Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 14.5s

Alternatives: 16

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

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

FPCore

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

Derivation

1. Initial program 99.1%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 99.2

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

4. Applied rewrites 99.2%

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

Alternative 2: 51.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{PI}\left(\right)}{s}\\ t_1 := \frac{1}{1 + e^{t\_0}}\\ \mathbf{if}\;\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - t\_1\right) + t\_1} - 1\right) \leq -2.9999999050033628 \cdot 10^{-15}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + \mathsf{fma}\left(-\mathsf{fma}\left(t\_0 \cdot \mathsf{PI}\left(\right), -0.5, \mathsf{PI}\left(\right)\right), \frac{1}{s}, 1\right)} - t\_1\right) + t\_1} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + \mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot \mathsf{PI}\left(\right), t\_0, \mathsf{PI}\left(\right)\right), \frac{-1}{s}, 1\right)} - t\_1\right) + t\_1} - 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ (PI) s)) (t_1 (/ 1.0 (+ 1.0 (exp t_0)))))
   (if (<=
        (*
         (- s)
         (log
          (-
           (/ 1.0 (+ (* u (- (/ 1.0 (+ 1.0 (exp (/ (- (PI)) s)))) t_1)) t_1))
           1.0)))
        -2.9999999050033628e-15)
     (*
      (- s)
      (log
       (-
        (/
         1.0
         (+
          (*
           u
           (-
            (/
             1.0
             (+ 1.0 (fma (- (fma (* t_0 (PI)) -0.5 (PI))) (/ 1.0 s) 1.0)))
            t_1))
          t_1))
        1.0)))
     (*
      (- s)
      (log
       (-
        (/
         1.0
         (+
          (*
           u
           (-
            (/ 1.0 (+ 1.0 (fma (fma (* -0.5 (PI)) t_0 (PI)) (/ -1.0 s) 1.0)))
            t_1))
          t_1))
        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)))) < -2.99999991e-15

   1. Initial program 99.0%

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

   3. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f32 N/A

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

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. lower-PI.f32 N/A

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

      14. lower-PI.f32 9.9

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

   5. Applied rewrites 10.7%

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

   7. Applied rewrites 50.0%

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

   if -2.99999991e-15 < (*.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 99.2%

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

   3. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f32 N/A

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

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. lower-PI.f32 N/A

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

      14. lower-PI.f32 0.0

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

   5. Applied rewrites 0.0%

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

   7. Applied rewrites 0.0%

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

   9. Applied rewrites 48.6%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \leq -2.9999999050033628 \cdot 10^{-15}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + \mathsf{fma}\left(-\mathsf{fma}\left(\frac{\mathsf{PI}\left(\right)}{s} \cdot \mathsf{PI}\left(\right), -0.5, \mathsf{PI}\left(\right)\right), \frac{1}{s}, 1\right)} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + \mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot \mathsf{PI}\left(\right), \frac{\mathsf{PI}\left(\right)}{s}, \mathsf{PI}\left(\right)\right), \frac{-1}{s}, 1\right)} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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 99.1%

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

Alternative 4: 52.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{PI}\left(\right)}{s}\\ t_1 := \frac{1}{1 + e^{t\_0}}\\ \mathbf{if}\;s \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + \mathsf{fma}\left(-\mathsf{fma}\left(-0.5 \cdot \mathsf{PI}\left(\right), t\_0, \mathsf{PI}\left(\right)\right), \frac{1}{s}, 1\right)} - t\_1\right) + t\_1} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + \mathsf{fma}\left(-\mathsf{fma}\left(t\_0 \cdot \mathsf{PI}\left(\right), -0.5, \mathsf{PI}\left(\right)\right), \frac{1}{s}, 1\right)} - t\_1\right) + t\_1} - 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ (PI) s)) (t_1 (/ 1.0 (+ 1.0 (exp t_0)))))
   (if (<= s 9.999999998199587e-24)
     (*
      (- s)
      (log
       (-
        (/
         1.0
         (+
          (*
           u
           (-
            (/
             1.0
             (+ 1.0 (fma (- (fma (* -0.5 (PI)) t_0 (PI))) (/ 1.0 s) 1.0)))
            t_1))
          t_1))
        1.0)))
     (*
      (- s)
      (log
       (-
        (/
         1.0
         (+
          (*
           u
           (-
            (/
             1.0
             (+ 1.0 (fma (- (fma (* t_0 (PI)) -0.5 (PI))) (/ 1.0 s) 1.0)))
            t_1))
          t_1))
        1.0))))))

