Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 16.2s

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: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.7%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. clear-num N/A

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

   4. div-inv N/A

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

   5. clear-num N/A

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

   6. lift-/.f32 N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f32 N/A

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

   9. exp-1-e N/A

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

   10. lower-E.f32 98.7

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

4. Applied rewrites 98.7%

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

Alternative 2: 8.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.5 \cdot \mathsf{PI}\left(\right), u, -0.25 \cdot \mathsf{PI}\left(\right)\right)\\ t_1 := \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\\ t_2 := \frac{t\_0}{s}\\ \mathbf{if}\;\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - t\_1\right) + t\_1} - 1\right) \leq -4.999999999099794 \cdot 10^{-24}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(\left(1 - \frac{\mathsf{fma}\left({t\_0}^{2}, -8, 0\right)}{s \cdot s}\right) - t\_2 \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(\mathsf{fma}\left(t\_2, -4, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (fma (* 0.5 (PI)) u (* -0.25 (PI))))
        (t_1 (/ 1.0 (+ 1.0 (exp (/ (PI) s)))))
        (t_2 (/ t_0 s)))
   (if (<=
        (*
         (- s)
         (log
          (-
           (/ 1.0 (+ (* u (- (/ 1.0 (+ 1.0 (exp (/ (- (PI)) s)))) t_1)) t_1))
           1.0)))
        -4.999999999099794e-24)
     (*
      (- s)
      (log (- (- 1.0 (/ (fma (pow t_0 2.0) -8.0 0.0) (* s s))) (* t_2 4.0))))
     (* (- s) (log (fma t_2 -4.0 1.0))))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32)))) < -5e-24

   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

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. neg-mul-1 N/A

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

      6. neg-mul-1 N/A

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

      7. distribute-lft-out N/A

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

      8. metadata-eval N/A

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

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

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. mul-1-neg N/A

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

      13. distribute-neg-frac N/A

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

      14. distribute-neg-frac2 N/A

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

      15. mul-1-neg N/A

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

   5. Applied rewrites 6.4%

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

   6. Taylor expanded in s around inf

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

      1. lower--.f32 N/A

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

   8. Applied rewrites 13.1%

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

   if -5e-24 < (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32))))

   1. Initial program 98.6%

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

   3. Taylor expanded in s around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. neg-mul-1 N/A

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

      6. neg-mul-1 N/A

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

      7. distribute-lft-out N/A

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

      8. metadata-eval N/A

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

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

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. mul-1-neg N/A

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

      13. distribute-neg-frac N/A

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

      14. distribute-neg-frac2 N/A

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

      15. mul-1-neg N/A

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

   5. Applied rewrites -0.0%

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

   6. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

   8. Applied rewrites 14.0%

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

Alternative 3: 18.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \mathsf{PI}\left(\right)\\ t_1 := \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\\ \mathbf{if}\;\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - t\_1\right) + t\_1} - 1\right) \leq -2.0000000390829628 \cdot 10^{-25}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(\left(\mathsf{fma}\left(-8, \frac{\mathsf{fma}\left(t\_0 \cdot u, 0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right)}{s}, 2\right) - \frac{\mathsf{PI}\left(\right) \cdot 12}{s} \cdot -0.25\right) - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(t\_0, u, -0.25 \cdot \mathsf{PI}\left(\right)\right)}{s}, -4, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (* 0.5 (PI))) (t_1 (/ 1.0 (+ 1.0 (exp (/ (PI) s))))))
   (if (<=
        (*
         (- s)
         (log
          (-
           (/ 1.0 (+ (* u (- (/ 1.0 (+ 1.0 (exp (/ (- (PI)) s)))) t_1)) t_1))
           1.0)))
        -2.0000000390829628e-25)
     (*
      (- s)
      (log
       (-
        (-
         (fma -8.0 (/ (fma (* t_0 u) 0.5 (* 0.25 (PI))) s) 2.0)
         (* (/ (* (PI) 12.0) s) -0.25))
        1.0)))
     (* (- s) (log (fma (/ (fma t_0 u (* -0.25 (PI))) s) -4.0 1.0))))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32)))) < -2.00000004e-25

   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-exp.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. clear-num N/A

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

      4. div-inv N/A

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

      5. clear-num N/A

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

      6. lift-/.f32 N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f32 N/A

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

      9. exp-1-e N/A

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

      10. lower-E.f32 98.7

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

   4. Applied rewrites 98.7%

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

   6. Applied rewrites 98.8%

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

   7. Taylor expanded in s around inf

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

      1. lower--.f32 N/A

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

   9. Applied rewrites 23.0%

      \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{\left(\mathsf{fma}\left(-8, \frac{\mathsf{fma}\left(\left(0.5 \cdot \mathsf{PI}\left(\right)\right) \cdot u, 0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right)}{s}, 2\right) - \frac{\mathsf{PI}\left(\right) \cdot 12}{s} \cdot -0.25\right)} - 1\right) \]

   if -2.00000004e-25 < (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32))))

