Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 15.7s

Alternatives: 11

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

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. Final simplification 98.9%

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

Alternative 2: 97.3% accurate, 1.2× speedup

Math

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

FPCore

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

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 97.7%

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

6. Final simplification 97.7%

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

Alternative 3: 97.2% accurate, 1.2× speedup

Math

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

FPCore

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

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 97.7%

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

6. Taylor expanded in s around 0

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

8. Applied rewrites 97.6%

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

9. Final simplification 97.6%

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

Alternative 4: 97.2% accurate, 1.2× speedup

Math

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

FPCore

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

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 97.7%

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

6. Taylor expanded in s around 0

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

8. Applied rewrites 97.6%

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

9. Final simplification 97.6%

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

Alternative 5: 94.9% accurate, 1.2× speedup

Math

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

FPCore

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

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 95.5

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

5. Applied rewrites 95.5%

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

6. Final simplification 95.5%

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

Alternative 6: 37.6% accurate, 1.3× speedup

Math

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

FPCore

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

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 38.2%

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

6. Final simplification 38.2%

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

Alternative 7: 6.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{PI}\left(\right)}{s}\\ \log \left(\frac{1}{\mathsf{fma}\left(\frac{0.5 - 0.25 \cdot t\_0}{u}, -1, -0.5 \cdot t\_0\right) \cdot \left(-u\right)} - 1\right) \cdot \left(-s\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ (PI) s)))
   (*
    (log
     (-
      (/ 1.0 (* (fma (/ (- 0.5 (* 0.25 t_0)) u) -1.0 (* -0.5 t_0)) (- u)))
      1.0))
    (- s))))

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

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

5. Applied rewrites 10.5%

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

7. Applied rewrites 1.2%

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

9. Applied rewrites -0.0%

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

10. Taylor expanded in u around -inf

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

12. Applied rewrites 5.5%

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

13. Final simplification 7.4%

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

Alternative 8: 9.8% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 2.4999999292951713 \cdot 10^{-10}:\\ \;\;\;\;\log \left(\frac{1}{\mathsf{fma}\left(-0.25, \frac{\mathsf{PI}\left(\right)}{s}, 0.5\right)} - 1\right) \cdot \left(-s\right)\\ \mathbf{else}:\\ \;\;\;\;\left(u \cdot u\right) \cdot \left(\frac{0}{s} - \frac{\frac{\mathsf{PI}\left(\right)}{u} + -2 \cdot \mathsf{PI}\left(\right)}{u}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (if (<= s 2.4999999292951713e-10)
   (* (log (- (/ 1.0 (fma -0.25 (/ (PI) s) 0.5)) 1.0)) (- s))
   (* (* u u) (- (/ 0.0 s) (/ (+ (/ (PI) u) (* -2.0 (PI))) u)))))

Derivation

1. Split input into 2 regimes

2. if s < 2.49999993e-10

   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

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

   5. Applied rewrites 11.6%

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

   7. Applied rewrites -0.0%

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

   8. Taylor expanded in u around 0

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

   10. Applied rewrites 11.5%

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

   if 2.49999993e-10 < s

   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

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

   5. Applied rewrites 5.7%

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

   6. Taylor expanded in u around -inf

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

   8. Applied rewrites 22.0%

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

4. Final simplification 14.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 2.4999999292951713 \cdot 10^{-10}:\\ \;\;\;\;\log \left(\frac{1}{\mathsf{fma}\left(-0.25, \frac{\mathsf{PI}\left(\right)}{s}, 0.5\right)} - 1\right) \cdot \left(-s\right)\\ \mathbf{else}:\\ \;\;\;\;\left(u \cdot u\right) \cdot \left(\frac{0}{s} - \frac{\frac{\mathsf{PI}\left(\right)}{u} + -2 \cdot \mathsf{PI}\left(\right)}{u}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 11.6% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \left(u \cdot u\right) \cdot \left(\frac{0}{s} - \frac{\frac{\mathsf{PI}\left(\right)}{u} + -2 \cdot \mathsf{PI}\left(\right)}{u}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* (* u u) (- (/ 0.0 s) (/ (+ (/ (PI) u) (* -2.0 (PI))) 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

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

5. Applied rewrites 6.1%

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

6. Taylor expanded in u around -inf

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

8. Applied rewrites 12.0%

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

9. Final simplification 12.0%

   \[\leadsto \left(u \cdot u\right) \cdot \left(\frac{0}{s} - \frac{\frac{\mathsf{PI}\left(\right)}{u} + -2 \cdot \mathsf{PI}\left(\right)}{u}\right) \]
10. Add Preprocessing

Alternative 10: 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 11.8

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

5. Applied rewrites 11.8%

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

Alternative 11: 10.3% accurate, 510.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (u s) :precision binary32 0.0)

C

float code(float u, float s) {
	return 0.0f;
}

Fortran

real(4) function code(u, s)
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = 0.0e0
end function

Julia

function code(u, s)
	return Float32(0.0)
end

MATLAB

function tmp = code(u, s)
	tmp = single(0.0);
end

Derivation

1. Initial program 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

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

5. Applied rewrites 6.4%

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

6. Taylor expanded in s around 0

   \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{-16 \cdot {\left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{1}{2} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + 16 \cdot {\left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{1}{2} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{s}} \]
7. Step-by-step derivation

8. Applied rewrites 10.5%

   \[\leadsto \frac{0}{\color{blue}{s}} \]

9. Taylor expanded in s around 0

   \[\leadsto 0 \]
10. Step-by-step derivation

11. Applied rewrites 10.5%

    \[\leadsto 0 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024284 
(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))))