Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 17.9s

Alternatives: 11

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

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

FPCore

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

Derivation

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

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 99.0

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

4. Applied rewrites 99.0%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

   5. lift-neg.f32 N/A

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

   6. /-rgt-identity N/A

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

   7. clear-num N/A

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

   8. inv-pow N/A

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

   9. pow-flip N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{{s}^{\color{blue}{\left(3 - 2\right)}}}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   12. pow-div N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{\color{blue}{\frac{{s}^{3}}{{s}^{2}}}}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   13. sqr-pow N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{\frac{\color{blue}{{s}^{\left(\frac{3}{2}\right)} \cdot {s}^{\left(\frac{3}{2}\right)}}}{{s}^{2}}}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   14. unpow-prod-down N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{\frac{\color{blue}{{\left(s \cdot s\right)}^{\left(\frac{3}{2}\right)}}}{{s}^{2}}}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   15. sqr-neg N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{\frac{{\color{blue}{\left(\left(\mathsf{neg}\left(s\right)\right) \cdot \left(\mathsf{neg}\left(s\right)\right)\right)}}^{\left(\frac{3}{2}\right)}}{{s}^{2}}}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   16. lift-neg.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{\frac{{\left(\color{blue}{\left(-s\right)} \cdot \left(\mathsf{neg}\left(s\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{{s}^{2}}}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   17. lift-neg.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{\frac{{\left(\left(-s\right) \cdot \color{blue}{\left(-s\right)}\right)}^{\left(\frac{3}{2}\right)}}{{s}^{2}}}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   18. pow-prod-down N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{\frac{\color{blue}{{\left(-s\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(-s\right)}^{\left(\frac{3}{2}\right)}}}{{s}^{2}}}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   19. sqr-pow N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{\frac{\color{blue}{{\left(-s\right)}^{3}}}{{s}^{2}}}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   20. pow2 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{\frac{{\left(-s\right)}^{3}}{\color{blue}{s \cdot s}}}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   21. sqr-neg N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{\frac{{\left(-s\right)}^{3}}{\color{blue}{\left(\mathsf{neg}\left(s\right)\right) \cdot \left(\mathsf{neg}\left(s\right)\right)}}}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   22. lift-neg.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{\frac{{\left(-s\right)}^{3}}{\color{blue}{\left(-s\right)} \cdot \left(\mathsf{neg}\left(s\right)\right)}}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   23. lift-neg.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{\frac{{\left(-s\right)}^{3}}{\left(-s\right) \cdot \color{blue}{\left(-s\right)}}}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   24. pow2 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{\frac{{\left(-s\right)}^{3}}{\color{blue}{{\left(-s\right)}^{2}}}}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   25. pow-div N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{\color{blue}{{\left(-s\right)}^{\left(3 - 2\right)}}}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   26. metadata-eval N/A

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

   27. unpow1 N/A

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

   28. lift-neg.f32 N/A

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

6. Applied rewrites 99.0%

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

7. Final simplification 99.0%

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

Alternative 2: 7.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\mathsf{PI}\left(\right)\\ t_1 := \mathsf{PI}\left(\right) \cdot u\\ t_2 := \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\\ t_3 := \mathsf{fma}\left(0.5, u, -0.25\right) \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;\log \left(\frac{1}{\left(\frac{1}{e^{\frac{t\_0}{s}} + 1} - t\_2\right) \cdot u + t\_2} - 1\right) \cdot \left(-s\right) \leq -1.999999936531045 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(u \cdot u\right) \cdot 0}{s}, -0.5, \mathsf{fma}\left(t\_1, 2, t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(t\_3 \cdot 16, t\_3, -16 \cdot {t\_3}^{2}\right)}{s}, -0.5, \mathsf{fma}\left(t\_1, 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (- (PI)))
        (t_1 (* (PI) u))
        (t_2 (/ 1.0 (+ (exp (/ (PI) s)) 1.0)))
        (t_3 (* (fma 0.5 u -0.25) (PI))))
   (if (<=
        (*
         (log
          (-
           (/ 1.0 (+ (* (- (/ 1.0 (+ (exp (/ t_0 s)) 1.0)) t_2) u) t_2))
           1.0))
         (- s))
        -1.999999936531045e-19)
     (fma (/ (* (* u u) 0.0) s) -0.5 (fma t_1 2.0 t_0))
     (fma
      (/ (fma (* t_3 16.0) t_3 (* -16.0 (pow t_3 2.0))) s)
      -0.5
      (* (fma t_1 0.5 (* -0.25 (PI))) 4.0)))))

