Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 14.5s

Alternatives: 12

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

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

FPCore

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

Derivation

1. Initial program 99.1%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. clear-num N/A

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

   4. div-inv N/A

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

   5. clear-num N/A

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

   6. lift-/.f32 N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f32 N/A

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

   9. exp-1-e N/A

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

   10. lower-E.f32 99.1

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

4. Applied rewrites 99.1%

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

5. Final simplification 99.1%

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

Alternative 2: 11.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\mathsf{PI}\left(\right)\\ t_1 := \frac{-1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\\ \mathbf{if}\;\log \left(-1 - \frac{-1}{\left(t\_1 - \frac{-1}{e^{\frac{t\_0}{s}} + 1}\right) \cdot u - t\_1}\right) \cdot \left(-s\right) \leq -1.3999999555717316 \cdot 10^{-19}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \mathsf{PI}\left(\right), 0\right) \cdot u\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (- (PI))) (t_1 (/ -1.0 (+ (exp (/ (PI) s)) 1.0))))
   (if (<=
        (*
         (log
          (-
           -1.0
           (/ -1.0 (- (* (- t_1 (/ -1.0 (+ (exp (/ t_0 s)) 1.0))) u) t_1))))
         (- s))
        -1.3999999555717316e-19)
     t_0
     (* (fma 2.0 (PI) 0.0) u))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32)))) < -1.39999996e-19

   1. Initial program 99.2%

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

   3. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 N/A

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

      3. lower-PI.f32 15.5

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

   5. Applied rewrites 15.5%

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

   if -1.39999996e-19 < (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32))))

   1. Initial program 99.0%

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

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

      2. lower-neg.f32 N/A

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

      3. lower-PI.f32 7.5

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

   5. Applied rewrites 7.5%

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

   6. Taylor expanded in s around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

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

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-rgt-out N/A

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

      11. lower-*.f32 N/A

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

      12. lower-PI.f32 N/A

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

      13. lower-fma.f32 7.5

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

   8. Applied rewrites 7.5%

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

   9. Taylor expanded in u around inf

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

   11. Applied rewrites 5.8%

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

   13. Applied rewrites 13.3%

       \[\leadsto \mathsf{fma}\left(2, \mathsf{PI}\left(\right), 0\right) \cdot u \]
3. Recombined 2 regimes into one program.

4. Final simplification 11.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(-1 - \frac{-1}{\left(\frac{-1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1} - \frac{-1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1}\right) \cdot u - \frac{-1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}}\right) \cdot \left(-s\right) \leq -1.3999999555717316 \cdot 10^{-19}:\\ \;\;\;\;-\mathsf{PI}\left(\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \mathsf{PI}\left(\right), 0\right) \cdot u\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.1%

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

3. Final simplification 99.1%

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

Alternative 4: 97.3% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

5. Applied rewrites 97.6%

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

6. Final simplification 97.6%

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

Alternative 5: 97.7% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in s around inf

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

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

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

   2. metadata-eval N/A

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

   3. associate-*r* N/A

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

   4. metadata-eval N/A

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

   5. distribute-rgt-neg-in N/A

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

   6. associate-*r* N/A

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

   7. metadata-eval N/A

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

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

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

   9. distribute-neg-frac N/A

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

   10. distribute-neg-frac2 N/A

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

   11. mul-1-neg N/A

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

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

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

   13. metadata-eval N/A

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

   14. associate-*r* N/A

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

   15. *-commutative N/A

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

   16. times-frac N/A

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

   17. metadata-eval N/A

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

5. Applied rewrites 3.1%

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

6. Taylor expanded in u around inf

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

   1. lower--.f32 N/A

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

8. Applied rewrites 97.6%

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

9. Final simplification 97.6%

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

Alternative 6: 9.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.5, u, 0.25\right) \cdot \mathsf{PI}\left(\right)\\ \log \left(1 - \frac{\mathsf{fma}\left(t\_0, -4, \mathsf{fma}\left(\frac{{t\_0}^{2}}{s}, -8, \frac{0}{s}\right)\right)}{s}\right) \cdot \left(-s\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (* (fma -0.5 u 0.25) (PI))))
   (*
    (log (- 1.0 (/ (fma t_0 -4.0 (fma (/ (pow t_0 2.0) s) -8.0 (/ 0.0 s))) s)))
    (- s))))

