Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 8.9s

Alternatives: 17

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

4. Applied rewrites 99.0%

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

Alternative 2: 98.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{PI}\left(\right)}{s}\\ \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(0.16666666666666666, t\_0 \cdot \left(t\_0 \cdot t\_0\right), \mathsf{fma}\left(0.5, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s \cdot s}, t\_0\right)\right)}, u, \frac{1}{e^{t\_0} + 1}\right)} - 1\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ (PI) s)))
   (*
    (- s)
    (log
     (-
      (/
       1.0
       (fma
        (-
         (/ 1.0 (+ (exp (/ (- (PI)) s)) 1.0))
         (/
          1.0
          (+
           2.0
           (fma
            0.16666666666666666
            (* t_0 (* t_0 t_0))
            (fma 0.5 (/ (* (PI) (PI)) (* s s)) t_0)))))
        u
        (/ 1.0 (+ (exp t_0) 1.0))))
      1.0)))))

Derivation

1. Initial program 99.0%

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in s around inf

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{\color{blue}{2 + \left(\frac{1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{{s}^{3}} + \left(\frac{1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{s}^{2}} + \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]
6. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \color{blue}{\left(\frac{1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{{s}^{3}} + \left(\frac{1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{s}^{2}} + \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   2. lower-fma.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, \color{blue}{\frac{{\mathsf{PI}\left(\right)}^{3}}{{s}^{3}}}, \frac{1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{s}^{2}} + \frac{\mathsf{PI}\left(\right)}{s}\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   3. cube-div-rev N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, {\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}^{\color{blue}{3}}, \frac{1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{s}^{2}} + \frac{\mathsf{PI}\left(\right)}{s}\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   4. lower-pow.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, {\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}^{\color{blue}{3}}, \frac{1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{s}^{2}} + \frac{\mathsf{PI}\left(\right)}{s}\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   5. lift-/.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, {\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}^{3}, \frac{1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{s}^{2}} + \frac{\mathsf{PI}\left(\right)}{s}\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   6. lift-PI.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, {\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}^{3}, \frac{1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{s}^{2}} + \frac{\mathsf{PI}\left(\right)}{s}\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   7. lower-fma.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, {\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}^{3}, \mathsf{fma}\left(\frac{1}{2}, \frac{{\mathsf{PI}\left(\right)}^{2}}{{s}^{2}}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   8. lower-/.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, {\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}^{3}, \mathsf{fma}\left(\frac{1}{2}, \frac{{\mathsf{PI}\left(\right)}^{2}}{{s}^{2}}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   9. unpow2 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, {\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}^{3}, \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{{s}^{2}}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   10. lower-*.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, {\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}^{3}, \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{{s}^{2}}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   11. lift-PI.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, {\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}^{3}, \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{{s}^{2}}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   12. lift-PI.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, {\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}^{3}, \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{{s}^{2}}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   13. unpow2 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, {\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}^{3}, \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s \cdot s}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   14. lower-*.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, {\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}^{3}, \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s \cdot s}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   15. lift-/.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, {\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}^{3}, \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s \cdot s}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   16. lift-PI.f32 98.0

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s \cdot s}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

7. Applied rewrites 98.0%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{\color{blue}{2 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s \cdot s}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]
8. Step-by-step derivation

