{neg}\left(\left(-s\right) \cdot s\right)}}}} - 1\right) \]

      13. distribute-rgt-neg-out N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{\mathsf{neg}\left(\left(-s\right) \cdot \mathsf{PI}\left(\right)\right)}}{\mathsf{neg}\left(\left(-s\right) \cdot s\right)}}}} - 1\right) \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{\left(\mathsf{neg}\left(\left(-s\right)\right)\right) \cdot \mathsf{PI}\left(\right)}}{\mathsf{neg}\left(\left(-s\right) \cdot s\right)}}}} - 1\right) \]

      15. lift-neg.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(s\right)\right)}\right)\right) \cdot \mathsf{PI}\left(\right)}{\mathsf{neg}\left(\left(-s\right) \cdot s\right)}}}} - 1\right) \]

      16. remove-double-neg N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{s} \cdot \mathsf{PI}\left(\right)}{\mathsf{neg}\left(\left(-s\right) \cdot s\right)}}}} - 1\right) \]

      17. lower-*.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{s \cdot \mathsf{PI}\left(\right)}}{\mathsf{neg}\left(\left(-s\right) \cdot s\right)}}}} - 1\right) \]

      18. lift-*.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{s \cdot \mathsf{PI}\left(\right)}{\mathsf{neg}\left(\color{blue}{\left(-s\right) \cdot s}\right)}}}} - 1\right) \]

      19. lift-neg.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{s \cdot \mathsf{PI}\left(\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(s\right)\right)} \cdot s\right)}}}} - 1\right) \]

      20. distribute-lft-neg-out N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{s \cdot \mathsf{PI}\left(\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(s \cdot s\right)\right)}\right)}}}} - 1\right) \]

   6. Applied rewrites 99.0%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{s \cdot \mathsf{PI}\left(\right)}{s \cdot s}}}}} - 1\right) \]

   8. Applied rewrites 99.0%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{{\left(e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1\right)}^{-2} - {\left(\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right) \cdot u\right)}^{2}}{\frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1} - \left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right) \cdot u}}} - 1\right) \]

   9. Taylor expanded in s around -inf

      \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{\left(2 + -1 \cdot \frac{-4 \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right) - 2 \cdot \mathsf{PI}\left(\right)}{s}\right)} - 1\right) \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\left(2 + \color{blue}{\left(\mathsf{neg}\left(\frac{-4 \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right) - 2 \cdot \mathsf{PI}\left(\right)}{s}\right)\right)}\right) - 1\right) \]

       2. unsub-neg N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{\left(2 - \frac{-4 \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right) - 2 \cdot \mathsf{PI}\left(\right)}{s}\right)} - 1\right) \]

       3. lower--.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{\left(2 - \frac{-4 \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right) - 2 \cdot \mathsf{PI}\left(\right)}{s}\right)} - 1\right) \]

       4. lower-/.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\left(2 - \color{blue}{\frac{-4 \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right) - 2 \cdot \mathsf{PI}\left(\right)}{s}}\right) - 1\right) \]

   11. Applied rewrites 10.2%

       \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{\left(2 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot \mathsf{PI}\left(\right), u, -0.25 \cdot \mathsf{PI}\left(\right)\right), -4, -2 \cdot \mathsf{PI}\left(\right)\right)}{s}\right)} - 1\right) \]

   if 1.80000001e-25 < s

   1. Initial program 98.8%

      \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in s around -inf

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{-1 \cdot \frac{u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(\mathsf{neg}\left(\frac{u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s}\right)\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{neg}\left(s\right)}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      3. distribute-rgt-out-- N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{4} - \frac{1}{4}\right)\right)}}{\mathsf{neg}\left(s\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      4. metadata-eval N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{-1}{2}}\right)}{\mathsf{neg}\left(s\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{\color{blue}{\left(u \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{2}}}{\mathsf{neg}\left(s\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      6. mul-1-neg N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{\left(u \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{2}}{\color{blue}{-1 \cdot s}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      7. *-commutative N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{\left(u \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{2}}{\color{blue}{s \cdot -1}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      8. times-frac N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u \cdot \mathsf{PI}\left(\right)}{s} \cdot \frac{\frac{-1}{2}}{-1}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u \cdot \mathsf{PI}\left(\right)}{s} \cdot \color{blue}{\frac{1}{2}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      10. lower-*.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u \cdot \mathsf{PI}\left(\right)}{s} \cdot \frac{1}{2}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      11. lower-/.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u \cdot \mathsf{PI}\left(\right)}{s}} \cdot \frac{1}{2} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{\color{blue}{u \cdot \mathsf{PI}\left(\right)}}{s} \cdot \frac{1}{2} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      13. lower-PI.f32 6.1