Derivation

1. Split input into 2 regimes

2. if s < 1e-23

   1. Initial program 99.3%

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

   3. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f32 N/A

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

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. lower-PI.f32 N/A

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

      14. lower-PI.f32 0.0

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

   5. Applied rewrites 0.0%

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

   7. Applied rewrites 0.1%

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

   9. Applied rewrites 51.2%

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

   if 1e-23 < s

   1. Initial program 99.0%

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

   3. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f32 N/A

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

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. lower-PI.f32 N/A

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

      14. lower-PI.f32 5.3

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

   5. Applied rewrites 6.4%

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

   7. Applied rewrites 51.0%

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

Alternative 5: 53.8% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ (PI) s)) (t_1 (/ 1.0 (+ 1.0 (exp t_0)))))
   (if (<= s 4.999999999099794e-24)
     (*
      (- s)
      (log
       (-
        (/
         1.0
         (+
          (*
           u
           (-
            (/
             1.0
             (+ 1.0 (fma (- (fma (* -0.5 (PI)) t_0 (PI))) (/ 1.0 s) 1.0)))
            t_1))
          t_1))
        1.0)))
     (*
      (- s)
      (log
       (-
        (/
         1.0
         (+
          (*
           u
           (-
            (/ 1.0 (+ 1.0 (fma -1.0 (/ (fma (* t_0 (PI)) -0.5 (PI)) s) 1.0)))
            t_1))
          t_1))
        1.0))))))

Derivation

1. Split input into 2 regimes

2. if s < 5e-24

   1. Initial program 99.4%

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

   3. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f32 N/A

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

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. lower-PI.f32 N/A

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

      14. lower-PI.f32 0.0

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

   5. Applied rewrites 0.0%

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

   7. Applied rewrites 0.0%

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

   9. Applied rewrites 50.1%

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

   if 5e-24 < s

   1. Initial program 98.9%

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

   3. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f32 N/A

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

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. lower-PI.f32 N/A

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

      14. lower-PI.f32 4.4

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

   5. Applied rewrites 5.0%

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

   7. Applied rewrites 54.6%

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

Alternative 6: 55.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{PI}\left(\right)}{s}\\ t_1 := \frac{1}{1 + e^{t\_0}}\\ \mathbf{if}\;u \leq 0.0004600000102072954:\\ \;\;\;\;\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + \mathsf{fma}\left(\mathsf{fma}\left(t\_0 \cdot \mathsf{PI}\left(\right), -0.5, \mathsf{PI}\left(\right)\right), \frac{-1}{s}, 1\right)} - t\_1\right) + t\_1} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + \mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot \mathsf{PI}\left(\right), t\_0, \mathsf{PI}\left(\right)\right), \frac{-1}{s}, 1\right)} - t\_1\right) + t\_1} - 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ (PI) s)) (t_1 (/ 1.0 (+ 1.0 (exp t_0)))))
   (if (<= u 0.0004600000102072954)
     (*
      (- s)
      (log
       (-
        (/
         1.0
         (+
          (*
           u
           (-
            (/ 1.0 (+ 1.0 (fma (fma (* t_0 (PI)) -0.5 (PI)) (/ -1.0 s) 1.0)))
            t_1))
          t_1))
        1.0)))
     (*
      (- s)
      (log
       (-
        (/
         1.0
         (+
          (*
           u
           (-
            (/ 1.0 (+ 1.0 (fma (fma (* -0.5 (PI)) t_0 (PI)) (/ -1.0 s) 1.0)))
            t_1))
          t_1))
        1.0))))))