   1. lift-pow.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, {\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}^{\color{blue}{3}}, \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s \cdot s}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   2. lift-PI.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, {\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}^{3}, \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s \cdot s}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   3. lift-/.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, {\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}^{3}, \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s \cdot s}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   4. cube-mult N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, \frac{\mathsf{PI}\left(\right)}{s} \cdot \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{s} \cdot \frac{\mathsf{PI}\left(\right)}{s}\right)}, \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s \cdot s}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   5. times-frac N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, \frac{\mathsf{PI}\left(\right)}{s} \cdot \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{\color{blue}{s \cdot s}}, \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s \cdot s}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   6. lift-*.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, \frac{\mathsf{PI}\left(\right)}{s} \cdot \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{\color{blue}{s} \cdot s}, \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s \cdot s}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   7. lift-PI.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, \frac{\mathsf{PI}\left(\right)}{s} \cdot \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s \cdot s}, \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s \cdot s}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   8. lift-PI.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, \frac{\mathsf{PI}\left(\right)}{s} \cdot \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s \cdot s}, \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s \cdot s}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   9. lift-/.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, \frac{\mathsf{PI}\left(\right)}{s} \cdot \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{\color{blue}{s \cdot s}}, \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s \cdot s}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   10. lift-*.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, \frac{\mathsf{PI}\left(\right)}{s} \cdot \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s \cdot \color{blue}{s}}, \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s \cdot s}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   11. lower-*.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, \frac{\mathsf{PI}\left(\right)}{s} \cdot \color{blue}{\frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s \cdot s}}, \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s \cdot s}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   12. lift-/.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, \frac{\mathsf{PI}\left(\right)}{s} \cdot \frac{\color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}}{s \cdot s}, \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s \cdot s}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   13. lift-PI.f32 98.0

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(0.16666666666666666, \frac{\mathsf{PI}\left(\right)}{s} \cdot \frac{\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)}{s \cdot s}, \mathsf{fma}\left(0.5, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s \cdot s}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   14. lift-*.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, \frac{\mathsf{PI}\left(\right)}{s} \cdot \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s \cdot \color{blue}{s}}, \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s \cdot s}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   15. lift-/.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, \frac{\mathsf{PI}\left(\right)}{s} \cdot \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{\color{blue}{s \cdot s}}, \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s \cdot s}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   16. lift-PI.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, \frac{\mathsf{PI}\left(\right)}{s} \cdot \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s \cdot s}, \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s \cdot s}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   17. lift-PI.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, \frac{\mathsf{PI}\left(\right)}{s} \cdot \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s \cdot s}, \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s \cdot s}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   18. lift-*.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, \frac{\mathsf{PI}\left(\right)}{s} \cdot \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{\color{blue}{s} \cdot s}, \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s \cdot s}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   19. times-frac N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, \frac{\mathsf{PI}\left(\right)}{s} \cdot \left(\frac{\mathsf{PI}\left(\right)}{s} \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{s}}\right), \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s \cdot s}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   20. lower-*.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, \frac{\mathsf{PI}\left(\right)}{s} \cdot \left(\frac{\mathsf{PI}\left(\right)}{s} \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{s}}\right), \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s \cdot s}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

9. Applied rewrites 98.0%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(0.16666666666666666, \frac{\mathsf{PI}\left(\right)}{s} \cdot \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{s} \cdot \frac{\mathsf{PI}\left(\right)}{s}\right)}, \mathsf{fma}\left(0.5, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s \cdot s}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

10. Final simplification 98.0%

    \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(0.16666666666666666, \frac{\mathsf{PI}\left(\right)}{s} \cdot \left(\frac{\mathsf{PI}\left(\right)}{s} \cdot \frac{\mathsf{PI}\left(\right)}{s}\right), \mathsf{fma}\left(0.5, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s \cdot s}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]
11. Add Preprocessing

Alternative 3: 97.3% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in s around inf