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in s around inf

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

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

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

   2. metadata-eval N/A

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

   3. associate-*r* N/A

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

   4. metadata-eval N/A

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

   5. distribute-rgt-neg-in N/A

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

   6. associate-*r* N/A

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

   7. metadata-eval N/A

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

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

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

   9. distribute-neg-frac N/A

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

   10. distribute-neg-frac2 N/A

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

   11. mul-1-neg N/A

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

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

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

   13. metadata-eval N/A

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

   14. associate-*r* N/A

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

   15. *-commutative N/A

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

   16. times-frac N/A

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

   17. metadata-eval N/A

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

5. Applied rewrites 3.1%

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

6. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

8. Applied rewrites 14.4%

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

9. Final simplification 15.0%

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

Alternative 7: 14.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 1.9999999920083944 \cdot 10^{-12}:\\ \;\;\;\;\log \left(\frac{1}{\mathsf{fma}\left(-0.25, \frac{\mathsf{PI}\left(\right)}{s}, \mathsf{fma}\left(\frac{\left(\log \mathsf{E}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot u}{s}, 0.5, 0.5\right)\right)} - 1\right) \cdot \left(-s\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{0.25}{u} - 0.5\right) \cdot u\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (if (<= s 1.9999999920083944e-12)
   (*
    (log
     (-
      (/
       1.0
       (fma -0.25 (/ (PI) s) (fma (/ (* (* (log (E)) (PI)) u) s) 0.5 0.5)))
      1.0))
    (- s))
   (* (* (* (- (/ 0.25 u) 0.5) u) (PI)) -4.0)))

Derivation

1. Split input into 2 regimes

2. if s < 1.99999999e-12

   1. Initial program 99.0%

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

   3. Taylor expanded in s around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lower-PI.f32 N/A

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

      8. +-commutative N/A

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

   5. Applied rewrites 12.3%

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

   7. Applied rewrites 14.1%

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

   if 1.99999999e-12 < s

   1. Initial program 99.2%

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

   3. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 N/A

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

      3. lower-PI.f32 18.1

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

   5. Applied rewrites 18.1%

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

   6. Taylor expanded in s around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

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

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-rgt-out N/A

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

      11. lower-*.f32 N/A

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

      12. lower-PI.f32 N/A

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

      13. lower-fma.f32 18.1

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

   8. Applied rewrites 17.9%

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

   9. Taylor expanded in u around inf

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

   11. Applied rewrites 18.3%

       \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\left(\frac{0.25}{u} - 0.5\right) \cdot u\right)\right) \cdot -4 \]
3. Recombined 2 regimes into one program.

4. Final simplification 15.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 1.9999999920083944 \cdot 10^{-12}:\\ \;\;\;\;\log \left(\frac{1}{\mathsf{fma}\left(-0.25, \frac{\mathsf{PI}\left(\right)}{s}, \mathsf{fma}\left(\frac{\left(\log \mathsf{E}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot u}{s}, 0.5, 0.5\right)\right)} - 1\right) \cdot \left(-s\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{0.25}{u} - 0.5\right) \cdot u\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -4\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 11.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 4.00000018325482 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5, u, 0.25\right), -4 \cdot \mathsf{PI}\left(\right), \frac{-0.5}{s} \cdot \left({\left(\mathsf{fma}\left(-0.5, u, 0.25\right) \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot 0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{0.25}{u} - 0.5\right) \cdot u\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (if (<= s 4.00000018325482e-18)
   (fma
    (fma -0.5 u 0.25)
    (* -4.0 (PI))
    (* (/ -0.5 s) (* (pow (* (fma -0.5 u 0.25) (PI)) 2.0) 0.0)))
   (* (* (* (- (/ 0.25 u) 0.5) u) (PI)) -4.0)))

