Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 98.9%

Time: 17.8s

Alternatives: 9

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: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\mathsf{PI}\left(\right)}\\ \log \left(\frac{1}{\frac{1}{e^{\left(\frac{1}{s} \cdot t\_0\right) \cdot {t\_0}^{2}} + 1} + \left(\frac{-1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1} - \frac{-1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1}\right) \cdot u} - 1\right) \cdot \left(-s\right) \end{array} \end{array} \]

FPCore

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

Derivation

1. Initial program 98.9%

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

   1. lift-/.f32 N/A

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

   2. div-inv N/A

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

   3. lift-PI.f32 N/A

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

   4. add-cube-cbrt N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f32 N/A

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

   7. pow2 N/A

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

   8. lower-pow.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lower-cbrt.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. lower-cbrt.f32 N/A

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

   14. lower-/.f32 98.9

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

4. Applied rewrites 98.9%

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

5. Final simplification 98.9%

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

Alternative 2: 28.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{PI}\left(\right)}{s}\\ t_1 := \frac{-1}{e^{t\_0} + 1}\\ \mathbf{if}\;\log \left(-1 - \frac{-1}{\left(t\_1 - \frac{-1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1}\right) \cdot u - t\_1}\right) \cdot \left(-s\right) \leq -9.999999960041972 \cdot 10^{-12}:\\ \;\;\;\;\log \left(\left(\frac{t\_0 + 1}{u} - 2 \cdot t\_0\right) \cdot u\right) \cdot \left(-s\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot u, -0.5, {\left({\mathsf{PI}\left(\right)}^{3}\right)}^{0.3333333333333333} \cdot 0.25\right), \frac{4}{s}, 1\right)\right) \cdot \left(-s\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ (PI) s)) (t_1 (/ -1.0 (+ (exp t_0) 1.0))))
   (if (<=
        (*
         (log
          (-
           -1.0
           (/
            -1.0
            (- (* (- t_1 (/ -1.0 (+ (exp (/ (- (PI)) s)) 1.0))) u) t_1))))
         (- s))
        -9.999999960041972e-12)
     (* (log (* (- (/ (+ t_0 1.0) u) (* 2.0 t_0)) u)) (- s))
     (*
      (log
       (fma
        (fma (* (PI) u) -0.5 (* (pow (pow (PI) 3.0) 0.3333333333333333) 0.25))
        (/ 4.0 s)
        1.0))
      (- s)))))

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)))) < -9.99999996e-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

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

      1. associate-*r/ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f32 N/A

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

   5. Applied rewrites 6.9%

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

   6. Taylor expanded in u around -inf

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

   8. Applied rewrites 29.7%

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

   if -9.99999996e-12 < (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32))))

   1. Initial program 98.8%

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

   3. Taylor expanded in s around -inf

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

      1. associate-*r/ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f32 N/A

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

   5. Applied rewrites 11.9%

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

   7. Applied rewrites 28.9%

      \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{fma}\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot {\left({\mathsf{PI}\left(\right)}^{3}\right)}^{0.3333333333333333}\right), \frac{4}{s}, 1\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(-1 - \frac{-1}{\left(\frac{-1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1} - \frac{-1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1}\right) \cdot u - \frac{-1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}}\right) \cdot \left(-s\right) \leq -9.999999960041972 \cdot 10^{-12}:\\ \;\;\;\;\log \left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{s} + 1}{u} - 2 \cdot \frac{\mathsf{PI}\left(\right)}{s}\right) \cdot u\right) \cdot \left(-s\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot u, -0.5, {\left({\mathsf{PI}\left(\right)}^{3}\right)}^{0.3333333333333333} \cdot 0.25\right), \frac{4}{s}, 1\right)\right) \cdot \left(-s\right)\\ \end{array} \]
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 98.9%

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

3. Final simplification 98.9%

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

Alternative 4: 97.7% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in u around inf

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

   1. lower--.f32 N/A

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in u around 0

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

8. Applied rewrites 98.8%

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

9. Final simplification 98.8%

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

Alternative 5: 75.6% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in u around inf

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

   1. lower--.f32 N/A

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in u around 0

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

8. Applied rewrites 98.8%

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

9. Taylor expanded in u around 0

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

11. Applied rewrites 76.3%

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

12. Final simplification 76.3%

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

Alternative 6: 25.0% accurate, 4.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in 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. associate-*r/ N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-fma.f32 N/A

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

5. Applied rewrites 10.3%

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

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

8. Applied rewrites 25.1%

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

9. Final simplification 25.1%

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

Alternative 7: 13.9% accurate, 14.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in u around inf

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

   1. lower--.f32 N/A

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in u around 0

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

   1. lower-/.f32 N/A

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

   2. lower-PI.f32 11.0

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

8. Applied rewrites 11.0%

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

   1. lift-neg.f32 N/A

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

   2. neg-sub0 N/A

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

   3. flip-- N/A

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

   4. metadata-eval N/A

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

   5. lift-*.f32 N/A

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

   6. neg-sub0 N/A

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

   7. lower-/.f32 N/A

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

   8. lift-*.f32 N/A

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

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

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

   10. lift-neg.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-+.f32 13.4

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

10. Applied rewrites 13.4%

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

11. Final simplification 13.4%

    \[\leadsto \frac{\mathsf{PI}\left(\right)}{s} \cdot \frac{\left(-s\right) \cdot s}{s} \]
12. Add Preprocessing

Alternative 8: 11.5% accurate, 36.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in u around inf

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

   1. lower--.f32 N/A

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in u around 0

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

8. Applied rewrites 98.8%

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

9. Taylor expanded in s around inf

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

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

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

    2. metadata-eval N/A

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

    3. distribute-lft-in N/A

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

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

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

    5. metadata-eval N/A

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

    6. associate-*r* N/A

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

    7. *-commutative N/A

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

    8. associate-*r* N/A

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

    9. metadata-eval N/A

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

    10. associate-*r* N/A

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

    11. metadata-eval N/A

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

    12. lower-fma.f32 N/A

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

    13. *-commutative N/A

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

    14. lower-*.f32 N/A

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

    15. lower-PI.f32 N/A

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

    16. mul-1-neg N/A

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

    17. lower-neg.f32 N/A

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

    18. lower-PI.f32 11.0

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

11. Applied rewrites 11.0%

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

13. Applied rewrites 11.2%

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

Alternative 9: 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 98.9%

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

3. Taylor expanded in u around 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 2024331 
(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))))