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{\color{blue}{2 + \left(\frac{1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{{s}^{3}} + \left(\frac{1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{s}^{2}} + \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]
6. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \color{blue}{\left(\frac{1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{{s}^{3}} + \left(\frac{1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{s}^{2}} + \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   2. lower-fma.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, \color{blue}{\frac{{\mathsf{PI}\left(\right)}^{3}}{{s}^{3}}}, \frac{1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{s}^{2}} + \frac{\mathsf{PI}\left(\right)}{s}\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   3. cube-div-rev N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, {\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}^{\color{blue}{3}}, \frac{1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{s}^{2}} + \frac{\mathsf{PI}\left(\right)}{s}\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   4. lower-pow.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, {\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}^{\color{blue}{3}}, \frac{1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{s}^{2}} + \frac{\mathsf{PI}\left(\right)}{s}\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   5. lift-/.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, {\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}^{3}, \frac{1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{s}^{2}} + \frac{\mathsf{PI}\left(\right)}{s}\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   6. lift-PI.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, {\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}^{3}, \frac{1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{s}^{2}} + \frac{\mathsf{PI}\left(\right)}{s}\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   7. lower-fma.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, {\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}^{3}, \mathsf{fma}\left(\frac{1}{2}, \frac{{\mathsf{PI}\left(\right)}^{2}}{{s}^{2}}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   8. lower-/.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, {\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}^{3}, \mathsf{fma}\left(\frac{1}{2}, \frac{{\mathsf{PI}\left(\right)}^{2}}{{s}^{2}}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   9. unpow2 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, {\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}^{3}, \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{{s}^{2}}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   10. lower-*.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, {\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}^{3}, \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{{s}^{2}}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   11. lift-PI.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, {\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}^{3}, \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{{s}^{2}}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   12. lift-PI.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, {\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}^{3}, \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{{s}^{2}}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   13. unpow2 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, {\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}^{3}, \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s \cdot s}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   14. lower-*.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, {\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}^{3}, \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s \cdot s}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   15. lift-/.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(\frac{1}{6}, {\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}^{3}, \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s \cdot s}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

   16. lift-PI.f32 98.0

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s \cdot s}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

7. Applied rewrites 98.0%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{\color{blue}{2 + \mathsf{fma}\left(0.16666666666666666, {\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s \cdot s}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

8. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. pow2 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lift-PI.f32 97.3

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \frac{\mathsf{PI}\left(\right) + 0.5 \cdot \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s}}{s}}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]

10. Applied rewrites 97.3%

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

11. Final simplification 97.3%

    \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{2 + \frac{\mathsf{PI}\left(\right) + 0.5 \cdot \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s}}{s}}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]
12. Add Preprocessing

Alternative 4: 97.8% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in u around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 96.6%

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

Alternative 5: 37.5% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 38.6%

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

8. Final simplification 38.6%

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

Alternative 6: 37.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{PI}\left(\right)}{s}\\ \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(0.5 - \frac{1}{2 + \mathsf{fma}\left(0.5, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s \cdot s}, t\_0\right)}\right) + \frac{1}{1 + e^{t\_0}}} - 1\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ (PI) s)))
   (*
    (- s)
    (log
     (-
      (/
       1.0
       (+
        (* u (- 0.5 (/ 1.0 (+ 2.0 (fma 0.5 (/ (* (PI) (PI)) (* s s)) t_0)))))
        (/ 1.0 (+ 1.0 (exp t_0)))))
      1.0)))))

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-PI.f32 4.9

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

5. Applied rewrites 4.9%

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

6. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-PI.f32 4.9

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\mathsf{fma}\left(0.25, \frac{\mathsf{PI}\left(\right)}{s}, 0.5\right) - \frac{1}{2 + \frac{\mathsf{PI}\left(\right)}{s}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

8. Applied rewrites 4.9%

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

9. Taylor expanded in s around inf

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

11. Applied rewrites 38.6%

    \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(0.5 - \frac{1}{2 + \frac{\mathsf{PI}\left(\right)}{s}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

12. Taylor expanded in s around inf

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

    1. lower-+.f32 N/A

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

    2. lower-fma.f32 N/A

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

    3. lower-/.f32 N/A

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

    4. unpow2 N/A

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

    5. lower-*.f32 N/A

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

    6. lift-PI.f32 N/A

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

    7. lift-PI.f32 N/A

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

    8. unpow2 N/A

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

    9. lower-*.f32 N/A

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

    10. lift-/.f32 N/A

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

    11. lift-PI.f32 38.6

        \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(0.5 - \frac{1}{2 + \mathsf{fma}\left(0.5, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s \cdot s}, \frac{\mathsf{PI}\left(\right)}{s}\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

