Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 23.4s

Alternatives: 10

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

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

FPCore

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

Derivation

1. Initial program 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-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 98.9

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

4. Applied rewrites 98.9%

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

5. Final simplification 98.9%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ -1.0 (- (exp (/ (PI) s)) -1.0))))
   (*
    (log
     (-
      -1.0
      (/ -1.0 (- (* (- t_0 (/ -1.0 (- (exp (/ (- (PI)) s)) -1.0))) u) 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 3: 97.9% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 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 inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 97.5%

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

6. Final simplification 97.5%

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

Alternative 4: 16.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 1.9999999774532045 \cdot 10^{-26}:\\ \;\;\;\;\log \left(\frac{1}{\mathsf{fma}\left(-0.25, \mathsf{fma}\left(\frac{1}{s}, {\left(\frac{1}{\mathsf{PI}\left(\right)}\right)}^{-1}, 0\right), 0.5\right)} - 1\right) \cdot \left(-s\right)\\ \mathbf{else}:\\ \;\;\;\;\left(u \cdot u\right) \cdot \frac{\frac{\mathsf{PI}\left(\right)}{u} + -2 \cdot \mathsf{PI}\left(\right)}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (if (<= s 1.9999999774532045e-26)
   (*
    (log
     (-
      (/ 1.0 (fma -0.25 (fma (/ 1.0 s) (pow (/ 1.0 (PI)) -1.0) 0.0) 0.5))
      1.0))
    (- s))
   (* (* u u) (/ (+ (/ (PI) u) (* -2.0 (PI))) (- u)))))

Derivation

1. Split input into 2 regimes

2. if s < 1.99999998e-26

   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%

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

   6. Taylor expanded in u around 0

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

   8. Applied rewrites 14.7%

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

   10. Applied rewrites 23.0%

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

   if 1.99999998e-26 < 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

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

   5. Applied rewrites 6.9%

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

   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 13.7%

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

4. Final simplification 17.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 1.9999999774532045 \cdot 10^{-26}:\\ \;\;\;\;\log \left(\frac{1}{\mathsf{fma}\left(-0.25, \mathsf{fma}\left(\frac{1}{s}, {\left(\frac{1}{\mathsf{PI}\left(\right)}\right)}^{-1}, 0\right), 0.5\right)} - 1\right) \cdot \left(-s\right)\\ \mathbf{else}:\\ \;\;\;\;\left(u \cdot u\right) \cdot \frac{\frac{\mathsf{PI}\left(\right)}{u} + -2 \cdot \mathsf{PI}\left(\right)}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 11.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{PI}\left(\right)}{u}\\ t_1 := \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s}\\ \mathbf{if}\;s \leq 4.999999999099794 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-4, \frac{t\_1}{u} + t\_1, \mathsf{fma}\left(t\_0, \frac{\mathsf{PI}\left(\right)}{s}, t\_1\right) \cdot 4\right) \cdot \left(u \cdot u\right), -0.5, \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot u, -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(u \cdot u\right) \cdot \frac{t\_0 + -2 \cdot \mathsf{PI}\left(\right)}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ (PI) u)) (t_1 (/ (* (PI) (PI)) s)))
   (if (<= s 4.999999999099794e-24)
     (fma
      (* (fma -4.0 (+ (/ t_1 u) t_1) (* (fma t_0 (/ (PI) s) t_1) 4.0)) (* u u))
      -0.5
      (* (fma (* (PI) u) -0.5 (* 0.25 (PI))) -4.0))
     (* (* u u) (/ (+ t_0 (* -2.0 (PI))) (- u))))))