Derivation

1. Split input into 2 regimes

2. if s < 4.00000018e-18

   1. Initial program 99.0%

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

   3. Taylor expanded in s around -inf

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

   5. Applied rewrites 3.2%

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

   7. Applied rewrites 6.6%

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

   9. Applied rewrites 7.6%

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

   if 4.00000018e-18 < s

   1. Initial program 99.2%

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

   3. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 N/A

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

      3. lower-PI.f32 16.3

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

   5. Applied rewrites 16.3%

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

   6. Taylor expanded in s around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

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

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-rgt-out N/A

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

      11. lower-*.f32 N/A

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

      12. lower-PI.f32 N/A

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

      13. lower-fma.f32 16.3

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

   8. Applied rewrites 16.3%

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

   9. Taylor expanded in u around inf

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

   11. Applied rewrites 16.5%

       \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\left(\frac{0.25}{u} - 0.5\right) \cdot u\right)\right) \cdot -4 \]
3. Recombined 2 regimes into one program.

4. Final simplification 10.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 4.00000018325482 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5, u, 0.25\right), -4 \cdot \mathsf{PI}\left(\right), \frac{-0.5}{s} \cdot \left({\left(\mathsf{fma}\left(-0.5, u, 0.25\right) \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot 0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{0.25}{u} - 0.5\right) \cdot u\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -4\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 11.9% accurate, 14.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 2.1999999301841496 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(2, \mathsf{PI}\left(\right), 0\right) \cdot u\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{0.25}{u} - 0.5\right) \cdot u\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (if (<= s 2.1999999301841496e-20)
   (* (fma 2.0 (PI) 0.0) u)
   (* (* (* (- (/ 0.25 u) 0.5) u) (PI)) -4.0)))

Derivation

1. Split input into 2 regimes

2. if s < 2.19999993e-20

   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

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

      2. lower-neg.f32 N/A

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

      3. lower-PI.f32 7.6

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

   5. Applied rewrites 7.6%

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

   6. Taylor expanded in s around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

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

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-rgt-out N/A

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

      11. lower-*.f32 N/A

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

      12. lower-PI.f32 N/A

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

      13. lower-fma.f32 7.6

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

   8. Applied rewrites 7.6%

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

   9. Taylor expanded in u around inf

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

   11. Applied rewrites 5.7%

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

   13. Applied rewrites 13.3%

       \[\leadsto \mathsf{fma}\left(2, \mathsf{PI}\left(\right), 0\right) \cdot u \]

   if 2.19999993e-20 < s

   1. Initial program 99.3%

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

   3. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 N/A

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

      3. lower-PI.f32 15.5

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

   5. Applied rewrites 15.5%

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

   6. Taylor expanded in s around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

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

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-rgt-out N/A

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

      11. lower-*.f32 N/A

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

      12. lower-PI.f32 N/A

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

      13. lower-fma.f32 15.5

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

   8. Applied rewrites 15.4%

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

   9. Taylor expanded in u around inf

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

   11. Applied rewrites 15.6%

       \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\left(\frac{0.25}{u} - 0.5\right) \cdot u\right)\right) \cdot -4 \]
3. Recombined 2 regimes into one program.