14. Applied rewrites 38.6%

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

Alternative 7: 37.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{PI}\left(\right)}{s}\\ \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(u, 0.5 - \frac{1}{2 + t\_0}, \frac{1}{1 + e^{t\_0}}\right)} - 1\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ (PI) s)))
   (*
    (- s)
    (log
     (-
      (/ 1.0 (fma u (- 0.5 (/ 1.0 (+ 2.0 t_0))) (/ 1.0 (+ 1.0 (exp t_0)))))
      1.0)))))

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-PI.f32 4.9

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

5. Applied rewrites 4.9%

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

6. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-PI.f32 4.9

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\mathsf{fma}\left(0.25, \frac{\mathsf{PI}\left(\right)}{s}, 0.5\right) - \frac{1}{2 + \frac{\mathsf{PI}\left(\right)}{s}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

8. Applied rewrites 4.9%

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

9. Taylor expanded in s around inf

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

11. Applied rewrites 38.6%

    \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(0.5 - \frac{1}{2 + \frac{\mathsf{PI}\left(\right)}{s}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
12. Step-by-step derivation

13. Applied rewrites 38.6%

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

Alternative 8: 37.4% accurate, 1.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-PI.f32 4.9

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

5. Applied rewrites 4.9%

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

6. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-PI.f32 4.9

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\mathsf{fma}\left(0.25, \frac{\mathsf{PI}\left(\right)}{s}, 0.5\right) - \frac{1}{2 + \frac{\mathsf{PI}\left(\right)}{s}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

8. Applied rewrites 4.9%

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

9. Taylor expanded in s around inf

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

11. Applied rewrites 38.6%

    \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(0.5 - \frac{1}{2 + \frac{\mathsf{PI}\left(\right)}{s}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

12. Taylor expanded in s around 0

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

    1. lift-/.f32 N/A

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

    2. lift-PI.f32 38.6

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

14. Applied rewrites 38.6%

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

Alternative 9: 36.7% accurate, 2.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-PI.f32 4.9

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

5. Applied rewrites 4.9%

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

6. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-PI.f32 4.9

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\mathsf{fma}\left(0.25, \frac{\mathsf{PI}\left(\right)}{s}, 0.5\right) - \frac{1}{2 + \frac{\mathsf{PI}\left(\right)}{s}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

8. Applied rewrites 4.9%

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

9. Taylor expanded in s around inf

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

11. Applied rewrites 38.6%

    \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(0.5 - \frac{1}{2 + \frac{\mathsf{PI}\left(\right)}{s}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

12. Taylor expanded in s around -inf

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

    1. lower-+.f32 N/A

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

    2. lower-*.f32 N/A

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

    3. lower-/.f32 N/A

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

    4. lower-fma.f32 N/A

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

    5. lift-PI.f32 N/A

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

    6. lower-*.f32 N/A

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

    7. lower-/.f32 N/A

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

    8. unpow2 N/A

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

    9. lower-*.f32 N/A

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

    10. lift-PI.f32 N/A

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

    11. lift-PI.f32 37.1

        \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(0.5 - \frac{1}{2 + \frac{\mathsf{PI}\left(\right)}{s}}\right) + \frac{1}{1 + \left(1 + -1 \cdot \frac{\mathsf{fma}\left(-1, \mathsf{PI}\left(\right), -0.5 \cdot \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s}\right)}{s}\right)}} - 1\right) \]

14. Applied rewrites 37.1%

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

15. Final simplification 37.1%

    \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(0.5 - \frac{1}{2 + \frac{\mathsf{PI}\left(\right)}{s}}\right) + \frac{1}{1 + \left(1 - \frac{\mathsf{fma}\left(-1, \mathsf{PI}\left(\right), -0.5 \cdot \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s}\right)}{s}\right)}} - 1\right) \]
16. Add Preprocessing