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-19

   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

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{-16 \cdot {\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -2 \cdot \left(-8 \cdot {\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{s} + 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)} \]

   5. Applied rewrites 9.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, -16, \mathsf{fma}\left(16, {\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, 0\right)\right)}{s}, -0.5, \mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot 4\right)} \]

   6. Taylor expanded in u around inf

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

   8. Applied rewrites 15.6%

      \[\leadsto \mathsf{fma}\left(\frac{0 \cdot \left(u \cdot u\right)}{s}, -0.5, \mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot 4\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 8.8%

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

   if -1.99999994e-19 < (*.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 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

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{-16 \cdot {\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -2 \cdot \left(-8 \cdot {\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{s} + 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)} \]

   5. Applied rewrites 6.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, -16, \mathsf{fma}\left(16, {\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, 0\right)\right)}{s}, -0.5, \mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot 4\right)} \]

   6. Taylor expanded in u around 0

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{1}{2} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, -16, \mathsf{fma}\left(16, {\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), \frac{1}{2}, \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, 0\right)\right)}{s}, \frac{-1}{2}, \mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), \frac{1}{2}, \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot 4\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 6.0%

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(0.5, u, -0.25\right)\right)}^{2}, -16, \mathsf{fma}\left(16, {\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, 0\right)\right)}{s}, -0.5, \mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot 4\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 8.1%

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(16 \cdot \left(\mathsf{fma}\left(0.5, u, -0.25\right) \cdot \mathsf{PI}\left(\right)\right), \mathsf{fma}\left(0.5, u, -0.25\right) \cdot \mathsf{PI}\left(\right), {\left(\mathsf{fma}\left(0.5, u, -0.25\right) \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot -16\right)}{s}, -0.5, \mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot 4\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 7.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(\frac{1}{\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right) \cdot u + \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}} - 1\right) \cdot \left(-s\right) \leq -1.999999936531045 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(u \cdot u\right) \cdot 0}{s}, -0.5, \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot u, 2, -\mathsf{PI}\left(\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(0.5, u, -0.25\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 16, \mathsf{fma}\left(0.5, u, -0.25\right) \cdot \mathsf{PI}\left(\right), -16 \cdot {\left(\mathsf{fma}\left(0.5, u, -0.25\right) \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)}{s}, -0.5, \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot u, 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot 4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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 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. Final simplification 99.0%

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

Alternative 4: 97.5% 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 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 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 97.8%

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

6. Final simplification 97.8%

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

Alternative 5: 8.5% 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 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

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

5. Applied rewrites 10.4%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{0.5}} - 1\right) \]

6. Taylor expanded in s around -inf

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

8. Applied rewrites 13.7%

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

9. Final simplification 12.8%

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

Alternative 6: 8.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot u\\ t_1 := \mathsf{fma}\left(0.5, u, -0.25\right) \cdot \mathsf{PI}\left(\right)\\ t_2 := {t\_1}^{2}\\ \mathbf{if}\;s \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(t\_1 \cdot 16, t\_1, -16 \cdot t\_2\right)}{s}, -0.5, \mathsf{fma}\left(t\_0, 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(1 - \frac{\mathsf{fma}\left(-8, t\_2, 0\right)}{s \cdot s}\right) - \frac{\mathsf{fma}\left(t\_0, 2, -\mathsf{PI}\left(\right)\right)}{s}\right) \cdot \left(-s\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (* (PI) u)) (t_1 (* (fma 0.5 u -0.25) (PI))) (t_2 (pow t_1 2.0)))
   (if (<= s 1.9999999996399175e-23)
     (fma
      (/ (fma (* t_1 16.0) t_1 (* -16.0 t_2)) s)
      -0.5
      (* (fma t_0 0.5 (* -0.25 (PI))) 4.0))
     (*
      (log
       (- (- 1.0 (/ (fma -8.0 t_2 0.0) (* s s))) (/ (fma t_0 2.0 (- (PI))) s)))
      (- s)))))