Derivation

1. Split input into 2 regimes

2. if s < 5e-24

   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

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

   5. Applied rewrites 7.5%

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

   7. Applied rewrites 12.0%

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

   8. Taylor expanded in u around inf

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

   10. Applied rewrites 7.3%

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

   if 5e-24 < 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

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

   5. Applied rewrites 6.9%

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

   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 14.4%

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

4. Final simplification 11.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 4.999999999099794 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-4, \frac{\frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s}}{u} + \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s}, \mathsf{fma}\left(\frac{\mathsf{PI}\left(\right)}{u}, \frac{\mathsf{PI}\left(\right)}{s}, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s}\right) \cdot 4\right) \cdot \left(u \cdot u\right), -0.5, \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot u, -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(u \cdot u\right) \cdot \frac{\frac{\mathsf{PI}\left(\right)}{u} + -2 \cdot \mathsf{PI}\left(\right)}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 14.1% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 1.0999999650920748 \cdot 10^{-20}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(u \cdot u\right) \cdot \frac{\frac{\mathsf{PI}\left(\right)}{u} + -2 \cdot \mathsf{PI}\left(\right)}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (if (<= s 1.0999999650920748e-20)
   0.0
   (* (* u u) (/ (+ (/ (PI) u) (* -2.0 (PI))) (- u)))))

Derivation

1. Split input into 2 regimes

2. if s < 1.09999997e-20

   1. Initial program 99.1%

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

   3. Taylor expanded in s around -inf

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

   5. Applied rewrites 7.1%

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

   6. Taylor expanded in s around 0

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

   8. Applied rewrites 13.3%

      \[\leadsto 0 \]

   if 1.09999997e-20 < s

   1. Initial program 98.7%

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

   3. Taylor expanded in s around -inf

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

   5. Applied rewrites 7.4%

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

   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 15.7%

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

4. Final simplification 14.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 1.0999999650920748 \cdot 10^{-20}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(u \cdot u\right) \cdot \frac{\frac{\mathsf{PI}\left(\right)}{u} + -2 \cdot \mathsf{PI}\left(\right)}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 14.1% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 1.0999999650920748 \cdot 10^{-20}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{u}\right) \cdot u\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (if (<= s 1.0999999650920748e-20) 0.0 (* (- (* 2.0 (PI)) (/ (PI) u)) u)))

Derivation

1. Split input into 2 regimes

2. if s < 1.09999997e-20

   1. Initial program 99.1%

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

   3. Taylor expanded in s around -inf

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

   5. Applied rewrites 7.1%

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

   6. Taylor expanded in s around 0

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

   8. Applied rewrites 13.3%

      \[\leadsto 0 \]

   if 1.09999997e-20 < s

   1. Initial program 98.7%

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

      1. lift-neg.f32 N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{\left(0 - s\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(0 + \left(\mathsf{neg}\left(s\right)\right)\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      4. lift-neg.f32 N/A

         \[\leadsto \left(0 + \color{blue}{\left(-s\right)}\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      5. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{0}^{3} + {\left(-s\right)}^{3}}{0 \cdot 0 + \left(\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)\right)}} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      6. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{{0}^{3} + {\left(-s\right)}^{3}}{0 \cdot 0 + \left(\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)\right)}} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0} + {\left(-s\right)}^{3}}{0 \cdot 0 + \left(\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      8. lower-+.f32 N/A

         \[\leadsto \frac{\color{blue}{0 + {\left(-s\right)}^{3}}}{0 \cdot 0 + \left(\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      9. lower-pow.f32 N/A

         \[\leadsto \frac{0 + \color{blue}{{\left(-s\right)}^{3}}}{0 \cdot 0 + \left(\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      10. metadata-eval N/A

          \[\leadsto \frac{0 + {\left(-s\right)}^{3}}{\color{blue}{0} + \left(\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      11. lower-+.f32 N/A

          \[\leadsto \frac{0 + {\left(-s\right)}^{3}}{\color{blue}{0 + \left(\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)\right)}} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      12. lower--.f32 N/A

          \[\leadsto \frac{0 + {\left(-s\right)}^{3}}{0 + \color{blue}{\left(\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)\right)}} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \frac{0 + {\left(-s\right)}^{3}}{0 + \left(\color{blue}{\left(-s\right) \cdot \left(-s\right)} - 0 \cdot \left(-s\right)\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      14. lower-*.f32 74.6

          \[\leadsto \frac{0 + {\left(-s\right)}^{3}}{0 + \left(\left(-s\right) \cdot \left(-s\right) - \color{blue}{0 \cdot \left(-s\right)}\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   4. Applied rewrites 74.6%

      \[\leadsto \color{blue}{\frac{0 + {\left(-s\right)}^{3}}{0 + \left(\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)\right)}} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
   5. Step-by-step derivation