Alternative 10: 36.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{PI}\left(\right)}{s}\\ \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(0.5 - \frac{1}{2 + t\_0}\right) + \frac{1}{1 + \left(1 + \mathsf{fma}\left(0.5, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s \cdot s}, t\_0\right)\right)}} - 1\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ (PI) s)))
   (*
    (- s)
    (log
     (-
      (/
       1.0
       (+
        (* u (- 0.5 (/ 1.0 (+ 2.0 t_0))))
        (/ 1.0 (+ 1.0 (+ 1.0 (fma 0.5 (/ (* (PI) (PI)) (* s s)) t_0))))))
      1.0)))))

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-PI.f32 4.9

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

5. Applied rewrites 4.9%

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

6. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-PI.f32 4.9

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\mathsf{fma}\left(0.25, \frac{\mathsf{PI}\left(\right)}{s}, 0.5\right) - \frac{1}{2 + \frac{\mathsf{PI}\left(\right)}{s}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

8. Applied rewrites 4.9%

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

9. Taylor expanded in s around inf

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

11. Applied rewrites 38.6%

    \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(0.5 - \frac{1}{2 + \frac{\mathsf{PI}\left(\right)}{s}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

12. Taylor expanded in s around inf

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

    1. lower-+.f32 N/A

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

    2. lower-fma.f32 N/A

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

    3. lower-/.f32 N/A

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

    4. unpow2 N/A

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

    5. lower-*.f32 N/A

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

    6. lift-PI.f32 N/A

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

    7. lift-PI.f32 N/A

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

    8. unpow2 N/A

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

    9. lower-*.f32 N/A

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

    10. lift-/.f32 N/A

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

    11. lift-PI.f32 37.1

        \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(0.5 - \frac{1}{2 + \frac{\mathsf{PI}\left(\right)}{s}}\right) + \frac{1}{1 + \left(1 + \mathsf{fma}\left(0.5, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s \cdot s}, \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}} - 1\right) \]

14. Applied rewrites 37.1%

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

Alternative 11: 36.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{PI}\left(\right)}{s}\\ \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(0.5 - \frac{1}{2 + t\_0}\right) + \frac{1}{1 + \left(1 + t\_0\right)}} - 1\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ (PI) s)))
   (*
    (- s)
    (log
     (-
      (/ 1.0 (+ (* u (- 0.5 (/ 1.0 (+ 2.0 t_0)))) (/ 1.0 (+ 1.0 (+ 1.0 t_0)))))
      1.0)))))

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-PI.f32 4.9

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

5. Applied rewrites 4.9%

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

6. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-PI.f32 4.9

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\mathsf{fma}\left(0.25, \frac{\mathsf{PI}\left(\right)}{s}, 0.5\right) - \frac{1}{2 + \frac{\mathsf{PI}\left(\right)}{s}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

8. Applied rewrites 4.9%

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

9. Taylor expanded in s around inf

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

11. Applied rewrites 38.6%

    \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(0.5 - \frac{1}{2 + \frac{\mathsf{PI}\left(\right)}{s}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

12. Taylor expanded in s around inf

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

    1. lower-+.f32 N/A

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

    2. lift-/.f32 N/A

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

    3. lift-PI.f32 36.4

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(0.5 - \frac{1}{2 + \frac{\mathsf{PI}\left(\right)}{s}}\right) + \frac{1}{1 + \left(1 + \frac{\mathsf{PI}\left(\right)}{s}\right)}} - 1\right) \]

14. Applied rewrites 36.4%

    \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(0.5 - \frac{1}{2 + \frac{\mathsf{PI}\left(\right)}{s}}\right) + \frac{1}{1 + \color{blue}{\left(1 + \frac{\mathsf{PI}\left(\right)}{s}\right)}}} - 1\right) \]
15. Add Preprocessing

Alternative 12: 25.0% accurate, 4.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