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u \cdot \color{blue}{\mathsf{PI}\left(\right)}}{s} \cdot 0.5 + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   5. Applied rewrites 6.1%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u \cdot \mathsf{PI}\left(\right)}{s} \cdot 0.5} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   6. Taylor expanded in s around inf

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\left(1 + -1 \cdot \frac{-8 \cdot {\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{{s}^{2}}\right) - 4 \cdot \frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)}{s}\right)} \]
   7. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\left(1 + -1 \cdot \frac{-8 \cdot {\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{{s}^{2}}\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)}{s}\right)} \]

      2. metadata-eval N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\left(1 + -1 \cdot \frac{-8 \cdot {\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{{s}^{2}}\right) + \color{blue}{-4} \cdot \frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)}{s}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(-4 \cdot \frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)}{s} + \left(1 + -1 \cdot \frac{-8 \cdot {\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{{s}^{2}}\right)\right)} \]

   8. Applied rewrites 6.8%

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.25, \mathsf{PI}\left(\right), 0.5 \cdot \left(\mathsf{PI}\left(\right) \cdot u\right)\right)}{s}, -4, 1 - \frac{\frac{\mathsf{fma}\left(-8, {\left(\mathsf{fma}\left(-0.25, \mathsf{PI}\left(\right), 0.5 \cdot \left(\mathsf{PI}\left(\right) \cdot u\right)\right)\right)}^{2}, 0\right)}{s}}{s}\right)\right)} \]

   9. Taylor expanded in u around inf

      \[\leadsto \left(-s\right) \cdot \log \left(2 \cdot \color{blue}{\frac{{u}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{{s}^{2}}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 15.3%

       \[\leadsto \left(-s\right) \cdot \log \left(\left(2 \cdot \left(u \cdot u\right)\right) \cdot \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{s} \cdot \frac{\mathsf{PI}\left(\right)}{s}\right)}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 19.3%

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{\left(\left(2 \cdot u\right) \cdot \left(u \cdot \frac{\mathsf{PI}\left(\right)}{s}\right)\right) \cdot \mathsf{PI}\left(\right)}{s}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 15.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 1.8000000105227632 \cdot 10^{-25}:\\ \;\;\;\;\log \left(\left(2 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot \mathsf{PI}\left(\right), u, -0.25 \cdot \mathsf{PI}\left(\right)\right), -4, -2 \cdot \mathsf{PI}\left(\right)\right)}{s}\right) - 1\right) \cdot \left(-s\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{\left(\left(\frac{\mathsf{PI}\left(\right)}{s} \cdot u\right) \cdot \left(2 \cdot u\right)\right) \cdot \mathsf{PI}\left(\right)}{s}\right) \cdot \left(-s\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 8.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \log \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.25, \mathsf{PI}\left(\right), \left(\mathsf{PI}\left(\right) \cdot u\right) \cdot 0.5\right)}{s}, -4, 1\right)\right) \cdot \left(-s\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* (log (fma (/ (fma -0.25 (PI) (* (* (PI) u) 0.5)) s) -4.0 1.0)) (- s)))