Derivation

1. Split input into 2 regimes

2. if s < 2e-23

   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

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{-16 \cdot {\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -2 \cdot \left(-8 \cdot {\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{s} + 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)} \]

   5. Applied rewrites 6.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, -16, \mathsf{fma}\left(16, {\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, 0\right)\right)}{s}, -0.5, \mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot 4\right)} \]

   6. Taylor expanded in u around 0

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{1}{2} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, -16, \mathsf{fma}\left(16, {\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), \frac{1}{2}, \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, 0\right)\right)}{s}, \frac{-1}{2}, \mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), \frac{1}{2}, \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot 4\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 6.9%

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(0.5, u, -0.25\right)\right)}^{2}, -16, \mathsf{fma}\left(16, {\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, 0\right)\right)}{s}, -0.5, \mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot 4\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 7.9%

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(16 \cdot \left(\mathsf{fma}\left(0.5, u, -0.25\right) \cdot \mathsf{PI}\left(\right)\right), \mathsf{fma}\left(0.5, u, -0.25\right) \cdot \mathsf{PI}\left(\right), {\left(\mathsf{fma}\left(0.5, u, -0.25\right) \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot -16\right)}{s}, -0.5, \mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot 4\right) \]

   if 2e-23 < s

   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

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

   5. Applied rewrites 7.8%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{0.5}} - 1\right) \]

   6. Taylor expanded in s around inf

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

   8. Applied rewrites 14.6%

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

4. Final simplification 9.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(0.5, u, -0.25\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 16, \mathsf{fma}\left(0.5, u, -0.25\right) \cdot \mathsf{PI}\left(\right), -16 \cdot {\left(\mathsf{fma}\left(0.5, u, -0.25\right) \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)}{s}, -0.5, \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot u, 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(1 - \frac{\mathsf{fma}\left(-8, {\left(\mathsf{fma}\left(0.5, u, -0.25\right) \cdot \mathsf{PI}\left(\right)\right)}^{2}, 0\right)}{s \cdot s}\right) - \frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot u, 2, -\mathsf{PI}\left(\right)\right)}{s}\right) \cdot \left(-s\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 8.6% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\left(u \cdot u\right) \cdot 0}{s}, -0.5, \left(2 \cdot \mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{u}\right) \cdot u\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (fma (/ (* (* u u) 0.0) s) -0.5 (* (- (* 2.0 (PI)) (/ (PI) u)) u)))

Derivation

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

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{-16 \cdot {\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -2 \cdot \left(-8 \cdot {\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{s} + 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)} \]

5. Applied rewrites 8.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, -16, \mathsf{fma}\left(16, {\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, 0\right)\right)}{s}, -0.5, \mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot 4\right)} \]

6. Taylor expanded in u around inf

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

8. Applied rewrites 11.5%

   \[\leadsto \mathsf{fma}\left(\frac{0 \cdot \left(u \cdot u\right)}{s}, -0.5, \mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot 4\right) \]

9. Taylor expanded in u around inf

   \[\leadsto \mathsf{fma}\left(\frac{0 \cdot \left(u \cdot u\right)}{s}, \frac{-1}{2}, u \cdot \left(-1 \cdot \frac{\mathsf{PI}\left(\right)}{u} + 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]
10. Step-by-step derivation

