Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 10.5s

Alternatives: 8

Speedup: N/A×

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, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\mathsf{PI}\left(\right)}{s}}\\ t_1 := e^{-\mathsf{log1p}\left(t\_0\right)}\\ t_2 := \left(e^{-\mathsf{log1p}\left(e^{\frac{-\mathsf{PI}\left(\right)}{s}}\right)} - t\_1\right) \cdot u\\ \left(-s\right) \cdot \log \left(\frac{1}{\frac{{\left(t\_0 - -1\right)}^{-2} - {t\_2}^{2}}{t\_1 - t\_2}} - 1\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (exp (/ (PI) s)))
        (t_1 (exp (- (log1p t_0))))
        (t_2 (* (- (exp (- (log1p (exp (/ (- (PI)) s))))) t_1) u)))
   (*
    (- s)
    (log
     (-
      (/ 1.0 (/ (- (pow (- t_0 -1.0) -2.0) (pow t_2 2.0)) (- t_1 t_2)))
      1.0)))))

Derivation

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

4. Applied rewrites 99.1%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{{\left(e^{\frac{\mathsf{PI}\left(\right)}{s}} - -1\right)}^{-2} - {\left(\left(e^{-\mathsf{log1p}\left(e^{\frac{-\mathsf{PI}\left(\right)}{s}}\right)} - e^{-\mathsf{log1p}\left(e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)}\right) \cdot u\right)}^{2}}{e^{-\mathsf{log1p}\left(e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)} - \left(e^{-\mathsf{log1p}\left(e^{\frac{-\mathsf{PI}\left(\right)}{s}}\right)} - e^{-\mathsf{log1p}\left(e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)}\right) \cdot u}}} - 1\right) \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

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 0

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

   1. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Applied rewrites 99.1%

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

6. Final simplification 99.1%

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

Alternative 3: 97.6% accurate, 1.3× speedup

Math

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

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))))

Derivation

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 s around inf

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

   1. lower-/.f32 N/A

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

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

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

   3. metadata-eval N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 3.3

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

5. Applied rewrites 3.3%

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

6. Taylor expanded in u around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-+.f32 N/A

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

   6. lower-exp.f32 N/A

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

   7. associate-*r/ N/A

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

   8. lower-/.f32 N/A

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

   9. mul-1-neg N/A

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

   10. lower-neg.f32 N/A

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

   11. lower-PI.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. lower-+.f32 N/A

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

8. Applied rewrites 98.1%

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

Alternative 4: 92.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\mathsf{PI}\left(\right)\\ t_1 := \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\\ \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{t\_0}{s}}} - \frac{1}{1 + \left(1 + \mathsf{fma}\left(0.5, \frac{t\_1}{s \cdot s}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}\right) + \frac{1}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{t\_1}{s}, -0.5, t\_0\right)}{s}, -1, 2\right)}} - 1\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (- (PI))) (t_1 (* (PI) (PI))))
   (*
    (- s)
    (log
     (-
      (/
       1.0
       (+
        (*
         u
         (-
          (/ 1.0 (+ 1.0 (exp (/ t_0 s))))
          (/ 1.0 (+ 1.0 (+ 1.0 (fma 0.5 (/ t_1 (* s s)) (/ (PI) s)))))))
        (/ 1.0 (fma (/ (fma (/ t_1 s) -0.5 t_0) s) -1.0 2.0))))
      1.0)))))

Derivation

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 s around -inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 N/A

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

   12. lower-PI.f32 N/A

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

   13. mul-1-neg N/A

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

   14. lower-neg.f32 N/A

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

   15. lower-PI.f32 94.5

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

5. Applied rewrites 94.5%

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

6. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 N/A

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

   7. lower-PI.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-/.f32 N/A

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

   11. lower-PI.f32 94.5

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

8. Applied rewrites 94.5%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + \color{blue}{\left(1 + \mathsf{fma}\left(0.5, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s \cdot s}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}}\right) + \frac{1}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s}, -0.5, -\mathsf{PI}\left(\right)\right)}{s}, -1, 2\right)}} - 1\right) \]
9. Add Preprocessing

Alternative 5: 92.2% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (- (PI))))
   (*
    (- s)
    (log
     (-
      (/
       1.0
       (+
        (*
         u
         (-
          (/ 1.0 (+ 1.0 (exp (/ t_0 s))))
          (/ 1.0 (+ 1.0 (+ 1.0 (/ (PI) s))))))
        (/ 1.0 (fma (/ (fma (/ (* (PI) (PI)) s) -0.5 t_0) s) -1.0 2.0))))
      1.0)))))

Derivation

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 s around -inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 N/A

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

   12. lower-PI.f32 N/A

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

   13. mul-1-neg N/A

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

   14. lower-neg.f32 N/A

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

   15. lower-PI.f32 94.5

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

5. Applied rewrites 94.5%

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

6. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-PI.f32 94.0

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

8. Applied rewrites 94.0%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + \color{blue}{\left(1 + \frac{\mathsf{PI}\left(\right)}{s}\right)}}\right) + \frac{1}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s}, -0.5, -\mathsf{PI}\left(\right)\right)}{s}, -1, 2\right)}} - 1\right) \]
9. Add Preprocessing

Alternative 6: 24.9% accurate, 3.6× speedup

Math

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

FPCore

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

Derivation

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 s around inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

5. Applied rewrites 24.7%

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

Alternative 7: 11.6% accurate, 30.0× speedup

Math

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

FPCore

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

Derivation

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 s around inf

   \[\leadsto \color{blue}{4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)} \]
4. 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. fp-cancel-sub-sign-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. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f32 N/A

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

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

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

   8. metadata-eval N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 11.4

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

5. Applied rewrites 11.4%

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

6. Taylor expanded in u around 0

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

8. Applied rewrites 11.4%

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

Alternative 8: 11.4% accurate, 170.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 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 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 11.2

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

5. Applied rewrites 11.2%

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

Reproduce

herbie shell --seed 2025017 
(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))))