Derivation

1. Initial program 98.8%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
2. Add Preprocessing

3. Taylor expanded in s around -inf

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{-1 \cdot \frac{u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(\mathsf{neg}\left(\frac{u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s}\right)\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{neg}\left(s\right)}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   3. distribute-rgt-out-- N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{4} - \frac{1}{4}\right)\right)}}{\mathsf{neg}\left(s\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   4. metadata-eval N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{-1}{2}}\right)}{\mathsf{neg}\left(s\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   5. associate-*r* N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{\color{blue}{\left(u \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{2}}}{\mathsf{neg}\left(s\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   6. mul-1-neg N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{\left(u \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{2}}{\color{blue}{-1 \cdot s}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   7. *-commutative N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{\left(u \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{2}}{\color{blue}{s \cdot -1}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   8. times-frac N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u \cdot \mathsf{PI}\left(\right)}{s} \cdot \frac{\frac{-1}{2}}{-1}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   9. metadata-eval N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u \cdot \mathsf{PI}\left(\right)}{s} \cdot \color{blue}{\frac{1}{2}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   10. lower-*.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u \cdot \mathsf{PI}\left(\right)}{s} \cdot \frac{1}{2}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   11. lower-/.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u \cdot \mathsf{PI}\left(\right)}{s}} \cdot \frac{1}{2} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   12. lower-*.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{\color{blue}{u \cdot \mathsf{PI}\left(\right)}}{s} \cdot \frac{1}{2} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   13. lower-PI.f32 3.8

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u \cdot \color{blue}{\mathsf{PI}\left(\right)}}{s} \cdot 0.5 + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

5. Applied rewrites 3.8%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u \cdot \mathsf{PI}\left(\right)}{s} \cdot 0.5} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

6. Taylor expanded in s around inf

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(1 + -4 \cdot \frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)}{s}\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(-4 \cdot \frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)}{s} + 1\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{\frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)}{s} \cdot -4} + 1\right) \]

   3. lower-fma.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\mathsf{fma}\left(\frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)}{s}, -4, 1\right)\right)} \]

8. Applied rewrites 10.4%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.25, \mathsf{PI}\left(\right), 0.5 \cdot \left(\mathsf{PI}\left(\right) \cdot u\right)\right)}{s}, -4, 1\right)\right)} \]

9. Final simplification 10.4%

   \[\leadsto \log \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.25, \mathsf{PI}\left(\right), \left(\mathsf{PI}\left(\right) \cdot u\right) \cdot 0.5\right)}{s}, -4, 1\right)\right) \cdot \left(-s\right) \]
10. Add Preprocessing

Alternative 13: 11.6% accurate, 17.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(\frac{0.25}{u} - 0.5\right) \cdot u\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -4 \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (* (* (* (- (/ 0.25 u) 0.5) u) (PI)) -4.0))

Derivation

1. Initial program 98.8%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
2. Add Preprocessing

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{-1 \cdot \mathsf{PI}\left(\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)} \]

   2. lower-neg.f32 N/A

      \[\leadsto \color{blue}{-\mathsf{PI}\left(\right)} \]

   3. lower-PI.f32 11.0

      \[\leadsto -\color{blue}{\mathsf{PI}\left(\right)} \]

5. Applied rewrites 11.0%

   \[\leadsto \color{blue}{-\mathsf{PI}\left(\right)} \]

6. Taylor expanded in s around -inf

   \[\leadsto \color{blue}{-4 \cdot \left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot -4} \]

   2. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot -4} \]

   3. cancel-sign-sub-inv N/A

      \[\leadsto \color{blue}{\left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot -4 \]

   4. metadata-eval N/A

      \[\leadsto \left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) + \color{blue}{\frac{1}{4}} \cdot \mathsf{PI}\left(\right)\right) \cdot -4 \]

   5. distribute-rgt-out-- N/A

      \[\leadsto \left(u \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{4} - \frac{1}{4}\right)\right)} + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot -4 \]

   6. metadata-eval N/A

      \[\leadsto \left(u \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{-1}{2}}\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot -4 \]

   7. *-commutative N/A

      \[\leadsto \left(u \cdot \color{blue}{\left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)} + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot -4 \]

   8. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(u \cdot \frac{-1}{2}\right) \cdot \mathsf{PI}\left(\right)} + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot -4 \]

   9. *-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\frac{-1}{2} \cdot u\right)} \cdot \mathsf{PI}\left(\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot -4 \]

   10. distribute-rgt-out N/A

       \[\leadsto \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{2} \cdot u + \frac{1}{4}\right)\right)} \cdot -4 \]

   11. lower-*.f32 N/A

       \[\leadsto \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{2} \cdot u + \frac{1}{4}\right)\right)} \cdot -4 \]

