Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 17.1s

Alternatives: 7

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 98.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (log
   (-
    (/
     1.0
     (+
      (/ 1.0 (+ (exp (/ 1.0 (/ s (PI)))) 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. 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. clear-num N/A

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

   3. lower-/.f32 N/A

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

   4. lower-/.f32 99.0

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

4. Applied rewrites 99.0%

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

5. Final simplification 99.0%

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

Alternative 2: 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 3: 97.6% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 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 98.1%

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

6. Final simplification 98.1%

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

Alternative 4: 2.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \mathsf{PI}\left(\right)\\ \frac{\mathsf{fma}\left({t\_0}^{2}, u \cdot u, \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -0.0625\right) \cdot \frac{-4}{s}}{\mathsf{fma}\left(t\_0, u, 0.25 \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(-s\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (* 0.5 (PI))))
   (*
    (/
     (* (fma (pow t_0 2.0) (* u u) (* (* (PI) (PI)) -0.0625)) (/ -4.0 s))
     (fma t_0 u (* 0.25 (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 s around inf

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f32 N/A

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

5. Applied rewrites 11.4%

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

7. Applied rewrites 11.4%

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

9. Applied rewrites 9.1%

   \[\leadsto \left(-s\right) \cdot \frac{\frac{-4}{s} \cdot \left({\left(\left(\mathsf{PI}\left(\right) \cdot u\right) \cdot 0.5\right)}^{2} - 0.0625 \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{\mathsf{fma}\left(0.5 \cdot \mathsf{PI}\left(\right), u, 0.25 \cdot \mathsf{PI}\left(\right)\right)}} \]
10. Step-by-step derivation

11. Applied rewrites 2.8%

    \[\leadsto \left(-s\right) \cdot \frac{\frac{-4}{s} \cdot \mathsf{fma}\left({\left(0.5 \cdot \mathsf{PI}\left(\right)\right)}^{2}, u \cdot u, -0.0625 \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)}{\mathsf{fma}\left(0.5 \cdot \color{blue}{\mathsf{PI}\left(\right)}, u, 0.25 \cdot \mathsf{PI}\left(\right)\right)} \]

12. Final simplification 2.8%

    \[\leadsto \frac{\mathsf{fma}\left({\left(0.5 \cdot \mathsf{PI}\left(\right)\right)}^{2}, u \cdot u, \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -0.0625\right) \cdot \frac{-4}{s}}{\mathsf{fma}\left(0.5 \cdot \mathsf{PI}\left(\right), u, 0.25 \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(-s\right) \]
13. Add Preprocessing

Alternative 5: 11.5% accurate, 21.3× speedup

Math

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

FPCore

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

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 11.4

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

5. Applied rewrites 11.4%

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

6. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

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

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

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

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

   5. metadata-eval N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. metadata-eval N/A

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

   9. associate-*r* N/A

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

   10. distribute-rgt-out N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 N/A

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

   13. lower-fma.f32 11.4

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

8. Applied rewrites 11.4%

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

10. Applied rewrites 11.6%

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

11. Final simplification 11.6%

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

Alternative 6: 11.5% accurate, 26.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 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 11.4

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

5. Applied rewrites 11.4%

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

6. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

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

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

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

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

   5. metadata-eval N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. metadata-eval N/A

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

   9. associate-*r* N/A

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

   10. distribute-rgt-out N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 N/A

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

   13. lower-fma.f32 11.4

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

8. Applied rewrites 11.4%

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

10. Applied rewrites 11.6%

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

11. Final simplification 11.6%

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

Alternative 7: 11.3% 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 11.4

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

5. Applied rewrites 11.4%

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

Reproduce

herbie shell --seed 2024242 
(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))))