Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 13.4s

Alternatives: 10

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: 99.0% 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: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

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

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

   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. distribute-frac-neg N/A

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

   4. lift--.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. mul0-lft N/A

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

   7. neg-sub0 N/A

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

   8. frac-2neg N/A

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

   9. distribute-frac-neg2 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

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

6. Applied rewrites 99.0%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

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

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 1.7%

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

7. Applied rewrites 99.0%

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

Alternative 3: 99.0% 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)))))

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. Add Preprocessing

Alternative 4: 97.5% accurate, 1.3× speedup

Math

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

FPCore

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

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

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 1.7%

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

7. Applied rewrites 99.0%

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

8. Taylor expanded in u around inf

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

10. Applied rewrites 97.7%

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

Alternative 5: 16.4% accurate, 1.9× speedup

Math

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

FPCore

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

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 1.7%

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

7. Applied rewrites 99.0%

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

8. Taylor expanded in s around -inf

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

10. Applied rewrites 15.9%

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

11. Final simplification 16.4%

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

Alternative 6: 14.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{PI}\left(\right)}\\ t_1 := \frac{\mathsf{PI}\left(\right)}{u}\\ \mathbf{if}\;s \leq 4.999999841327613 \cdot 10^{-22}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{u}{s}, 0, \mathsf{fma}\left(u, \frac{\mathsf{fma}\left(t\_0, t\_0 \cdot 0.5, t\_1 \cdot -0.25\right)}{s}, 0.5\right)\right)} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \mathsf{PI}\left(\right) + t\_1}{-u} \cdot \left(u \cdot u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (sqrt (PI))) (t_1 (/ (PI) u)))
   (if (<= s 4.999999841327613e-22)
     (*
      (- s)
      (log
       (-
        (/
         1.0
         (fma
          (/ u s)
          0.0
          (fma u (/ (fma t_0 (* t_0 0.5) (* t_1 -0.25)) s) 0.5)))
        1.0)))
     (* (/ (+ (* -2.0 (PI)) t_1) (- u)) (* u u)))))

Derivation

1. Split input into 2 regimes

2. if s < 4.9999998e-22

   1. Initial program 99.1%

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

   3. Taylor expanded in u around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 1.2%

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

   6. Taylor expanded in s around inf

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

   8. Applied rewrites 13.8%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{u}{s}, \color{blue}{\frac{0}{s}}, \mathsf{fma}\left(u, \frac{\mathsf{fma}\left(-0.25, \frac{\mathsf{PI}\left(\right)}{u}, 0.5 \cdot \mathsf{PI}\left(\right)\right)}{s}, 0.5\right)\right)} - 1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 13.8%

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

   if 4.9999998e-22 < 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

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

   5. Applied rewrites 7.9%

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

   6. Taylor expanded in u around -inf

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

   8. Applied rewrites 14.6%

      \[\leadsto \left(\frac{0}{s} - \frac{-2 \cdot \mathsf{PI}\left(\right) - \frac{-\mathsf{PI}\left(\right)}{u}}{u}\right) \cdot \color{blue}{\left(u \cdot u\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 14.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 4.999999841327613 \cdot 10^{-22}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{u}{s}, 0, \mathsf{fma}\left(u, \frac{\mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \sqrt{\mathsf{PI}\left(\right)} \cdot 0.5, \frac{\mathsf{PI}\left(\right)}{u} \cdot -0.25\right)}{s}, 0.5\right)\right)} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \mathsf{PI}\left(\right) + \frac{\mathsf{PI}\left(\right)}{u}}{-u} \cdot \left(u \cdot u\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 14.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{PI}\left(\right)}{u}\\ \mathbf{if}\;s \leq 5.200000093474659 \cdot 10^{-19}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(t\_0, -0.25, 0.5 \cdot \mathsf{PI}\left(\right)\right)}{s}, u, 0.5\right)} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \mathsf{PI}\left(\right) + t\_0}{-u} \cdot \left(u \cdot u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ (PI) u)))
   (if (<= s 5.200000093474659e-19)
     (*
      (- s)
      (log (- (/ 1.0 (fma (/ (fma t_0 -0.25 (* 0.5 (PI))) s) u 0.5)) 1.0)))
     (* (/ (+ (* -2.0 (PI)) t_0) (- u)) (* u u)))))

Derivation

1. Split input into 2 regimes

2. if s < 5.20000009e-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 u around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 1.1%

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

   6. Taylor expanded in s around inf

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

   8. Applied rewrites 13.2%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{u}{s}, \color{blue}{\frac{0}{s}}, \mathsf{fma}\left(u, \frac{\mathsf{fma}\left(-0.25, \frac{\mathsf{PI}\left(\right)}{u}, 0.5 \cdot \mathsf{PI}\left(\right)\right)}{s}, 0.5\right)\right)} - 1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 13.2%

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

   if 5.20000009e-19 < 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

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

   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

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

   8. Applied rewrites 15.3%

      \[\leadsto \left(\frac{0}{s} - \frac{-2 \cdot \mathsf{PI}\left(\right) - \frac{-\mathsf{PI}\left(\right)}{u}}{u}\right) \cdot \color{blue}{\left(u \cdot u\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 14.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 5.200000093474659 \cdot 10^{-19}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\mathsf{PI}\left(\right)}{u}, -0.25, 0.5 \cdot \mathsf{PI}\left(\right)\right)}{s}, u, 0.5\right)} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \mathsf{PI}\left(\right) + \frac{\mathsf{PI}\left(\right)}{u}}{-u} \cdot \left(u \cdot u\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 11.7% accurate, 11.9× speedup

Math

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

FPCore

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

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

5. Applied rewrites 7.4%

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

6. Taylor expanded in u around -inf

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

8. Applied rewrites 11.5%

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

9. Final simplification 11.5%

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

Alternative 9: 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 11.3

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

5. Applied rewrites 11.3%

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

Alternative 10: 10.3% 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

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

5. Applied rewrites 7.6%

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

6. Taylor expanded in s around 0

   \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{-16 \cdot {\left(\frac{-1}{2} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + 16 \cdot {\left(\frac{-1}{2} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{s}} \]
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 2024313 
(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))))