5. Applied rewrites 25.0%

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

6. Taylor expanded in u around 0

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

   1. lower-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-PI.f32 25.2

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

8. Applied rewrites 25.2%

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

Alternative 13: 11.5% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \frac{\mathsf{fma}\left(4, u \cdot \left(-0.25 \cdot \mathsf{PI}\left(\right) - 0.25 \cdot \mathsf{PI}\left(\right)\right), 4 \cdot \mathsf{fma}\left(-0.5, \mathsf{PI}\left(\right), -0.0625 \cdot \mathsf{fma}\left(-8, \mathsf{PI}\left(\right), -4 \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{s} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (/
   (fma
    4.0
    (* u (- (* -0.25 (PI)) (* 0.25 (PI))))
    (* 4.0 (fma -0.5 (PI) (* -0.0625 (fma -8.0 (PI) (* -4.0 (PI)))))))
   s)))

Derivation

1. Initial program 99.0%

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in s around -inf

   \[\leadsto \left(-s\right) \cdot \color{blue}{\frac{4 \cdot \left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right) + 4 \cdot \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right) + \frac{-1}{16} \cdot \left(-8 \cdot \mathsf{PI}\left(\right) + -4 \cdot \mathsf{PI}\left(\right)\right)\right)}{s}} \]
6. Step-by-step derivation

7. Applied rewrites 12.5%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(4, u \cdot \left(-0.25 \cdot \mathsf{PI}\left(\right) - 0.25 \cdot \mathsf{PI}\left(\right)\right), 4 \cdot \mathsf{fma}\left(-0.5, \mathsf{PI}\left(\right), -0.0625 \cdot \mathsf{fma}\left(-8, \mathsf{PI}\left(\right), -4 \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{s}} \]
8. Add Preprocessing

Alternative 14: 11.5% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(4, u \cdot \left(0.25 \cdot \mathsf{PI}\left(\right) - -0.25 \cdot \mathsf{PI}\left(\right)\right), 4 \cdot \mathsf{fma}\left(-0.0625, \mathsf{fma}\left(4, \mathsf{PI}\left(\right), 8 \cdot \mathsf{PI}\left(\right)\right), 0.5 \cdot \mathsf{PI}\left(\right)\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (fma
  4.0
  (* u (- (* 0.25 (PI)) (* -0.25 (PI))))
  (* 4.0 (fma -0.0625 (fma 4.0 (PI) (* 8.0 (PI))) (* 0.5 (PI))))))

Derivation

1. Initial program 99.0%

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in s around inf

   \[\leadsto \color{blue}{4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)\right) + 4 \cdot \left(\frac{-1}{16} \cdot \left(4 \cdot \mathsf{PI}\left(\right) + 8 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]
6. Step-by-step derivation

7. Applied rewrites 12.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(4, u \cdot \left(0.25 \cdot \mathsf{PI}\left(\right) - -0.25 \cdot \mathsf{PI}\left(\right)\right), 4 \cdot \mathsf{fma}\left(-0.0625, \mathsf{fma}\left(4, \mathsf{PI}\left(\right), 8 \cdot \mathsf{PI}\left(\right)\right), 0.5 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
8. Add Preprocessing

Alternative 15: 11.5% accurate, 18.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 12.5%

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

6. Taylor expanded in u around inf

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

   1. lower-*.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 12.5

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

8. Applied rewrites 12.5%

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

Alternative 16: 11.5% accurate, 30.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 12.5%

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

6. Taylor expanded in u around 0

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

   1. lower-fma.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 12.5

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

8. Applied rewrites 12.5%

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

Alternative 17: 11.2% accurate, 170.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in u around 0

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

   1. mul-1-neg N/A

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

   2. lift-neg.f32 N/A

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

   3. lift-PI.f32 12.3

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

5. Applied rewrites 12.3%

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

Reproduce

herbie shell --seed 2025037 
(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))))