Derivation

1. Split input into 2 regimes

2. if u < 4.6000001e-4

   1. Initial program 99.2%

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

   3. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f32 N/A

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

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. lower-PI.f32 N/A

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

      14. lower-PI.f32 3.8

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

   5. Applied rewrites 3.8%

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

   7. Applied rewrites 51.7%

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

   if 4.6000001e-4 < u

   1. Initial program 98.8%

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

   3. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f32 N/A

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

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. lower-PI.f32 N/A

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

      14. lower-PI.f32 0.6

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

   5. Applied rewrites 0.6%

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

   7. Applied rewrites 1.2%

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

   9. Applied rewrites 44.9%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq 0.0004600000102072954:\\ \;\;\;\;\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + \mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{PI}\left(\right)}{s} \cdot \mathsf{PI}\left(\right), -0.5, \mathsf{PI}\left(\right)\right), \frac{-1}{s}, 1\right)} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + \mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot \mathsf{PI}\left(\right), \frac{\mathsf{PI}\left(\right)}{s}, \mathsf{PI}\left(\right)\right), \frac{-1}{s}, 1\right)} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{PI}\left(\right)}{s}\\ t_1 := \frac{1}{1 + e^{t\_0}}\\ \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(t\_0 \cdot \mathsf{PI}\left(\right), -0.5, \mathsf{PI}\left(\right)\right)}{s}, 1\right)} - t\_1\right) + t\_1} - 1\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ (PI) s)) (t_1 (/ 1.0 (+ 1.0 (exp t_0)))))
   (*
    (- s)
    (log
     (-
      (/
       1.0
       (+
        (*
         u
         (-
          (/ 1.0 (+ 1.0 (fma -1.0 (/ (fma (* t_0 (PI)) -0.5 (PI)) s) 1.0)))
          t_1))
        t_1))
      1.0)))))

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. unpow2 N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. lower-PI.f32 N/A

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

   14. lower-PI.f32 2.6

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

5. Applied rewrites 2.3%

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

7. Applied rewrites 53.4%

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

Alternative 8: 56.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{PI}\left(\right)}{s}\\ t_1 := \frac{1}{1 + e^{t\_0}}\\ \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-0.5 \cdot \mathsf{PI}\left(\right), t\_0, \mathsf{PI}\left(\right)\right)}{s}, 1\right)} - t\_1\right) + t\_1} - 1\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ (PI) s)) (t_1 (/ 1.0 (+ 1.0 (exp t_0)))))
   (*
    (- s)
    (log
     (-
      (/
       1.0
       (+
        (*
         u
         (-
          (/ 1.0 (+ 1.0 (fma -1.0 (/ (fma (* -0.5 (PI)) t_0 (PI)) s) 1.0)))
          t_1))
        t_1))
      1.0)))))

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. unpow2 N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. lower-PI.f32 N/A

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

   14. lower-PI.f32 2.6

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

5. Applied rewrites 2.5%

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

7. Applied rewrites 2.5%

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

9. Applied rewrites 50.9%

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

Alternative 9: 37.6% accurate, 1.3× speedup

Math

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

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 1.0)) t_0)) t_0)) 1.0)))))

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in s around inf

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

5. Applied rewrites 37.8%

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

Alternative 10: 21.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(\left(\mathsf{fma}\left(-4, \frac{\mathsf{fma}\left(0.5 \cdot \mathsf{PI}\left(\right), u, 0.25 \cdot \mathsf{PI}\left(\right)\right)}{s}, 2\right) - -2 \cdot \frac{\mathsf{PI}\left(\right)}{s}\right) - 1\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (-
    (-
     (fma -4.0 (/ (fma (* 0.5 (PI)) u (* 0.25 (PI))) s) 2.0)
     (* -2.0 (/ (PI) s)))
    1.0))))