      1. lift-+.f32 N/A

         \[\leadsto \frac{\color{blue}{0 + {\left(-s\right)}^{3}}}{0 + \left(\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      2. +-lft-identity 74.6

         \[\leadsto \frac{\color{blue}{{\left(-s\right)}^{3}}}{0 + \left(\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      3. lift-+.f32 N/A

         \[\leadsto \frac{{\left(-s\right)}^{3}}{\color{blue}{0 + \left(\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)\right)}} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      4. +-lft-identity 74.6

         \[\leadsto \frac{{\left(-s\right)}^{3}}{\color{blue}{\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)}} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      5. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{{\left(-s\right)}^{3}}{\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)}} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      6. lift-pow.f32 N/A

         \[\leadsto \frac{\color{blue}{{\left(-s\right)}^{3}}}{\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      7. sqr-pow N/A

         \[\leadsto \frac{\color{blue}{{\left(-s\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(-s\right)}^{\left(\frac{3}{2}\right)}}}{\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      8. pow-prod-down N/A

         \[\leadsto \frac{\color{blue}{{\left(\left(-s\right) \cdot \left(-s\right)\right)}^{\left(\frac{3}{2}\right)}}}{\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      9. lift-neg.f32 N/A

         \[\leadsto \frac{{\left(\color{blue}{\left(\mathsf{neg}\left(s\right)\right)} \cdot \left(-s\right)\right)}^{\left(\frac{3}{2}\right)}}{\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      10. lift-neg.f32 N/A

          \[\leadsto \frac{{\left(\left(\mathsf{neg}\left(s\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(s\right)\right)}\right)}^{\left(\frac{3}{2}\right)}}{\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      11. sqr-neg N/A

          \[\leadsto \frac{{\color{blue}{\left(s \cdot s\right)}}^{\left(\frac{3}{2}\right)}}{\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      12. unpow-prod-down N/A

          \[\leadsto \frac{\color{blue}{{s}^{\left(\frac{3}{2}\right)} \cdot {s}^{\left(\frac{3}{2}\right)}}}{\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      13. sqr-pow N/A

          \[\leadsto \frac{\color{blue}{{s}^{3}}}{\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      14. lift--.f32 N/A

          \[\leadsto \frac{{s}^{3}}{\color{blue}{\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)}} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      15. lift-*.f32 N/A

          \[\leadsto \frac{{s}^{3}}{\color{blue}{\left(-s\right) \cdot \left(-s\right)} - 0 \cdot \left(-s\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      16. lift-*.f32 N/A

          \[\leadsto \frac{{s}^{3}}{\left(-s\right) \cdot \left(-s\right) - \color{blue}{0 \cdot \left(-s\right)}} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

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

          \[\leadsto \frac{{s}^{3}}{\color{blue}{\left(-s\right) \cdot \left(\left(-s\right) - 0\right)}} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      18. --rgt-identity N/A

          \[\leadsto \frac{{s}^{3}}{\left(-s\right) \cdot \color{blue}{\left(-s\right)}} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      19. lift-neg.f32 N/A

          \[\leadsto \frac{{s}^{3}}{\color{blue}{\left(\mathsf{neg}\left(s\right)\right)} \cdot \left(-s\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      20. lift-neg.f32 N/A

          \[\leadsto \frac{{s}^{3}}{\left(\mathsf{neg}\left(s\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(s\right)\right)}} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   6. Applied rewrites 98.6%

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

   7. Taylor expanded in s around -inf

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

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

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

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

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. associate-*r* N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 N/A