   1. Initial program 98.6%

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

   3. Taylor expanded in s around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. neg-mul-1 N/A

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

      6. neg-mul-1 N/A

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

      7. distribute-lft-out N/A

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

      8. metadata-eval N/A

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

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

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. mul-1-neg N/A

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

      13. distribute-neg-frac N/A

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

      14. distribute-neg-frac2 N/A

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

      15. mul-1-neg N/A

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

   5. Applied rewrites -0.0%

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

   6. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

   8. Applied rewrites 14.7%

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

4. Final simplification 18.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \leq -2.0000000390829628 \cdot 10^{-25}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(\left(\mathsf{fma}\left(-8, \frac{\mathsf{fma}\left(\left(0.5 \cdot \mathsf{PI}\left(\right)\right) \cdot u, 0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right)}{s}, 2\right) - \frac{\mathsf{PI}\left(\right) \cdot 12}{s} \cdot -0.25\right) - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.5 \cdot \mathsf{PI}\left(\right), u, -0.25 \cdot \mathsf{PI}\left(\right)\right)}{s}, -4, 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 10.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\\ t_1 := -0.25 \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - t\_0\right) + t\_0} - 1\right) \leq -6.000000240508405 \cdot 10^{-26}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(\left(2 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(-0.5 \cdot \mathsf{PI}\left(\right)\right) \cdot u, 0.5, t\_1\right), -8, 0.25 \cdot \left(\mathsf{PI}\left(\right) \cdot -12\right)\right)}{s}\right) - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.5 \cdot \mathsf{PI}\left(\right), u, t\_1\right)}{s}, -4, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ 1.0 (exp (/ (PI) s))))) (t_1 (* -0.25 (PI))))
   (if (<=
        (*
         (- s)
         (log
          (-
           (/ 1.0 (+ (* u (- (/ 1.0 (+ 1.0 (exp (/ (- (PI)) s)))) t_0)) t_0))
           1.0)))
        -6.000000240508405e-26)
     (*
      (- s)
      (log
       (-
        (-
         2.0
         (/
          (fma (fma (* (* -0.5 (PI)) u) 0.5 t_1) -8.0 (* 0.25 (* (PI) -12.0)))
          s))
        1.0)))
     (* (- s) (log (fma (/ (fma (* 0.5 (PI)) u t_1) s) -4.0 1.0))))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32)))) < -6.00000024e-26

   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-exp.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. clear-num N/A

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

      4. div-inv N/A

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

      5. clear-num N/A

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

      6. lift-/.f32 N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f32 N/A

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

      9. exp-1-e N/A

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

      10. lower-E.f32 98.7

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

   4. Applied rewrites 98.7%

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

   6. Applied rewrites 98.6%

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

   7. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

   9. Applied rewrites 10.6%

      \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{\left(2 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(-0.5 \cdot \mathsf{PI}\left(\right)\right) \cdot u, 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right), -8, 0.25 \cdot \left(\mathsf{PI}\left(\right) \cdot -12\right)\right)}{s}\right)} - 1\right) \]

   if -6.00000024e-26 < (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32))))

   1. Initial program 98.6%

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

   3. Taylor expanded in s around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. neg-mul-1 N/A

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

      6. neg-mul-1 N/A

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

      7. distribute-lft-out N/A

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

      8. metadata-eval N/A

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

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

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. mul-1-neg N/A

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

      13. distribute-neg-frac N/A

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

      14. distribute-neg-frac2 N/A

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

      15. mul-1-neg N/A

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

   5. Applied rewrites -0.0%

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

   6. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

   8. Applied rewrites 14.8%

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

Alternative 5: 9.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\\ t_1 := \left(4 \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(0.5, u, -0.25\right)\\ \mathbf{if}\;\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - t\_0\right) + t\_0} - 1\right) \leq -5.000000097707407 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left({t\_1}^{2}, \frac{1}{t\_1}, 0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.5 \cdot \mathsf{PI}\left(\right), u, -0.25 \cdot \mathsf{PI}\left(\right)\right)}{s}, -4, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ 1.0 (exp (/ (PI) s)))))
        (t_1 (* (* 4.0 (PI)) (fma 0.5 u -0.25))))
   (if (<=
        (*
         (- s)
         (log
          (-
           (/ 1.0 (+ (* u (- (/ 1.0 (+ 1.0 (exp (/ (- (PI)) s)))) t_0)) t_0))
           1.0)))
        -5.000000097707407e-25)
     (fma (pow t_1 2.0) (/ 1.0 t_1) 0.0)
     (*
      (- s)
      (log (fma (/ (fma (* 0.5 (PI)) u (* -0.25 (PI))) s) -4.0 1.0))))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32)))) < -5.0000001e-25