Derivation

1. Initial program 99.1%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. lower-/.f32 N/A

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

   4. lower-/.f32 99.1

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

4. Applied rewrites 99.1%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. clear-num N/A

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

   6. lift-/.f32 N/A

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

   7. flip-+ N/A

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

6. Applied rewrites 99.1%

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

7. Taylor expanded in s around inf

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

   1. lower--.f32 N/A

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

9. Applied rewrites 21.9%

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

Alternative 11: 6.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 3.0000000095132306 \cdot 10^{-30}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(-\mathsf{PI}\left(\right), \frac{\mathsf{fma}\left(-0.5, u, 0.25\right)}{s}, 0.5\right)} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{\mathsf{fma}\left(-0.5, u, 0.25\right)}{-s}, 0.5\right)} - 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (if (<= s 3.0000000095132306e-30)
   (* (- s) (log (- (/ 1.0 (fma (- (PI)) (/ (fma -0.5 u 0.25) s) 0.5)) 1.0)))
   (* (- s) (log (- (/ 1.0 (fma (PI) (/ (fma -0.5 u 0.25) (- s)) 0.5)) 1.0)))))

Derivation

1. Split input into 2 regimes

2. if s < 3e-30

   1. Initial program 99.3%

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

   3. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

   5. Applied rewrites -0.0%

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

   7. Applied rewrites -0.0%

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

   9. Applied rewrites 15.2%

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

   if 3e-30 < s

   1. Initial program 99.1%

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

   3. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

   5. Applied rewrites -0.0%

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

   7. Applied rewrites -0.0%

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

   9. Applied rewrites 8.6%

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

4. Final simplification 10.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 3.0000000095132306 \cdot 10^{-30}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(-\mathsf{PI}\left(\right), \frac{\mathsf{fma}\left(-0.5, u, 0.25\right)}{s}, 0.5\right)} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{\mathsf{fma}\left(-0.5, u, 0.25\right)}{-s}, 0.5\right)} - 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 6.2% accurate, 3.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

5. Applied rewrites -0.0%

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

7. Applied rewrites -0.0%

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

9. Applied rewrites 10.1%

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

10. Final simplification 10.1%

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

Alternative 13: 11.5% accurate, 17.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.1%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 99.2

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

4. Applied rewrites 99.2%

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

5. Taylor expanded in s around -inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

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

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

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

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

   5. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. metadata-eval N/A

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

   11. *-commutative N/A

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

   12. distribute-lft-in N/A

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

   13. +-commutative N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 N/A

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

   16. +-commutative N/A

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

   17. lower-fma.f32 N/A

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

   18. lower-PI.f32 11.9

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

7. Applied rewrites 11.9%

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

8. Taylor expanded in u around inf

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

10. Applied rewrites 12.1%

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

Alternative 14: 11.5% accurate, 26.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.1%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 99.2

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

4. Applied rewrites 99.2%

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

5. Taylor expanded in s around -inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

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

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

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

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

   5. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. metadata-eval N/A

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

   11. *-commutative N/A

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

   12. distribute-lft-in N/A

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

   13. +-commutative N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 N/A

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

   16. +-commutative N/A

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

   17. lower-fma.f32 N/A

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

   18. lower-PI.f32 11.9

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

7. Applied rewrites 11.9%

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

9. Applied rewrites 12.1%

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

Alternative 15: 11.3% accurate, 170.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in 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 11.9

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

5. Applied rewrites 11.9%

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

6. Final simplification 11.9%

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

Alternative 16: 10.3% accurate, 510.0× speedup

Math

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

FPCore

(FPCore (u s) :precision binary32 0.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

5. Applied rewrites -0.0%

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

   1. lift--.f32 N/A

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

7. Applied rewrites -0.0%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\mathsf{expm1}\left(-\log \left(0.5 - \frac{\mathsf{fma}\left(-0.5, u, 0.25\right) \cdot \mathsf{PI}\left(\right)}{s}\right)\right)\right)} \]

8. Taylor expanded in s around inf

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

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

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

   4. exp-neg N/A

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

   5. rem-exp-log N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

      \[\leadsto \left(\mathsf{neg}\left(\log \color{blue}{1}\right)\right) \cdot s \]

   8. metadata-eval N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{0}\right)\right) \cdot s \]

   9. metadata-eval N/A

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

   10. lower-*.f32 10.1

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

10. Applied rewrites 10.1%

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

11. Taylor expanded in s around 0

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

13. Applied rewrites 10.1%

    \[\leadsto 0 \]

14. Final simplification 10.1%

    \[\leadsto 0 \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024309 
(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))))