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

      15. lower-PI.f32 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f32 N/A

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

      18. lower-PI.f32 15.4

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

   9. Applied rewrites 15.4%

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

   10. Taylor expanded in u around inf

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

   12. Applied rewrites 15.7%

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

Alternative 8: 14.1% accurate, 25.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 1.0999999650920748 \cdot 10^{-20}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{PI}\left(\right) \cdot u\right) \cdot 2 - \mathsf{PI}\left(\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (if (<= s 1.0999999650920748e-20) 0.0 (- (* (* (PI) u) 2.0) (PI))))

Derivation

1. Split input into 2 regimes

2. if s < 1.09999997e-20

   1. Initial program 99.1%

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

   3. Taylor expanded in s around -inf

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

   5. Applied rewrites 7.1%

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

   6. Taylor expanded in s around 0

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

   8. Applied rewrites 13.3%

      \[\leadsto 0 \]

   if 1.09999997e-20 < s

   1. Initial program 98.7%

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

      1. lift-neg.f32 N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{\left(0 - s\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(0 + \left(\mathsf{neg}\left(s\right)\right)\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      4. lift-neg.f32 N/A

         \[\leadsto \left(0 + \color{blue}{\left(-s\right)}\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      5. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{0}^{3} + {\left(-s\right)}^{3}}{0 \cdot 0 + \left(\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)\right)}} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      6. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{{0}^{3} + {\left(-s\right)}^{3}}{0 \cdot 0 + \left(\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)\right)}} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0} + {\left(-s\right)}^{3}}{0 \cdot 0 + \left(\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      8. lower-+.f32 N/A

         \[\leadsto \frac{\color{blue}{0 + {\left(-s\right)}^{3}}}{0 \cdot 0 + \left(\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      9. lower-pow.f32 N/A

         \[\leadsto \frac{0 + \color{blue}{{\left(-s\right)}^{3}}}{0 \cdot 0 + \left(\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      10. metadata-eval N/A

          \[\leadsto \frac{0 + {\left(-s\right)}^{3}}{\color{blue}{0} + \left(\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      11. lower-+.f32 N/A

          \[\leadsto \frac{0 + {\left(-s\right)}^{3}}{\color{blue}{0 + \left(\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)\right)}} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      12. lower--.f32 N/A

          \[\leadsto \frac{0 + {\left(-s\right)}^{3}}{0 + \color{blue}{\left(\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)\right)}} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \frac{0 + {\left(-s\right)}^{3}}{0 + \left(\color{blue}{\left(-s\right) \cdot \left(-s\right)} - 0 \cdot \left(-s\right)\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      14. lower-*.f32 74.6

          \[\leadsto \frac{0 + {\left(-s\right)}^{3}}{0 + \left(\left(-s\right) \cdot \left(-s\right) - \color{blue}{0 \cdot \left(-s\right)}\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   4. Applied rewrites 74.6%

      \[\leadsto \color{blue}{\frac{0 + {\left(-s\right)}^{3}}{0 + \left(\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)\right)}} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
   5. Step-by-step derivation

      1. lift-+.f32 N/A

         \[\leadsto \frac{\color{blue}{0 + {\left(-s\right)}^{3}}}{0 + \left(\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      2. +-lft-identity 74.6

         \[\leadsto \frac{\color{blue}{{\left(-s\right)}^{3}}}{0 + \left(\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      3. lift-+.f32 N/A

         \[\leadsto \frac{{\left(-s\right)}^{3}}{\color{blue}{0 + \left(\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)\right)}} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      4. +-lft-identity 74.6

         \[\leadsto \frac{{\left(-s\right)}^{3}}{\color{blue}{\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)}} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      5. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{{\left(-s\right)}^{3}}{\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)}} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      6. lift-pow.f32 N/A

         \[\leadsto \frac{\color{blue}{{\left(-s\right)}^{3}}}{\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      7. sqr-pow N/A