4. Final simplification 14.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 2.1999999301841496 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(2, \mathsf{PI}\left(\right), 0\right) \cdot u\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{0.25}{u} - 0.5\right) \cdot u\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -4\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 11.9% accurate, 17.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 2.1999999301841496 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(2, \mathsf{PI}\left(\right), 0\right) \cdot u\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \mathsf{PI}\left(\right) + \left(\mathsf{PI}\left(\right) \cdot u\right) \cdot -0.5\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (if (<= s 2.1999999301841496e-20)
   (* (fma 2.0 (PI) 0.0) u)
   (* (+ (* 0.25 (PI)) (* (* (PI) u) -0.5)) -4.0)))

Derivation

1. Split input into 2 regimes

2. if s < 2.19999993e-20

   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

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

      2. lower-neg.f32 N/A

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

      3. lower-PI.f32 7.6

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

   5. Applied rewrites 7.6%

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

   6. Taylor expanded in s around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

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

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-rgt-out N/A

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

      11. lower-*.f32 N/A

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

      12. lower-PI.f32 N/A

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

      13. lower-fma.f32 7.6

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

   8. Applied rewrites 7.6%

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

   9. Taylor expanded in u around inf

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

   11. Applied rewrites 5.7%

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

   13. Applied rewrites 13.3%

       \[\leadsto \mathsf{fma}\left(2, \mathsf{PI}\left(\right), 0\right) \cdot u \]

   if 2.19999993e-20 < s

   1. Initial program 99.3%

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

   3. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 N/A

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

      3. lower-PI.f32 15.5

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

   5. Applied rewrites 15.5%

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

   6. Taylor expanded in s around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

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

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-rgt-out N/A

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

      11. lower-*.f32 N/A

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

      12. lower-PI.f32 N/A

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

      13. lower-fma.f32 15.5

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

   8. Applied rewrites 15.4%

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

   10. Applied rewrites 15.6%

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot u\right) \cdot -0.5 + 0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot -4 \]
3. Recombined 2 regimes into one program.

4. Final simplification 14.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 2.1999999301841496 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(2, \mathsf{PI}\left(\right), 0\right) \cdot u\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \mathsf{PI}\left(\right) + \left(\mathsf{PI}\left(\right) \cdot u\right) \cdot -0.5\right) \cdot -4\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 11.9% accurate, 20.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 2.1999999301841496 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(2, \mathsf{PI}\left(\right), 0\right) \cdot u\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.5 \cdot u + 0.25\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (if (<= s 2.1999999301841496e-20)
   (* (fma 2.0 (PI) 0.0) u)
   (* (* (+ (* -0.5 u) 0.25) (PI)) -4.0)))

Derivation

1. Split input into 2 regimes

2. if s < 2.19999993e-20

   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

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

      2. lower-neg.f32 N/A

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

      3. lower-PI.f32 7.6

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

   5. Applied rewrites 7.6%

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

   6. Taylor expanded in s around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

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

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-rgt-out N/A

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

      11. lower-*.f32 N/A

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

      12. lower-PI.f32 N/A

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

      13. lower-fma.f32 7.6

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

   8. Applied rewrites 7.6%

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

   9. Taylor expanded in u around inf

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

   11. Applied rewrites 5.7%

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

   13. Applied rewrites 13.3%

       \[\leadsto \mathsf{fma}\left(2, \mathsf{PI}\left(\right), 0\right) \cdot u \]

   if 2.19999993e-20 < s

   1. Initial program 99.3%

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

   3. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 N/A

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

      3. lower-PI.f32 15.5

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

   5. Applied rewrites 15.5%

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

   6. Taylor expanded in s around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

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

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-rgt-out N/A

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

      11. lower-*.f32 N/A

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

      12. lower-PI.f32 N/A

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

      13. lower-fma.f32 15.5

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

   8. Applied rewrites 15.5%

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

   10. Applied rewrites 15.6%

       \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(-0.5 \cdot u + 0.25\right)\right) \cdot -4 \]
3. Recombined 2 regimes into one program.

4. Final simplification 14.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 2.1999999301841496 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(2, \mathsf{PI}\left(\right), 0\right) \cdot u\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.5 \cdot u + 0.25\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -4\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 11.3% accurate, 170.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in u around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 N/A

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

   3. lower-PI.f32 11.0

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

5. Applied rewrites 11.0%

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

Reproduce

herbie shell --seed 2024296 
(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))))