Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 99.0%

Time: 13.1s

Alternatives: 15

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: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.9%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. clear-num N/A

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

   4. div-inv N/A

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

   5. clear-num N/A

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

   6. lift-/.f32 N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f32 N/A

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

   9. exp-1-e N/A

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

   10. lower-E.f32 98.9

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

4. Applied rewrites 98.9%

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

Alternative 2: 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)))))

Derivation

1. Initial program 98.9%

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

Alternative 3: 97.2% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

5. Applied rewrites 96.9%

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

Alternative 4: 94.9% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-PI.f32 94.3

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

5. Applied rewrites 94.3%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 94.3

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

7. Applied rewrites 94.3%

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

Alternative 5: 94.9% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-PI.f32 94.3

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

5. Applied rewrites 94.3%

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

Alternative 6: 37.6% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in s around inf

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

5. Applied rewrites 37.2%

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

Alternative 7: 6.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{PI}\left(\right)}{s}\\ \mathbf{if}\;s \leq 1.0000000168623835 \cdot 10^{-16}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\left(-0.5 \cdot u\right) \cdot \left(-\mathsf{PI}\left(\right)\right), \frac{1}{s}, 0.5\right)} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(\frac{1}{\left(-u\right) \cdot \mathsf{fma}\left(\frac{0.5 - 0.25 \cdot t\_0}{u}, -1, -0.5 \cdot t\_0\right)} - 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ (PI) s)))
   (if (<= s 1.0000000168623835e-16)
     (*
      (- s)
      (log (- (/ 1.0 (fma (* (* -0.5 u) (- (PI))) (/ 1.0 s) 0.5)) 1.0)))
     (*
      (- s)
      (log
       (-
        (/ 1.0 (* (- u) (fma (/ (- 0.5 (* 0.25 t_0)) u) -1.0 (* -0.5 t_0))))
        1.0))))))

Derivation

1. Split input into 2 regimes

2. if s < 1.00000002e-16

   1. Initial program 99.0%

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

   3. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

   5. Applied rewrites -0.0%

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

   7. Applied rewrites 12.3%

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

   8. Taylor expanded in u around inf

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

   10. Applied rewrites 12.2%

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

   11. Taylor expanded in u around inf

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

   13. Applied rewrites 12.2%

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

   if 1.00000002e-16 < s

   1. Initial program 98.8%

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

   3. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

   5. Applied rewrites -0.0%

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

   6. Taylor expanded in u around -inf

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

   8. Applied rewrites 7.9%

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

Alternative 8: 6.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\mathsf{PI}\left(\right)\\ \mathbf{if}\;s \leq 3.0000001167615996 \cdot 10^{-16}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\left(-0.5 \cdot u\right) \cdot t\_0, \frac{1}{s}, 0.5\right)} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, u, 0.25\right), \frac{t\_0}{s}, 0.5\right)} - 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (- (PI))))
   (if (<= s 3.0000001167615996e-16)
     (* (- s) (log (- (/ 1.0 (fma (* (* -0.5 u) t_0) (/ 1.0 s) 0.5)) 1.0)))
     (* (- s) (log (- (/ 1.0 (fma (fma -0.5 u 0.25) (/ t_0 s) 0.5)) 1.0))))))

Derivation

1. Split input into 2 regimes

2. if s < 3.0000001e-16

   1. Initial program 99.0%

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

   3. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

   5. Applied rewrites -0.0%

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

   7. Applied rewrites 12.2%

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

   8. Taylor expanded in u around inf

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

   10. Applied rewrites 12.1%

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

   11. Taylor expanded in u around inf

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

   13. Applied rewrites 12.1%

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

   if 3.0000001e-16 < s

   1. Initial program 98.8%

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

   3. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

   5. Applied rewrites -0.0%

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

   7. Applied rewrites 7.2%

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

   9. Applied rewrites 7.2%

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

Alternative 9: 5.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 2.000000033724767 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \mathsf{PI}\left(\right), u, -0.25 \cdot {\mathsf{PI}\left(\right)}^{1}\right), 4, \frac{0}{s} \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, u, 0.25\right), \frac{-\mathsf{PI}\left(\right)}{s}, 0.5\right)} - 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (if (<= s 2.000000033724767e-16)
   (fma (fma (* 0.5 (PI)) u (* -0.25 (pow (PI) 1.0))) 4.0 (* (/ 0.0 s) -0.5))
   (* (- s) (log (- (/ 1.0 (fma (fma -0.5 u 0.25) (/ (- (PI)) s) 0.5)) 1.0)))))