         \[\leadsto \frac{\color{blue}{{\left(-s\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(-s\right)}^{\left(\frac{3}{2}\right)}}}{\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      8. pow-prod-down N/A

         \[\leadsto \frac{\color{blue}{{\left(\left(-s\right) \cdot \left(-s\right)\right)}^{\left(\frac{3}{2}\right)}}}{\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      9. lift-neg.f32 N/A

         \[\leadsto \frac{{\left(\color{blue}{\left(\mathsf{neg}\left(s\right)\right)} \cdot \left(-s\right)\right)}^{\left(\frac{3}{2}\right)}}{\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      10. lift-neg.f32 N/A

          \[\leadsto \frac{{\left(\left(\mathsf{neg}\left(s\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(s\right)\right)}\right)}^{\left(\frac{3}{2}\right)}}{\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      11. sqr-neg N/A

          \[\leadsto \frac{{\color{blue}{\left(s \cdot s\right)}}^{\left(\frac{3}{2}\right)}}{\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      12. unpow-prod-down N/A

          \[\leadsto \frac{\color{blue}{{s}^{\left(\frac{3}{2}\right)} \cdot {s}^{\left(\frac{3}{2}\right)}}}{\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      13. sqr-pow N/A

          \[\leadsto \frac{\color{blue}{{s}^{3}}}{\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      14. lift--.f32 N/A

          \[\leadsto \frac{{s}^{3}}{\color{blue}{\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)}} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      15. lift-*.f32 N/A

          \[\leadsto \frac{{s}^{3}}{\color{blue}{\left(-s\right) \cdot \left(-s\right)} - 0 \cdot \left(-s\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      16. lift-*.f32 N/A

          \[\leadsto \frac{{s}^{3}}{\left(-s\right) \cdot \left(-s\right) - \color{blue}{0 \cdot \left(-s\right)}} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

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

          \[\leadsto \frac{{s}^{3}}{\color{blue}{\left(-s\right) \cdot \left(\left(-s\right) - 0\right)}} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      18. --rgt-identity N/A

          \[\leadsto \frac{{s}^{3}}{\left(-s\right) \cdot \color{blue}{\left(-s\right)}} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      19. lift-neg.f32 N/A

          \[\leadsto \frac{{s}^{3}}{\color{blue}{\left(\mathsf{neg}\left(s\right)\right)} \cdot \left(-s\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      20. lift-neg.f32 N/A

          \[\leadsto \frac{{s}^{3}}{\left(\mathsf{neg}\left(s\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(s\right)\right)}} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   6. Applied rewrites 98.6%

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

   7. Taylor expanded in s around -inf

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

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

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

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

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. associate-*r* N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 N/A

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

      15. lower-PI.f32 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f32 N/A

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

      18. lower-PI.f32 15.4

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

   9. Applied rewrites 15.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \mathsf{PI}\left(\right) \cdot u, -\mathsf{PI}\left(\right)\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 15.7%

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

4. Final simplification 14.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 1.0999999650920748 \cdot 10^{-20}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{PI}\left(\right) \cdot u\right) \cdot 2 - \mathsf{PI}\left(\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 13.9% accurate, 56.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 1.0999999650920748 \cdot 10^{-20}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{PI}\left(\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (if (<= s 1.0999999650920748e-20) 0.0 (- (PI))))

Derivation

1. Split input into 2 regimes

2. if s < 1.09999997e-20

   1. Initial program 99.1%

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

   3. Taylor expanded in s around -inf

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

   5. Applied rewrites 7.1%

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

   6. Taylor expanded in s around 0

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

   8. Applied rewrites 13.3%

      \[\leadsto 0 \]

   if 1.09999997e-20 < s

   1. Initial program 98.7%

      \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
   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 15.4

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

   5. Applied rewrites 15.4%

      \[\leadsto \color{blue}{-\mathsf{PI}\left(\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 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

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

5. Applied rewrites 7.3%

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

6. Taylor expanded in s around 0

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

8. Applied rewrites 10.8%

   \[\leadsto 0 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024249 
(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))))