   12. lower-PI.f32 N/A

       \[\leadsto \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\frac{-1}{2} \cdot u + \frac{1}{4}\right)\right) \cdot -4 \]

   13. lower-fma.f32 11.0

       \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5, u, 0.25\right)}\right) \cdot -4 \]

8. Applied rewrites 11.0%

   \[\leadsto \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-0.5, u, 0.25\right)\right) \cdot -4} \]

9. Taylor expanded in u around inf

   \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(u \cdot \left(\frac{1}{4} \cdot \frac{1}{u} - \frac{1}{2}\right)\right)\right) \cdot -4 \]
10. Step-by-step derivation

11. Applied rewrites 11.3%

    \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\left(\frac{0.25}{u} - 0.5\right) \cdot u\right)\right) \cdot -4 \]

12. Final simplification 11.3%

    \[\leadsto \left(\left(\left(\frac{0.25}{u} - 0.5\right) \cdot u\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -4 \]
13. Add Preprocessing

Alternative 14: 11.6% accurate, 26.8× speedup

Math

\[\begin{array}{l} \\ \left(\left(-0.5 \cdot u + 0.25\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -4 \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (* (* (+ (* -0.5 u) 0.25) (PI)) -4.0))

Derivation

1. Initial program 98.8%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
2. Add Preprocessing

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{-1 \cdot \mathsf{PI}\left(\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)} \]

   2. lower-neg.f32 N/A

      \[\leadsto \color{blue}{-\mathsf{PI}\left(\right)} \]

   3. lower-PI.f32 11.0

      \[\leadsto -\color{blue}{\mathsf{PI}\left(\right)} \]

5. Applied rewrites 11.0%

   \[\leadsto \color{blue}{-\mathsf{PI}\left(\right)} \]

6. Taylor expanded in s around -inf

   \[\leadsto \color{blue}{-4 \cdot \left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot -4} \]

   2. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot -4} \]

   3. cancel-sign-sub-inv N/A

      \[\leadsto \color{blue}{\left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot -4 \]

   4. metadata-eval N/A

      \[\leadsto \left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) + \color{blue}{\frac{1}{4}} \cdot \mathsf{PI}\left(\right)\right) \cdot -4 \]

   5. distribute-rgt-out-- N/A

      \[\leadsto \left(u \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{4} - \frac{1}{4}\right)\right)} + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot -4 \]

   6. metadata-eval N/A

      \[\leadsto \left(u \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{-1}{2}}\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot -4 \]

   7. *-commutative N/A

      \[\leadsto \left(u \cdot \color{blue}{\left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)} + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot -4 \]

   8. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(u \cdot \frac{-1}{2}\right) \cdot \mathsf{PI}\left(\right)} + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot -4 \]

   9. *-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\frac{-1}{2} \cdot u\right)} \cdot \mathsf{PI}\left(\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot -4 \]

   10. distribute-rgt-out N/A

       \[\leadsto \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{2} \cdot u + \frac{1}{4}\right)\right)} \cdot -4 \]

   11. lower-*.f32 N/A

       \[\leadsto \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{2} \cdot u + \frac{1}{4}\right)\right)} \cdot -4 \]

   12. lower-PI.f32 N/A

       \[\leadsto \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\frac{-1}{2} \cdot u + \frac{1}{4}\right)\right) \cdot -4 \]

   13. lower-fma.f32 11.0

       \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5, u, 0.25\right)}\right) \cdot -4 \]

8. Applied rewrites 11.0%

   \[\leadsto \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-0.5, u, 0.25\right)\right) \cdot -4} \]
9. Step-by-step derivation

10. Applied rewrites 11.3%

    \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(-0.5 \cdot u + 0.25\right)\right) \cdot -4 \]

11. Final simplification 11.3%

    \[\leadsto \left(\left(-0.5 \cdot u + 0.25\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -4 \]
12. Add Preprocessing

Alternative 15: 9.2% accurate, 42.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(2, u, -1\right) \cdot \mathsf{PI}\left(\right) \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (* (fma 2.0 u -1.0) (PI)))

Derivation

1. Initial program 98.8%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
2. Add Preprocessing

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{-1 \cdot \mathsf{PI}\left(\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)} \]

