Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 99.0%

Time: 15.2s

Alternatives: 10

Speedup: 0.3×

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, 0.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

   1. lift--.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. metadata-eval N/A

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

   4. frac-sub N/A

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

   5. lower-/.f32 N/A

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

4. Applied rewrites 99.1%

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

Alternative 2: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\mathsf{log1p}\left(e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)}\\ \left(-s\right) \cdot \log \left(\mathsf{expm1}\left(-\log \left(\mathsf{fma}\left(e^{-\mathsf{log1p}\left(e^{\frac{-\mathsf{PI}\left(\right)}{s}}\right)} - t\_0, u, t\_0\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (exp (- (log1p (exp (/ (PI) s)))))))
   (*
    (- s)
    (log
     (expm1
      (- (log (fma (- (exp (- (log1p (exp (/ (- (PI)) s))))) t_0) u t_0))))))))

Derivation

1. Initial program 99.0%

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

4. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\mathsf{expm1}\left(-\log \left(\mathsf{fma}\left(e^{-\mathsf{log1p}\left(e^{\frac{-\mathsf{PI}\left(\right)}{s}}\right)} - e^{-\mathsf{log1p}\left(e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)}, u, e^{-\mathsf{log1p}\left(e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)}\right)\right)\right)\right)} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in u around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-lft-neg-in N/A

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

   3. lower-*.f32 N/A

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

   4. lower-neg.f32 N/A

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

   5. lower--.f32 N/A

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

5. Applied rewrites 99.0%

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

6. Final simplification 99.0%

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

Alternative 4: 98.9% accurate, 1.0× speedup

Math

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

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 99.0%

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

3. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Applied rewrites 99.0%

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

6. Final simplification 99.0%

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

Alternative 5: 97.6% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in u around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 98.4%

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

6. Final simplification 98.4%

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

Alternative 6: 25.0% accurate, 3.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

3. 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

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

5. Applied rewrites 25.0%

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

6. Taylor expanded in u around 0

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

   1. lower-*.f32 N/A

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

   2. lower-PI.f32 25.2

      \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{fma}\left(\frac{-0.25 \cdot \mathsf{PI}\left(\right)}{s}, -4, 1\right)\right) \]

8. Applied rewrites 25.2%

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

Alternative 7: 11.6% accurate, 20.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 11.1%

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

6. Taylor expanded in u around 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

   1. lower-*.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. associate-*r/ N/A

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

   6. lower-/.f32 N/A

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

   7. mul-1-neg N/A

      \[\leadsto u \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right), \frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{u}\right) \]

   8. lower-neg.f32 N/A

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

   9. lower-PI.f32 11.1

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

8. Applied rewrites 11.1%

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

Alternative 8: 11.6% accurate, 23.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in 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

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 11.1%

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

Alternative 9: 11.6% accurate, 36.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 11.1%

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

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. mul-1-neg N/A

      \[\leadsto \mathsf{fma}\left(2, u \cdot \mathsf{PI}\left(\right), \mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \]

   6. lower-neg.f32 N/A

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

   7. lower-PI.f32 11.1

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

8. Applied rewrites 11.1%

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

Alternative 10: 11.3% accurate, 170.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in u around 0

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

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\mathsf{PI}\left(\right)\right) \]

   2. lower-neg.f32 N/A

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

   3. lower-PI.f32 10.8

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

5. Applied rewrites 10.8%

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

Reproduce

herbie shell --seed 2025026 
(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))))