11. Applied rewrites 11.6%

    \[\leadsto \mathsf{fma}\left(\frac{0 \cdot \left(u \cdot u\right)}{s}, -0.5, \left(2 \cdot \mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{u}\right) \cdot u\right) \]

12. Final simplification 11.7%

    \[\leadsto \mathsf{fma}\left(\frac{\left(u \cdot u\right) \cdot 0}{s}, -0.5, \left(2 \cdot \mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{u}\right) \cdot u\right) \]
13. Add Preprocessing

Alternative 8: 7.5% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\left(u \cdot u\right) \cdot 0}{s}, -0.5, \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot u, 2, -\mathsf{PI}\left(\right)\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (fma (/ (* (* u u) 0.0) s) -0.5 (fma (* (PI) u) 2.0 (- (PI)))))

Derivation

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

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{-16 \cdot {\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -2 \cdot \left(-8 \cdot {\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{s} + 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)} \]

5. Applied rewrites 8.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, -16, \mathsf{fma}\left(16, {\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, 0\right)\right)}{s}, -0.5, \mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot 4\right)} \]

6. Taylor expanded in u around inf

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

8. Applied rewrites 11.5%

   \[\leadsto \mathsf{fma}\left(\frac{0 \cdot \left(u \cdot u\right)}{s}, -0.5, \mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot 4\right) \]
9. Step-by-step derivation

10. Applied rewrites 8.2%

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

11. Final simplification 11.5%

    \[\leadsto \mathsf{fma}\left(\frac{\left(u \cdot u\right) \cdot 0}{s}, -0.5, \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot u, 2, -\mathsf{PI}\left(\right)\right)\right) \]
12. Add Preprocessing

Alternative 9: 11.7% accurate, 36.4× speedup

Math

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

FPCore

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

Derivation

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

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{-16 \cdot {\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -2 \cdot \left(-8 \cdot {\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{s} + 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)} \]

5. Applied rewrites 7.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, -16, \mathsf{fma}\left(16, {\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, 0\right)\right)}{s}, -0.5, \mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot 4\right)} \]
6. Step-by-step derivation

7. Applied rewrites 8.3%

   \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, -16, \left(4 \cdot \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot u, 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(4 \cdot \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot u, 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{s}, -0.5, \mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot 4\right) \]

9. Applied rewrites 11.7%

   \[\leadsto \left(0 + \left(\mathsf{PI}\left(\right) \cdot u\right) \cdot 2\right) + \color{blue}{\left(-\mathsf{PI}\left(\right)\right)} \]

10. Final simplification 11.7%

    \[\leadsto \left(\mathsf{PI}\left(\right) \cdot u\right) \cdot 2 - \mathsf{PI}\left(\right) \]
11. Add Preprocessing

Alternative 10: 11.5% 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 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 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 11.5

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

5. Applied rewrites 11.5%

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

Alternative 11: 10.2% accurate, 510.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (u s) :precision binary32 0.0)

C

float code(float u, float s) {
	return 0.0f;
}

Fortran

real(4) function code(u, s)
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = 0.0e0
end function

Julia

function code(u, s)
	return Float32(0.0)
end

MATLAB

function tmp = code(u, s)
	tmp = single(0.0);
end

Derivation

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

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{-16 \cdot {\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -2 \cdot \left(-8 \cdot {\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{s} + 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)} \]

5. Applied rewrites 8.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, -16, \mathsf{fma}\left(16, {\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, 0\right)\right)}{s}, -0.5, \mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot 4\right)} \]

6. Taylor expanded in s around 0

   \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{-16 \cdot {\left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{1}{2} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + 16 \cdot {\left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{1}{2} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{s}} \]
7. Step-by-step derivation

8. Applied rewrites 10.4%

   \[\leadsto \frac{0}{\color{blue}{s}} \]

9. Taylor expanded in s around 0

   \[\leadsto 0 \]
10. Step-by-step derivation

11. Applied rewrites 10.4%

    \[\leadsto 0 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024283 
(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))))