   2. lower-neg.f32 N/A

      \[\leadsto \color{blue}{-\mathsf{PI}\left(\right)} \]

   3. lower-PI.f32 11.0

      \[\leadsto -\color{blue}{\mathsf{PI}\left(\right)} \]

5. Applied rewrites 11.0%

   \[\leadsto \color{blue}{-\mathsf{PI}\left(\right)} \]

6. Taylor expanded in s around -inf

   \[\leadsto \color{blue}{-4 \cdot \left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot -4} \]

   2. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot -4} \]

   3. cancel-sign-sub-inv N/A

      \[\leadsto \color{blue}{\left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot -4 \]

   4. metadata-eval N/A

      \[\leadsto \left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) + \color{blue}{\frac{1}{4}} \cdot \mathsf{PI}\left(\right)\right) \cdot -4 \]

   5. distribute-rgt-out-- N/A

      \[\leadsto \left(u \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{4} - \frac{1}{4}\right)\right)} + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot -4 \]

   6. metadata-eval N/A

      \[\leadsto \left(u \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{-1}{2}}\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot -4 \]

   7. *-commutative N/A

      \[\leadsto \left(u \cdot \color{blue}{\left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)} + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot -4 \]

   8. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(u \cdot \frac{-1}{2}\right) \cdot \mathsf{PI}\left(\right)} + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot -4 \]

   9. *-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\frac{-1}{2} \cdot u\right)} \cdot \mathsf{PI}\left(\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot -4 \]

   10. distribute-rgt-out N/A

       \[\leadsto \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{2} \cdot u + \frac{1}{4}\right)\right)} \cdot -4 \]

   11. lower-*.f32 N/A

       \[\leadsto \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{2} \cdot u + \frac{1}{4}\right)\right)} \cdot -4 \]

   12. lower-PI.f32 N/A

       \[\leadsto \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\frac{-1}{2} \cdot u + \frac{1}{4}\right)\right) \cdot -4 \]

   13. lower-fma.f32 11.0

       \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5, u, 0.25\right)}\right) \cdot -4 \]

8. Applied rewrites 11.0%

   \[\leadsto \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-0.5, u, 0.25\right)\right) \cdot -4} \]

9. Taylor expanded in u around inf

   \[\leadsto 2 \cdot \color{blue}{\left(u \cdot \mathsf{PI}\left(\right)\right)} \]
10. Step-by-step derivation

11. Applied rewrites 5.1%

    \[\leadsto 2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot u\right)} \]

12. Taylor expanded in u around 0

    \[\leadsto -1 \cdot \mathsf{PI}\left(\right) + \color{blue}{2 \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)} \]
13. Step-by-step derivation

14. Applied rewrites 11.0%

    \[\leadsto \mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{fma}\left(2, u, -1\right)} \]

15. Final simplification 10.0%

    \[\leadsto \mathsf{fma}\left(2, u, -1\right) \cdot \mathsf{PI}\left(\right) \]
16. Add Preprocessing

Alternative 16: 11.4% accurate, 170.0× speedup

Math

\[\begin{array}{l} \\ -\mathsf{PI}\left(\right) \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (- (PI)))

Derivation

1. Initial program 98.8%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
2. Add Preprocessing

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{-1 \cdot \mathsf{PI}\left(\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)} \]

   2. lower-neg.f32 N/A

      \[\leadsto \color{blue}{-\mathsf{PI}\left(\right)} \]

   3. lower-PI.f32 11.0

      \[\leadsto -\color{blue}{\mathsf{PI}\left(\right)} \]

5. Applied rewrites 11.0%

   \[\leadsto \color{blue}{-\mathsf{PI}\left(\right)} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024288 
(FPCore (u s)
  :name "Sample trimmed logistic on [-pi, pi]"
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0)) (and (<= 0.0 s) (<= s 1.0651631)))
  (* (- s) (log (- (/ 1.0 (+ (* u (- (/ 1.0 (+ 1.0 (exp (/ (- (PI)) s)))) (/ 1.0 (+ 1.0 (exp (/ (PI) s)))))) (/ 1.0 (+ 1.0 (exp (/ (PI) s)))))) 1.0))))