   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

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

   5. Applied rewrites 8.3%

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

   7. Applied rewrites 4.2%

      \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot {\left(\frac{\mathsf{fma}\left(0.5, u, 0.25\right)}{\mathsf{fma}\left(u \cdot u, 0.25, -0.0625\right)}\right)}^{-1}, 4, 0\right) \]

   9. Applied rewrites 9.8%

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

   if -5.0000001e-25 < (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32))))

   1. Initial program 98.6%

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

   3. Taylor expanded in s around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. neg-mul-1 N/A

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

      6. neg-mul-1 N/A

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

      7. distribute-lft-out N/A

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

      8. metadata-eval N/A

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

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

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. mul-1-neg N/A

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

      13. distribute-neg-frac N/A

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

      14. distribute-neg-frac2 N/A

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

      15. mul-1-neg N/A

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

   5. Applied rewrites -0.0%

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

   6. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

   8. Applied rewrites 14.5%

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

Alternative 6: 12.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\\ \mathbf{if}\;\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - t\_0\right) + t\_0} - 1\right) \leq -7.00000013679037 \cdot 10^{-25}:\\ \;\;\;\;\left(2 \cdot \mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{u}\right) \cdot u\\ \mathbf{else}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.5 \cdot \mathsf{PI}\left(\right), u, -0.25 \cdot \mathsf{PI}\left(\right)\right)}{s}, -4, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ 1.0 (exp (/ (PI) s))))))
   (if (<=
        (*
         (- s)
         (log
          (-
           (/ 1.0 (+ (* u (- (/ 1.0 (+ 1.0 (exp (/ (- (PI)) s)))) t_0)) t_0))
           1.0)))
        -7.00000013679037e-25)
     (* (- (* 2.0 (PI)) (/ (PI) u)) u)
     (*
      (- s)
      (log (fma (/ (fma (* 0.5 (PI)) u (* -0.25 (PI))) s) -4.0 1.0))))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32)))) < -7.00000014e-25

   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

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

   5. Applied rewrites 8.3%

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

   6. Taylor expanded in u around inf

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

   8. Applied rewrites 13.6%

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

   10. Applied rewrites 13.7%

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

   if -7.00000014e-25 < (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32))))

   1. Initial program 98.6%

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

   3. Taylor expanded in s around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. neg-mul-1 N/A

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

      6. neg-mul-1 N/A

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

      7. distribute-lft-out N/A

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

      8. metadata-eval N/A

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

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

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. mul-1-neg N/A

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

      13. distribute-neg-frac N/A

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

      14. distribute-neg-frac2 N/A

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

      15. mul-1-neg N/A

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

   5. Applied rewrites -0.0%

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

   6. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

   8. Applied rewrites 14.4%

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

Alternative 7: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.7%

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

Alternative 8: 97.6% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

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

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

   1. *-commutative N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

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

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

   5. neg-mul-1 N/A

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

   6. neg-mul-1 N/A

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

   7. distribute-lft-out N/A

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

   8. metadata-eval N/A

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

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

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. mul-1-neg N/A

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

   13. distribute-neg-frac N/A

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

   14. distribute-neg-frac2 N/A

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

   15. mul-1-neg N/A

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

5. Applied rewrites 3.6%

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

6. Taylor expanded in u around inf

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

   1. lower--.f32 N/A

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

8. Applied rewrites 97.2%

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

Alternative 9: 11.6% accurate, 20.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in s around inf

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

5. Applied rewrites 10.6%

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

6. Taylor expanded in u around inf

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

8. Applied rewrites 11.1%

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

10. Applied rewrites 11.4%

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

Alternative 10: 11.6% accurate, 36.4× speedup

Math

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

FPCore

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

Derivation

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

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

5. Applied rewrites 10.6%

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

6. Taylor expanded in u around inf

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

8. Applied rewrites 5.1%

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

9. Taylor expanded in u around 0

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

11. Applied rewrites 11.1%

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

13. Applied rewrites 11.4%

    \[\leadsto \mathsf{PI}\left(\right) \cdot \left(2 \cdot u + -1\right) \]
14. Add Preprocessing

Alternative 11: 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 98.7%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
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 11.1

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

5. Applied rewrites 11.1%

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

Reproduce

herbie shell --seed 2024302 
(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))))