Derivation

1. Split input into 2 regimes

2. if s < 2.00000003e-16

   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-exp.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. clear-num N/A

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

      4. div-inv N/A

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

      5. clear-num N/A

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

      6. lift-/.f32 N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f32 N/A

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

      9. exp-1-e N/A

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

      10. lower-E.f32 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in s around inf

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

   7. Applied rewrites 6.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \mathsf{PI}\left(\right), u, -0.25 \cdot \mathsf{PI}\left(\right)\right), 4, \frac{\mathsf{fma}\left(\mathsf{fma}\left({\left(\mathsf{fma}\left(0.5 \cdot \mathsf{PI}\left(\right), u, -0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, -8, 0\right), -2, {\left(\mathsf{fma}\left(0.5 \cdot \mathsf{PI}\left(\right), u, -0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot -16\right)}{s} \cdot -0.5\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 7.2%

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

   10. Taylor expanded in u around 0

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

   12. Applied rewrites 4.4%

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

   if 2.00000003e-16 < s

   1. Initial program 98.8%

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

   3. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

   5. Applied rewrites -0.0%

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

   7. Applied rewrites 7.3%

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

   9. Applied rewrites 7.3%

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

4. Final simplification 2.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 2.000000033724767 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \mathsf{PI}\left(\right), u, -0.25 \cdot {\mathsf{PI}\left(\right)}^{1}\right), 4, \frac{0}{s} \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, u, 0.25\right), \frac{-\mathsf{PI}\left(\right)}{s}, 0.5\right)} - 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 11.6% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{PI}\left(\right) \cdot \frac{\left(-0.5 \cdot u\right) \cdot \left(-0.5 \cdot u\right) - 0.0625}{-0.5 \cdot u - 0.25}\right) \cdot -4 \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (* (PI) (/ (- (* (* -0.5 u) (* -0.5 u)) 0.0625) (- (* -0.5 u) 0.25)))
  -4.0))

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in s around -inf

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

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

   4. metadata-eval N/A

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

   5. *-commutative N/A

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

   6. *-commutative N/A

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

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

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

   8. metadata-eval N/A

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

   9. associate-*l* N/A

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

   10. distribute-lft-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 11.1

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

5. Applied rewrites 11.1%

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

7. Applied rewrites 11.4%

   \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \frac{\left(-0.5 \cdot u\right) \cdot \left(-0.5 \cdot u\right) - 0.0625}{-0.5 \cdot u - 0.25}\right) \cdot -4 \]
8. Add Preprocessing

Alternative 11: 11.6% accurate, 20.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in s around -inf

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

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

   4. metadata-eval N/A

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

   5. *-commutative N/A

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

   6. *-commutative N/A

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

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

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

   8. metadata-eval N/A

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

   9. associate-*l* N/A

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

   10. distribute-lft-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 11.1

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

5. Applied rewrites 11.1%

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

6. Taylor expanded in u around inf

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

8. Applied rewrites 11.4%

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

Alternative 12: 11.6% accurate, 21.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in s around -inf

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

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

   4. metadata-eval N/A

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

   5. *-commutative N/A

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

   6. *-commutative N/A

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

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

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

   8. metadata-eval N/A

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

   9. associate-*l* N/A

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

   10. distribute-lft-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 11.1

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

5. Applied rewrites 11.1%

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

7. Applied rewrites 11.4%

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

Alternative 13: 11.6% accurate, 26.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in s around -inf

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

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

   4. metadata-eval N/A

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

   5. *-commutative N/A

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

   6. *-commutative N/A

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

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

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

   8. metadata-eval N/A

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

   9. associate-*l* N/A

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

   10. distribute-lft-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 11.1

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

5. Applied rewrites 11.1%

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

7. Applied rewrites 11.4%

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

Alternative 14: 7.3% accurate, 30.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in s around -inf

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

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

   4. metadata-eval N/A

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

   5. *-commutative N/A

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

   6. *-commutative N/A

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

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

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

   8. metadata-eval N/A

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

   9. associate-*l* N/A

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

   10. distribute-lft-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 11.1

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

5. Applied rewrites 11.1%

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

Alternative 15: 11.4% accurate, 170.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in 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 11.1

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

5. Applied rewrites 11.1%

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

Reproduce

herbie shell --seed 2024322 
(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))))